The iISS Property for Globally Asymptotically Stable and Passive
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 8, SEPTEMBER 2008
[12] M. Krstic and A. Smyshlyaev, “Adaptive boundary control for unstable parabolic PDEs—Part I: Lyapunov design,” IEEE Trans. Autom. Control, vol. 53, no. 7, Aug. 2008. [13] Z. H. Li and M. Krstic, “Optimal design of adaptive tracking controllers for nonlinear systems,” Automatica, vol. 33, pp. 1459–1473, 1997. [14] W. Liu and M. Krstic, “Adaptive control of Burgers’ equation with unknown viscosity,” Int. J. Adaptive Control and Signal Process., vol. 15, pp. 745–766, 2001. [15] H. Logemann and B. Martensson, “Adaptive stabilization of infinitedimensional systems,” IEEE Trans. Autom. Control, vol. 37, no. 12, pp. 1869–1883, Dec. 1992. [16] P. J. Moylan and B. D. O. Anderson, “Nonlinear regulator theory and an inverse optimal control problem,” IEEE Trans. Autom. Control, vol. AC-18, pp. 460–465, 1973. [17] Y. Orlov, “Sliding mode observer-based synthesis of state derivative-free model reference adaptive control of distributed parameter systems,” J. Dynam. Syst., Meas., and Control, vol. 122, pp. 726–731, 2000. [18] R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. New York: Springer, 1997. [19] A. Smyshlyaev and M. Krstic, “Closed form boundary state feedbacks for a class of 1-D partial integro-differential equations,” IEEE Trans. Autom. Control, vol. 49, no. 12, pp. 2185–2202, Dec. 2004. [20] A. Smyshlyaev and M. Krstic, “Adaptive boundary control for unstable parabolic PDEs—Part II: Estimation-based designs,” Automatica, vol. 43, pp. 1543–1556, 2007. [21] A. Smyshlyaev and M. Krstic, “Adaptive boundary control for unstable parabolic PDEs—Part III: Output-feedback examples with swapping identifiers,” Automatica, vol. 43, pp. 1557–1564, 2007. [22] S. Townley, “Simple adaptive stabilization of output feedback stabilizable distributed parameter systems,” Dynam. Control, vol. 5, pp. 107–123, 1995. [23] J. T.-Y. Wen and M. J. Balas, “Robust adaptive control in Hilbert space,” J. Mathemat. Anal. Applic., vol. 143, pp. 1–26, 1989.
The iISS Property for Globally Asymptotically Stable and Passive Nonlinear Systems Chen Wang and George Weiss
Abstract—This paper investigates the integral input-to-state stability (iISS) property for passive nonlinear systems. We show that under mild technical assumptions, a passive nonlinear system which is globally asymptotically stable is also iISS. Moreover, the integral term from the norm). definition of the iISS property has a very simple form (like an We illustrate the result by proving that the drive-train of a wind turbine with quadratic torque control is iISS. Index Terms—Globally asymptotically stable, integral input-to-state stability (iISS), wind turbine.
I. INTRODUCTION The concept of passivity is important in control theory because it is a property shared by many physical systems and it is related to stability Manuscript received June 23, 2008; revised July 28, 2008. First published September 12, 2008; current version published September 24, 2008.Recommended by Associate Editor David Angeli. C. Wang is with the Control and Power Group, Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: [email protected]). G. Weiss is with the Department of Electrical Engineering—Systems, Faculty of Engineering, Tel Aviv University, Ramat Aviv 69978, Israel (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2008.929465
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(see Moylan [1], Hill and Moylan [2], Byrnes et al. [3]). Consider a dynamical system S described by
x_ = f (x; u) y = h(x; u)
(1.1)
n 2 m ! n is locally Lipschitz continuous and m m is continuous. Here x(t) is the state at time 2 ! h: n t, which is in ; u is the input signal and y is the output signal. Under
where
f
n
:
these assumptions, for every initial state x(0) and for every bounded input signal u, (1.1) has a unique solution on some time interval [0; "), with " > 0. S is said to be passive if there exists a continuously differentiable storage function or Hamiltonian H : n ! [0; 1) such that
H_ uT y;
_ = where H
@H f (x; u) @x
for all (x; u) 2 n 2 m . To investigate the Lyapunov stability of the equilibrium points of S corresponding to u = 0 we may use H as a Lyapunov function (see Willems [4] or Khalil [5]). The notion of input-to-state stability (ISS), as introduced in Sontag [6], implies that f (x; 0) = 0 iff x = 0 and for any initial state, if the input becomes uniformly very small after some time, then also the state becomes uniformly very small after some time (see Sontag [7]). Sontag and Teel [8] gives a characterization of supply rates for ISS systems. A strictly weaker variant of ISS is the concept of integral input-to-state stability (iISS), where the uniform smallness of the input is replaced by the smallness of a certain integral depending on the input, see Angeli, Sontag and Wang [9]. The formal definition of iISS is given in Section II. In this paper, we investigate the iISS property for a class of passive nonlinear systems. In our main result (stated in Section III), we show that under mild assumptions, a passive nonlinear system which is globally asymptotically stable (GAS) is also iISS. By combining our result with a recent result in Jayawardhana et al. [10], we can actually prove that under mild technical assumptions, a passive and GAS system satisfies the iISS type estimate with a very simple (L1 norm type) integral term. We will illustrate the result by proving the iISS property (with a simple integral term) for the drive-train of a wind turbine, in Section IV. II. BACKGROUND CONCEPTS In this section, we recall the background about the iISS property following [9]. Some commonly used terminology: A function H : n ! [0; 1) is called positive definite if H (x) = 0 iff x = 0. H is called proper if H (x) ! 1 when kxk ! 1. A continuous function : [0; a) ! [0; 1) belongs to the class K if it is strictly increasing and (0) = 0. Such a function is in the class K if a = 1 and (r) ! 1 as r ! 1. A continuous function : [0; a) 2 [0; 1) ! [0; 1) belongs to the class KL if, for each fixed s; (1; s) belongs to K and, for each fixed r , the mapping (r; 1) is decreasing and (r; s) ! 0 as s ! 1. Given any measurable and bounded control u and any 2 n , there is a unique solution of (1.1) with x(0) = . This solution (or state trajectory) is defined on some maximal interval [0; ), and it is denoted by x(1; ; u). Definition 2.1: The system (1.1) is integral input-to-state stable (iISS) if there exist a class K function , a class KL function and a class K function such that for every 2 n and for every measurable and bounded function u, the state trajectory x(t; ; u) is defined for all t 0 and
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1
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 8, SEPTEMBER 2008
(kx(t; ; u)k) (k k; t) +
t 0
(ku( )k) d 8t 0:
(2.1)
The function is called the iISS gain of the system (1.1). If u is such that 0 (ku( )k)d < 1, then the iISS estimate in (2.1) also implies that x(t; ; u) ! 0 as t ! 1. However, if the system is iISS and u is only bounded, then the state trajectory x(1; ; u) may be unbounded. It is easy to see that if the system (1.1) is iISS, then this system has a unique equilibrium point at zero (f (x; 0) = 0 iff x = 0) and the system is globally asymptotically stable (GAS), which means that it is Lyapunov stable, the trajectories x(t; ; 0) are defined for all t 0 and tend to zero (as t ! 1). Definition 2.2: The system (1.1) is zero-output dissipative, if there exists a continuously differentiable proper and positive definite function V , and a class K function , such that
1
@V f (x; u) (kuk) @x
8
(x; u)
2
n 2 m:
(2.2)
Theorem 2.3: The system (1.1) is iISS if and only if it is GAS and zero-output dissipative. This follows from ([9], Theorem 1) together with ([9], Remark II.3). Theorem 2.4: Assume that the system (1.1) is GAS, zero-output dissipative and satisfies the following: for every compact set M n , there exists m > 0 such that k
f (x; u)k m(1 + (kuk))8(x;u) 2 M 2 m
(2.3)
with the same function in (2.2) and (2.3). Then for every 2 n and for every measurable and bounded function u : [0; 1) ! m , the state trajectory x(t; ; u) (which is defined for all t 0 according to Theorem 2.3) satisfies (2.1) with = . This recent result on the iISS gain is due to Jayawardhana et al. ([10, Theor. 3.1]). III.
IISS
In this section, we consider the system 6 described by (1.1) together with a simpler output equation
y = h (x ) where h : n ! m is continuous. Our main results are the following. Theorem 3.1: We assume that 6 is passive and GAS, with the storage function H . Denote
c(r) = sup kh(x)k: kxkr
1 0
c
d
and = : +1 1
Then the system (1.1) is zero-output dissipative with hence it is iISS.
0,
Choose the Lyapunov function V (x) =
F 0 ( ) =
1 +1
c(R ) =
F (H (x)), where 8
( )
c
then obviously
when < 0 ; when 0 ;
1
c(R)+1
0:
It is easy to see that F 0 () is a non-increasing continuous function of (see Fig. 1). We remark that in the region of n where H (x) 0 , we have F () = =(c(R) + 1), so that V (x) = H (x)=(c(R) + 1). We want to show that f with V satisfy (2.2). We consider two cases depending on kxk. • Assume that x 2 n with kxk < R (case 1). Using the passivity of 6 and the Cauchy-Schwarz inequality, we obtain
1
V_ = F 0 (H (x))H_
1
c(R ) + 1
c(R ) + 1
1
c(R ) + 1 1 k h (x )k 1 k u k 1
1
uT y
c(kx k) 1 k u k:
V_
k
(3.2)
(r) = r , and
u k:
F 0 (H (x)) F 0 (kkxk ): This implies, using again the passivity of Schwarz inequality
(3.1)
0 and R 0 such that
H (x) kkxk for kxk R
0 = kR
Proof: Define
c((0=k)(1= ).
Since c(r) is a non-decreasing function of r 2 [0; 1); kxk implies that c(kxk) c(R). Using this inequality, we have
PROPERTY FOR GAS AND PASSIVE NONLINEAR SYSTEMS
We assume that there exist ; k
Fig. 1. Function F (), which is a non-increasing continuous function of . In this figure, we have assumed that > 0.
6 and the Cauchy-
F 0 (k x ) u T y 1 h (x ) u c( x ) + 1 1 c( x ) u u : c( x ) + 1 k
k
k
k
k
1 k
1
k
k1 k
k
k
1 k
1
k
k k
k
Thus we have proved that V_ kuk for all x 2 n . This means that if V is proper, then the system (1.1) is zero-output dissipative with (r) = r . To show that indeed V is proper, note that 0
1 0 F ()d = 1:
(3.3)
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Indeed, using (3.2), we have
1 0 1 0 F ()d F ()d 0 1 d =k
c
+1
= 1:
Since H is proper [this follows from (3.1)], we have
!1 F (H ) = Hlim !1 kxlim k!1 V (x) = Hlim
H 0
Fig. 2. Two-mass drive-train model with gearbox.
F 0 ( ) d = 1 :
[We have used (3.3).] Since 6 is GAS, applying Theorem 2.3 we conclude that 6 is iISS. Remark 3.2: If the output y of 6 is a linear function of the state x, i.e., h(x) = Cx, where C is a matrix of matching dimensions, then c(r) = kC kr and then (3.2) holds for every 1. Remark 3.3: If there exist k1 > 0 and r0 0 such that
c(r ) k 1 r
then it follows that (3.2) holds. Corollary 3.4: We assume that 6 is passive, GAS and satisfies (3.1) and (3.2). Suppose also that for every compact set M n , there exists m > 0 such that
m:
(3.4)
Then for every 2 u : [0; 1) ! m ; 6 has a unique state trajectory x(1; ; u) defined on [0; 1), and this satisfies
n and for every measurable and bounded function
(kx(t; ; u)k) (k k; t) +
t 0
ku( )kd 8t 2 [0; 1)
(3.5)
where 2 K and 2 KL are independent of and u. Note that (3.5) means that 6 is iISS, with the iISS gain (r) = r . Proof: Recall from Theorem 3.1 that the system (1.1) is zerooutput dissipative with (kuk) = kuk. Applying Theorems 2.3 and 2.4, we see that this system is iISS, with the iISS gain (kuk) = kuk. Remark 3.5: After seeing Corollary 3.4, it is tempting to conjecture that if 6 satisfies the assumptions in this corollary, then it has state trajectories defined for all t 0, for every 2 n and every u 2 L1 [0; 1). However, this is not correct, as can be seen from the example at the beginning of Section IV in Jayawardhana and Weiss [11]. The existence of global solutions is guaranteed only for bounded and measurable inputs.
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IV. EXAMPLES Example 4.1: We consider the drive-train of a wind turbine with quadratic torque control. The two-mass drive-train model with gearbox is shown in Fig. 2 (see [12] and [13]). The large turbine inertia JT corresponds to the blades and the hub, and the small inertia JG represents the induction generator. We take the state variables x, the external input variables w (disturbances) and the control input u as follows:
k x = !T ; w = T a ; u = 0 T e !m
x_ = Ax + B1 w + B2 u; y = Cx
(4.1)
where
8r r0 ;
kf (x; u)k m(1 + kuk) 8(x; u) 2 M 2
where k is the angular difference between the two shafts, !T is the turbine rotor speed, !m is the generator rotor speed, Ta is the aerodynamic torque, and Te is the generator electrical torque. The output variables are !m and !T , so that y = [!T !m ]T . Then the third-order state-space representation is
A=
0
0 KJ K J n
0 n1
1
0 JC C J n
C J n 0 J Cn 0 Jb
0
0 0 ; C = 00 10 01 : 1
B = [ B1 B2 ] = J1
0
J
Here ng is the gearbox ratio, Ks > 0 is the torsional stiffness of the low speed shaft, Cs 0 is the torsional damping of the low speed shaft and b 0 is the damping coefficient. For the derivation of the above model, we refer to our paper [14]. For the wind turbine operating in low to medium wind speed region, it is desirable to maximize the output power of the turbine. The standard quadratic torque control law is often used to achieve this control objective (see [15])
0 when !m < 0; 2 when !m 0; K!m
Teref = where K
(4.2)
> 0. We introduce du = Teref 0 Te
(4.3)
which is the error between the reference torque and the real generator torque. There are many good books on wind turbine technology, of which we mention Burton et al. [16] and Heier [17]. Consider the linear drive-train system S1 , described by the matrices (A; B; C ) from (4.1), with input u = [Ta ; 0Te ]T , state x = [k ; !T ; !m ]T and output y = [!T ; !m ]T . Then this system is passive with the storage function H (x) = xT P x, where
P=
0 0 0 JT 0 : 0 0 JG
Ks
(4.4)
For the simple proof of the above claim, we refer to our paper [14]. Proposition 4.2: Consider the closed-loop wind turbine system S2 described by (4.1), (4.2) and (4.3). We regard this system with input d = [Ta ; du ]T , state x = [k ; !T ; !m ]T and output y = [!T ; !m ]T .
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 8, SEPTEMBER 2008
for all t > 0, where 2 K1 and 2 KL. Proof: We know from Proposition 4.2 that the closed-loop turbine system S2 is passive. We also know from Proposition 4.3 that S2 is , which is the positive solution of the equation GAS with respect to x
0 = Ax + B1 ng c + B2 (0K x23 ): (4.7) In the region !m < 0 there is no equilibrium point. 3 ), using (4.7) we obtain In the region !m 0 (or 3 0x Fig. 3. Linear passive two-mass drive-train from (4.1) with the quadratic torque _ = Ax + B1 (ng c + dw ) + B2 0Kx32 + du controller from (4.2). This closed-loop system is called S in Theorem 4.4. = A( + x) + B1 (ng c + dw ) 0 B2 K (3 + x3 )2 + B2 du = A + B1 dw + B2 du 0 B2 K 32 + 2x3 3 : T Then S2 is passive with the storage function H (x) = x P x, with P In the region !m < 0 (or 3 0x 3 ), using (4.7), we obtain as in (4.4). Proof: We have seen a little earlier that S1 is passive with the _ = A( + x) + B1 (ng c + dw ) + B2 du storage function H defined as in the proposition. Therefore, along tra= A + B1 dw + B2 du + B2 K x23 : jectories of S2 , we have In both regions !m 0 and !m < 0, it can be seen that _ depends T y1 Ta ref du 0 Te y2 T y1 = Tda 0 Teref y2 : y2 u
H_ uT y =
We see from (4.2) that the term Teref y2 !m 0. Hence we have
H_
0 only exists when y2 =
d1 T y1 : d2 y2
Thus, S2 is passive. Proposition 4.3: Consider the closed-loop system S2 formed by the drive-train (4.1) with the feedback law (4.2), with du = Teref 0 Te = 0 and with Ta = ng c > 0 (a constant). Then this system is GAS with respect to the equilibrium point
nc K x1 pb +4 cK 0b : x = x2 = 2Kn p x3 b +4cK 0b 2K
(4.5)
Note that Ta is the aerodynamic torque and c is the turbine torque referred to the high speed shaft. The stability is not due to the damping coefficient b, and it is true also for b = 0. For the proof, we refer to our paper [14], where one of Lyapunov’s stability theorems (see [5, Theor. 4.2]) has been used. Now we consider the closed-loop system S2 consisting of the twomass model of the drive-train S1 , with the quadratic torque control (4.2), with a torque tracking error du as in (4.3) and with an aerodynamic torque Ta = ng c + dw , where ng c is a “steady state” value and dw is the deviation of Ta from this value, see Fig. 3. After changing of variables, we regard this system S2 with input d = [dw ; du ]T , state = x 0 x, where x is given in (4.5), and output y = [2 ; 3 ]T . The following is our main result for this example: Theorem 4.4: Consider the closed-loop wind turbine system S2 described by (4.1), (4.2), and (4.3), where Ta = ng c + dw ; c > 0 is a constant and du ; dw are disturbance signals. We regard this system , where x is given in (4.5) with input d = [dw ; du ]T , state = x 0 x and output y = [2 ; 3 ]T . Then this system is iISS, more precisely
(k (t)k) (k (0)k; t) +
t
0
kd( )kd
(4.6)
linearly on the input d, therefore the condition 3.4 in Corollary 3.4 holds. Choose the storage function H ( ) = T P , with P as in (4.4). Let min denote the smallest eigenvalue of P (i.e., the smallest of Ks ; JT and JG ). Then H ( ) min k k2 . Hence, (3.1) holds for = 2 and k = min . Since the output of S2 depends linearly on the state , it can be shown easily that (3.2) holds (see Remark 3.2). Applying Corollary 3.4, we conclude that (4.6) holds. We remark that a direct proof of the iISS property for the turbine drive-train has been given in our paper [14], where we chose the Lyapunov function
V ( ) =
1 T P 2 1 + 12 T P
and showed that S2 with V satisfy (2.2). However, in [14], the explicit form of the iISS gain was unknown. The main result of this paper has been arrived at by extracting and generalizing the abstract idea in the direct proof of [14], and this eliminates the need for finding a Lyapunov function, which usually involves a big effort. Moreover, we show that under the technical assumption (3.4), the function is known explicitly, i.e., (r) = r . Example 4.5: We give a very simple example of a system 6 which is passive and GAS, but not iISS. This shows that the conditions (3.1) and (3.2) cannot be omitted from Theorem 3.1. The equations of the system are
x_ = 0x + x3 u; y = x4 : It is very easy to verify that 6 is indeed GAS and passive with the storage function H (x) = (1=2)x2 . The estimate (3.1) holds with 2, but for all such the condition (3.2) does not hold. Thus, Theorem 3.1 does not apply. Consider the initial state x(0) = (1=2) and the input function u(t) = (2 0 t)2 + (2 0 t). Then the state trajectory is x(t) = 1=(2 0 t), which is defined only on the time interval [0,2). The finite-time blow-up of the solution means that the system is not iISS. V. CONCLUSION In this paper, we have shown that under some technical assumptions, a passive nonlinear system which is GAS is also iISS with a very simple (L1 norm type) integral term. Our main result eliminates the need for
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finding a Lyapunov function satisfying the estimate (2.2) for this class of systems.
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Weight Selection in Feedback Design With Degree Constraints Mir Shahrouz Takyar, Ali Nasiri Amini, and Tryphon T. Georgiou
ACKNOWLEDGMENT The authors would like to thank Dr. B. Jayawardhana for his comments on their main results in Section III, and an anonymous referee for the second example in Section IV.
REFERENCES [1] P. Moylan, “Implications of passivity in a class of nonlinear systems,” IEEE Trans. Automat. Control, vol. AC-19, no. 19, pp. 373–381, Aug. 1974. [2] D. Hill and P. Moylan, “The stability of nonlinear dissipative systems,” IEEE Trans. Automat. Control, vol. AC-21, no. 5, pp. 708–711, Oct. 1976. [3] C. Byrnes, A. Isidori, and J. Willems, “Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems,” IEEE Trans. Automat. Control, vol. 36, no. 11, pp. 1228–1240, Nov. 1991. [4] J. Willems, “The generation of Lyapunov function for input-output stable systems,” J. SIAM Contr., vol. 9, pp. 105–134, 1971. [5] H. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2000. [6] E. Sontag, “Smooth stabilization implies coprime factorization,” IEEE Trans. Automat. Control, vol. 34, no. 4, pp. 435–443, Apr. 1989. [7] E. Sontag, “Comments on integral variants of ISS,” Syst. Control Lett., vol. 34, pp. 93–100, 1998. [8] E. Sontag and A. Teel, “Changing supply functions in input/state stable systems,” IEEE Trans. Automat. Control, vol. 40, no. 8, pp. 1476–1478, Aug. 1995. [9] D. Angeli, E. Sontag, and Y. Wang, “A characterization of integral input- to-state stability,” IEEE Trans. Automat. Control, vol. 45, no. 6, pp. 1082–1097, Jun. 2000. [10] B. Jayawardhana, A. R. Teel, and E. P. Ryan, “iISS gain of dissipative systems,” in Proc. 46th IEEE Conf. Decision and Control, New Orleans, LA, 2007, pp. 3835–3840. [11] B. Jayawardhana and G. Weiss, “Convergence of the state of a passive nonlinear plant with l input,” in Proc. ECC 2007, Kos, Greece, Jul. 2007, CD-ROM. [12] A. Hansen, P. Soerensen, F. Blaabjerg, and J. Becho, “Dynamic modeling of wind farm grid interaction,” Wind Eng., vol. 26, no. 4, pp. 191–208, 2002. [13] Z. Lubosny, Wind Turbine Operation in Electric Power Systems. Berlin, Germany: Springer-Verlag, 2003. [14] C. Wang and G. Weiss, “Integral input-to-state stability of the drivetrain of a wind turbine,” in Proc. 46th IEEE Conf. Decision and Control, New Orleans, LA, 2007, pp. 6100–6105. [15] K. Johnson, L. Pao, M. Balas, and L. Fingersh, “Control of variablespeed wind turbines: Standard and adaptive techniques for maximizing energy capture,” IEEE Control Syst. Mag., vol. 26, no. 3, pp. 70–81, Jun. 2006. [16] T. Burton, D. Sharpe, N. Jenkins, and E. Bossanyi, Wind Energy Handbook. Chichester, U.K.: Wiley, 2001. [17] S. Heier, Wind Energy Conversion Systems. Chichester, U.K.: Wiley, 1998.
Abstract—We present an approach for feedback design which is based on recent developments in analytic interpolation with a degree constraint. Performance is cast as an interpolation problem with bounded analytic functions. Minimizers of a certain weighted-entropy functional provide interpolants having degree less than the number of constraints. The choice of weight parameterizes all such bounded degree solutions. However, the relationship between the weights and the shape of corresponding transfer functions is not direct. Thus, in this paper we develop a formalism that guides weight selection. Index Terms—Control synthesis, weighted entropy-like functionals.
I. INTRODUCTION Modern robust control design focuses on shaping the frequency response of closed-loop transfer functions. Performance is cast as a weighted optimization problem where weights relate to desired frequency responses [1], [2]. A drawback of standard H -based methodologies is that they result in a degree inflation for the controller and the feedback system beyond what is necessary for achieving performance. This paper is about a new formalism based on recent developments in analytic interpolation with a degree constraint [3]–[6]. Here, interpolants are obtained as minimizers of a weighted entropy-like functional and the choice of weight affects the shape of the optimal closedloop operator (interpolant). Although this approach allows some handle on the degree of interpolants, the relation between the weighting function and the shape of the corresponding interpolant is not direct. Thus, in this work, building on earlier studies by Nagamune, Blomqvist, and others (see e.g., [7]–[12]), we present an approach to address this issue. We formulate a quasi- convex optimization problem for weight selection based on a desired shape for the closed-loop response. We deal with sensitivity shaping of single-input/single-output systems and demonstrate the efficacy of the new methodology with illustrative examples.
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II. EXTREMA OF WEIGHTED ENTROPY FUNCTIONALS Given a nominal scalar plant model P (z ), in discrete-time, internal stability of the closed-loop system with a suitable control C (z ) can be expressed via interpolation conditions on the sensitivity function 01 [13]. The conditions are as follows: first S (z ) = (1 + P (z )C (z )) c , and S (z ) must be analytic in the complement of the open unit disk then S (zi )
=
0
when zi is a pole of P in 1 when zi is a root of P in
c c
(1)
for i = 0; 1; . . . ; n, i.e., the number of interpolation conditions is assumed to be n + 1. Multiple poles and zeros induce interpolation on the Manuscript received May 2, 2007; revised July 30, 2007. Current version published September 24, 2008. This work was supported in part by the National Science Foundation (NSF) and AFOSR. Recommended by Associate Editor G. Chesi. The authors are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2008.929874 0018-9286/$25.00 © 2008 IEEE
[12] M. Krstic and A. Smyshlyaev, “Adaptive boundary control for unstable parabolic PDEs—Part I: Lyapunov design,” IEEE Trans. Autom. Control, vol. 53, no. 7, Aug. 2008. [13] Z. H. Li and M. Krstic, “Optimal design of adaptive tracking controllers for nonlinear systems,” Automatica, vol. 33, pp. 1459–1473, 1997. [14] W. Liu and M. Krstic, “Adaptive control of Burgers’ equation with unknown viscosity,” Int. J. Adaptive Control and Signal Process., vol. 15, pp. 745–766, 2001. [15] H. Logemann and B. Martensson, “Adaptive stabilization of infinitedimensional systems,” IEEE Trans. Autom. Control, vol. 37, no. 12, pp. 1869–1883, Dec. 1992. [16] P. J. Moylan and B. D. O. Anderson, “Nonlinear regulator theory and an inverse optimal control problem,” IEEE Trans. Autom. Control, vol. AC-18, pp. 460–465, 1973. [17] Y. Orlov, “Sliding mode observer-based synthesis of state derivative-free model reference adaptive control of distributed parameter systems,” J. Dynam. Syst., Meas., and Control, vol. 122, pp. 726–731, 2000. [18] R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. New York: Springer, 1997. [19] A. Smyshlyaev and M. Krstic, “Closed form boundary state feedbacks for a class of 1-D partial integro-differential equations,” IEEE Trans. Autom. Control, vol. 49, no. 12, pp. 2185–2202, Dec. 2004. [20] A. Smyshlyaev and M. Krstic, “Adaptive boundary control for unstable parabolic PDEs—Part II: Estimation-based designs,” Automatica, vol. 43, pp. 1543–1556, 2007. [21] A. Smyshlyaev and M. Krstic, “Adaptive boundary control for unstable parabolic PDEs—Part III: Output-feedback examples with swapping identifiers,” Automatica, vol. 43, pp. 1557–1564, 2007. [22] S. Townley, “Simple adaptive stabilization of output feedback stabilizable distributed parameter systems,” Dynam. Control, vol. 5, pp. 107–123, 1995. [23] J. T.-Y. Wen and M. J. Balas, “Robust adaptive control in Hilbert space,” J. Mathemat. Anal. Applic., vol. 143, pp. 1–26, 1989.
The iISS Property for Globally Asymptotically Stable and Passive Nonlinear Systems Chen Wang and George Weiss
Abstract—This paper investigates the integral input-to-state stability (iISS) property for passive nonlinear systems. We show that under mild technical assumptions, a passive nonlinear system which is globally asymptotically stable is also iISS. Moreover, the integral term from the norm). definition of the iISS property has a very simple form (like an We illustrate the result by proving that the drive-train of a wind turbine with quadratic torque control is iISS. Index Terms—Globally asymptotically stable, integral input-to-state stability (iISS), wind turbine.
I. INTRODUCTION The concept of passivity is important in control theory because it is a property shared by many physical systems and it is related to stability Manuscript received June 23, 2008; revised July 28, 2008. First published September 12, 2008; current version published September 24, 2008.Recommended by Associate Editor David Angeli. C. Wang is with the Control and Power Group, Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: [email protected]). G. Weiss is with the Department of Electrical Engineering—Systems, Faculty of Engineering, Tel Aviv University, Ramat Aviv 69978, Israel (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2008.929465
1947
(see Moylan [1], Hill and Moylan [2], Byrnes et al. [3]). Consider a dynamical system S described by
x_ = f (x; u) y = h(x; u)
(1.1)
n 2 m ! n is locally Lipschitz continuous and m m is continuous. Here x(t) is the state at time 2 ! h: n t, which is in ; u is the input signal and y is the output signal. Under
where
f
n
:
these assumptions, for every initial state x(0) and for every bounded input signal u, (1.1) has a unique solution on some time interval [0; "), with " > 0. S is said to be passive if there exists a continuously differentiable storage function or Hamiltonian H : n ! [0; 1) such that
H_ uT y;
_ = where H
@H f (x; u) @x
for all (x; u) 2 n 2 m . To investigate the Lyapunov stability of the equilibrium points of S corresponding to u = 0 we may use H as a Lyapunov function (see Willems [4] or Khalil [5]). The notion of input-to-state stability (ISS), as introduced in Sontag [6], implies that f (x; 0) = 0 iff x = 0 and for any initial state, if the input becomes uniformly very small after some time, then also the state becomes uniformly very small after some time (see Sontag [7]). Sontag and Teel [8] gives a characterization of supply rates for ISS systems. A strictly weaker variant of ISS is the concept of integral input-to-state stability (iISS), where the uniform smallness of the input is replaced by the smallness of a certain integral depending on the input, see Angeli, Sontag and Wang [9]. The formal definition of iISS is given in Section II. In this paper, we investigate the iISS property for a class of passive nonlinear systems. In our main result (stated in Section III), we show that under mild assumptions, a passive nonlinear system which is globally asymptotically stable (GAS) is also iISS. By combining our result with a recent result in Jayawardhana et al. [10], we can actually prove that under mild technical assumptions, a passive and GAS system satisfies the iISS type estimate with a very simple (L1 norm type) integral term. We will illustrate the result by proving the iISS property (with a simple integral term) for the drive-train of a wind turbine, in Section IV. II. BACKGROUND CONCEPTS In this section, we recall the background about the iISS property following [9]. Some commonly used terminology: A function H : n ! [0; 1) is called positive definite if H (x) = 0 iff x = 0. H is called proper if H (x) ! 1 when kxk ! 1. A continuous function : [0; a) ! [0; 1) belongs to the class K if it is strictly increasing and (0) = 0. Such a function is in the class K if a = 1 and (r) ! 1 as r ! 1. A continuous function : [0; a) 2 [0; 1) ! [0; 1) belongs to the class KL if, for each fixed s; (1; s) belongs to K and, for each fixed r , the mapping (r; 1) is decreasing and (r; s) ! 0 as s ! 1. Given any measurable and bounded control u and any 2 n , there is a unique solution of (1.1) with x(0) = . This solution (or state trajectory) is defined on some maximal interval [0; ), and it is denoted by x(1; ; u). Definition 2.1: The system (1.1) is integral input-to-state stable (iISS) if there exist a class K function , a class KL function and a class K function such that for every 2 n and for every measurable and bounded function u, the state trajectory x(t; ; u) is defined for all t 0 and
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1
1
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 8, SEPTEMBER 2008
(kx(t; ; u)k) (k k; t) +
t 0
(ku( )k) d 8t 0:
(2.1)
The function is called the iISS gain of the system (1.1). If u is such that 0 (ku( )k)d < 1, then the iISS estimate in (2.1) also implies that x(t; ; u) ! 0 as t ! 1. However, if the system is iISS and u is only bounded, then the state trajectory x(1; ; u) may be unbounded. It is easy to see that if the system (1.1) is iISS, then this system has a unique equilibrium point at zero (f (x; 0) = 0 iff x = 0) and the system is globally asymptotically stable (GAS), which means that it is Lyapunov stable, the trajectories x(t; ; 0) are defined for all t 0 and tend to zero (as t ! 1). Definition 2.2: The system (1.1) is zero-output dissipative, if there exists a continuously differentiable proper and positive definite function V , and a class K function , such that
1
@V f (x; u) (kuk) @x
8
(x; u)
2
n 2 m:
(2.2)
Theorem 2.3: The system (1.1) is iISS if and only if it is GAS and zero-output dissipative. This follows from ([9], Theorem 1) together with ([9], Remark II.3). Theorem 2.4: Assume that the system (1.1) is GAS, zero-output dissipative and satisfies the following: for every compact set M n , there exists m > 0 such that k
f (x; u)k m(1 + (kuk))8(x;u) 2 M 2 m
(2.3)
with the same function in (2.2) and (2.3). Then for every 2 n and for every measurable and bounded function u : [0; 1) ! m , the state trajectory x(t; ; u) (which is defined for all t 0 according to Theorem 2.3) satisfies (2.1) with = . This recent result on the iISS gain is due to Jayawardhana et al. ([10, Theor. 3.1]). III.
IISS
In this section, we consider the system 6 described by (1.1) together with a simpler output equation
y = h (x ) where h : n ! m is continuous. Our main results are the following. Theorem 3.1: We assume that 6 is passive and GAS, with the storage function H . Denote
c(r) = sup kh(x)k: kxkr
1 0
c
d
and = : +1 1
Then the system (1.1) is zero-output dissipative with hence it is iISS.
0,
Choose the Lyapunov function V (x) =
F 0 ( ) =
1 +1
c(R ) =
F (H (x)), where 8
( )
c
then obviously
when < 0 ; when 0 ;
1
c(R)+1
0:
It is easy to see that F 0 () is a non-increasing continuous function of (see Fig. 1). We remark that in the region of n where H (x) 0 , we have F () = =(c(R) + 1), so that V (x) = H (x)=(c(R) + 1). We want to show that f with V satisfy (2.2). We consider two cases depending on kxk. • Assume that x 2 n with kxk < R (case 1). Using the passivity of 6 and the Cauchy-Schwarz inequality, we obtain
1
V_ = F 0 (H (x))H_
1
c(R ) + 1
c(R ) + 1
1
c(R ) + 1 1 k h (x )k 1 k u k 1
1
uT y
c(kx k) 1 k u k:
V_
k
(3.2)
(r) = r , and
u k:
F 0 (H (x)) F 0 (kkxk ): This implies, using again the passivity of Schwarz inequality
(3.1)
0 and R 0 such that
H (x) kkxk for kxk R
0 = kR
Proof: Define
c((0=k)(1= ).
Since c(r) is a non-decreasing function of r 2 [0; 1); kxk implies that c(kxk) c(R). Using this inequality, we have
PROPERTY FOR GAS AND PASSIVE NONLINEAR SYSTEMS
We assume that there exist ; k
Fig. 1. Function F (), which is a non-increasing continuous function of . In this figure, we have assumed that > 0.
6 and the Cauchy-
F 0 (k x ) u T y 1 h (x ) u c( x ) + 1 1 c( x ) u u : c( x ) + 1 k
k
k
k
k
1 k
1
k
k1 k
k
k
1 k
1
k
k k
k
Thus we have proved that V_ kuk for all x 2 n . This means that if V is proper, then the system (1.1) is zero-output dissipative with (r) = r . To show that indeed V is proper, note that 0
1 0 F ()d = 1:
(3.3)
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Indeed, using (3.2), we have
1 0 1 0 F ()d F ()d 0 1 d =k
c
+1
= 1:
Since H is proper [this follows from (3.1)], we have
!1 F (H ) = Hlim !1 kxlim k!1 V (x) = Hlim
H 0
Fig. 2. Two-mass drive-train model with gearbox.
F 0 ( ) d = 1 :
[We have used (3.3).] Since 6 is GAS, applying Theorem 2.3 we conclude that 6 is iISS. Remark 3.2: If the output y of 6 is a linear function of the state x, i.e., h(x) = Cx, where C is a matrix of matching dimensions, then c(r) = kC kr and then (3.2) holds for every 1. Remark 3.3: If there exist k1 > 0 and r0 0 such that
c(r ) k 1 r
then it follows that (3.2) holds. Corollary 3.4: We assume that 6 is passive, GAS and satisfies (3.1) and (3.2). Suppose also that for every compact set M n , there exists m > 0 such that
m:
(3.4)
Then for every 2 u : [0; 1) ! m ; 6 has a unique state trajectory x(1; ; u) defined on [0; 1), and this satisfies
n and for every measurable and bounded function
(kx(t; ; u)k) (k k; t) +
t 0
ku( )kd 8t 2 [0; 1)
(3.5)
where 2 K and 2 KL are independent of and u. Note that (3.5) means that 6 is iISS, with the iISS gain (r) = r . Proof: Recall from Theorem 3.1 that the system (1.1) is zerooutput dissipative with (kuk) = kuk. Applying Theorems 2.3 and 2.4, we see that this system is iISS, with the iISS gain (kuk) = kuk. Remark 3.5: After seeing Corollary 3.4, it is tempting to conjecture that if 6 satisfies the assumptions in this corollary, then it has state trajectories defined for all t 0, for every 2 n and every u 2 L1 [0; 1). However, this is not correct, as can be seen from the example at the beginning of Section IV in Jayawardhana and Weiss [11]. The existence of global solutions is guaranteed only for bounded and measurable inputs.
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IV. EXAMPLES Example 4.1: We consider the drive-train of a wind turbine with quadratic torque control. The two-mass drive-train model with gearbox is shown in Fig. 2 (see [12] and [13]). The large turbine inertia JT corresponds to the blades and the hub, and the small inertia JG represents the induction generator. We take the state variables x, the external input variables w (disturbances) and the control input u as follows:
k x = !T ; w = T a ; u = 0 T e !m
x_ = Ax + B1 w + B2 u; y = Cx
(4.1)
where
8r r0 ;
kf (x; u)k m(1 + kuk) 8(x; u) 2 M 2
where k is the angular difference between the two shafts, !T is the turbine rotor speed, !m is the generator rotor speed, Ta is the aerodynamic torque, and Te is the generator electrical torque. The output variables are !m and !T , so that y = [!T !m ]T . Then the third-order state-space representation is
A=
0
0 KJ K J n
0 n1
1
0 JC C J n
C J n 0 J Cn 0 Jb
0
0 0 ; C = 00 10 01 : 1
B = [ B1 B2 ] = J1
0
J
Here ng is the gearbox ratio, Ks > 0 is the torsional stiffness of the low speed shaft, Cs 0 is the torsional damping of the low speed shaft and b 0 is the damping coefficient. For the derivation of the above model, we refer to our paper [14]. For the wind turbine operating in low to medium wind speed region, it is desirable to maximize the output power of the turbine. The standard quadratic torque control law is often used to achieve this control objective (see [15])
0 when !m < 0; 2 when !m 0; K!m
Teref = where K
(4.2)
> 0. We introduce du = Teref 0 Te
(4.3)
which is the error between the reference torque and the real generator torque. There are many good books on wind turbine technology, of which we mention Burton et al. [16] and Heier [17]. Consider the linear drive-train system S1 , described by the matrices (A; B; C ) from (4.1), with input u = [Ta ; 0Te ]T , state x = [k ; !T ; !m ]T and output y = [!T ; !m ]T . Then this system is passive with the storage function H (x) = xT P x, where
P=
0 0 0 JT 0 : 0 0 JG
Ks
(4.4)
For the simple proof of the above claim, we refer to our paper [14]. Proposition 4.2: Consider the closed-loop wind turbine system S2 described by (4.1), (4.2) and (4.3). We regard this system with input d = [Ta ; du ]T , state x = [k ; !T ; !m ]T and output y = [!T ; !m ]T .
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 8, SEPTEMBER 2008
for all t > 0, where 2 K1 and 2 KL. Proof: We know from Proposition 4.2 that the closed-loop turbine system S2 is passive. We also know from Proposition 4.3 that S2 is , which is the positive solution of the equation GAS with respect to x
0 = Ax + B1 ng c + B2 (0K x23 ): (4.7) In the region !m < 0 there is no equilibrium point. 3 ), using (4.7) we obtain In the region !m 0 (or 3 0x Fig. 3. Linear passive two-mass drive-train from (4.1) with the quadratic torque _ = Ax + B1 (ng c + dw ) + B2 0Kx32 + du controller from (4.2). This closed-loop system is called S in Theorem 4.4. = A( + x) + B1 (ng c + dw ) 0 B2 K (3 + x3 )2 + B2 du = A + B1 dw + B2 du 0 B2 K 32 + 2x3 3 : T Then S2 is passive with the storage function H (x) = x P x, with P In the region !m < 0 (or 3 0x 3 ), using (4.7), we obtain as in (4.4). Proof: We have seen a little earlier that S1 is passive with the _ = A( + x) + B1 (ng c + dw ) + B2 du storage function H defined as in the proposition. Therefore, along tra= A + B1 dw + B2 du + B2 K x23 : jectories of S2 , we have In both regions !m 0 and !m < 0, it can be seen that _ depends T y1 Ta ref du 0 Te y2 T y1 = Tda 0 Teref y2 : y2 u
H_ uT y =
We see from (4.2) that the term Teref y2 !m 0. Hence we have
H_
0 only exists when y2 =
d1 T y1 : d2 y2
Thus, S2 is passive. Proposition 4.3: Consider the closed-loop system S2 formed by the drive-train (4.1) with the feedback law (4.2), with du = Teref 0 Te = 0 and with Ta = ng c > 0 (a constant). Then this system is GAS with respect to the equilibrium point
nc K x1 pb +4 cK 0b : x = x2 = 2Kn p x3 b +4cK 0b 2K
(4.5)
Note that Ta is the aerodynamic torque and c is the turbine torque referred to the high speed shaft. The stability is not due to the damping coefficient b, and it is true also for b = 0. For the proof, we refer to our paper [14], where one of Lyapunov’s stability theorems (see [5, Theor. 4.2]) has been used. Now we consider the closed-loop system S2 consisting of the twomass model of the drive-train S1 , with the quadratic torque control (4.2), with a torque tracking error du as in (4.3) and with an aerodynamic torque Ta = ng c + dw , where ng c is a “steady state” value and dw is the deviation of Ta from this value, see Fig. 3. After changing of variables, we regard this system S2 with input d = [dw ; du ]T , state = x 0 x, where x is given in (4.5), and output y = [2 ; 3 ]T . The following is our main result for this example: Theorem 4.4: Consider the closed-loop wind turbine system S2 described by (4.1), (4.2), and (4.3), where Ta = ng c + dw ; c > 0 is a constant and du ; dw are disturbance signals. We regard this system , where x is given in (4.5) with input d = [dw ; du ]T , state = x 0 x and output y = [2 ; 3 ]T . Then this system is iISS, more precisely
(k (t)k) (k (0)k; t) +
t
0
kd( )kd
(4.6)
linearly on the input d, therefore the condition 3.4 in Corollary 3.4 holds. Choose the storage function H ( ) = T P , with P as in (4.4). Let min denote the smallest eigenvalue of P (i.e., the smallest of Ks ; JT and JG ). Then H ( ) min k k2 . Hence, (3.1) holds for = 2 and k = min . Since the output of S2 depends linearly on the state , it can be shown easily that (3.2) holds (see Remark 3.2). Applying Corollary 3.4, we conclude that (4.6) holds. We remark that a direct proof of the iISS property for the turbine drive-train has been given in our paper [14], where we chose the Lyapunov function
V ( ) =
1 T P 2 1 + 12 T P
and showed that S2 with V satisfy (2.2). However, in [14], the explicit form of the iISS gain was unknown. The main result of this paper has been arrived at by extracting and generalizing the abstract idea in the direct proof of [14], and this eliminates the need for finding a Lyapunov function, which usually involves a big effort. Moreover, we show that under the technical assumption (3.4), the function is known explicitly, i.e., (r) = r . Example 4.5: We give a very simple example of a system 6 which is passive and GAS, but not iISS. This shows that the conditions (3.1) and (3.2) cannot be omitted from Theorem 3.1. The equations of the system are
x_ = 0x + x3 u; y = x4 : It is very easy to verify that 6 is indeed GAS and passive with the storage function H (x) = (1=2)x2 . The estimate (3.1) holds with 2, but for all such the condition (3.2) does not hold. Thus, Theorem 3.1 does not apply. Consider the initial state x(0) = (1=2) and the input function u(t) = (2 0 t)2 + (2 0 t). Then the state trajectory is x(t) = 1=(2 0 t), which is defined only on the time interval [0,2). The finite-time blow-up of the solution means that the system is not iISS. V. CONCLUSION In this paper, we have shown that under some technical assumptions, a passive nonlinear system which is GAS is also iISS with a very simple (L1 norm type) integral term. Our main result eliminates the need for
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finding a Lyapunov function satisfying the estimate (2.2) for this class of systems.
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Weight Selection in Feedback Design With Degree Constraints Mir Shahrouz Takyar, Ali Nasiri Amini, and Tryphon T. Georgiou
ACKNOWLEDGMENT The authors would like to thank Dr. B. Jayawardhana for his comments on their main results in Section III, and an anonymous referee for the second example in Section IV.
REFERENCES [1] P. Moylan, “Implications of passivity in a class of nonlinear systems,” IEEE Trans. Automat. Control, vol. AC-19, no. 19, pp. 373–381, Aug. 1974. [2] D. Hill and P. Moylan, “The stability of nonlinear dissipative systems,” IEEE Trans. Automat. Control, vol. AC-21, no. 5, pp. 708–711, Oct. 1976. [3] C. Byrnes, A. Isidori, and J. Willems, “Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems,” IEEE Trans. Automat. Control, vol. 36, no. 11, pp. 1228–1240, Nov. 1991. [4] J. Willems, “The generation of Lyapunov function for input-output stable systems,” J. SIAM Contr., vol. 9, pp. 105–134, 1971. [5] H. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2000. [6] E. Sontag, “Smooth stabilization implies coprime factorization,” IEEE Trans. Automat. Control, vol. 34, no. 4, pp. 435–443, Apr. 1989. [7] E. Sontag, “Comments on integral variants of ISS,” Syst. Control Lett., vol. 34, pp. 93–100, 1998. [8] E. Sontag and A. Teel, “Changing supply functions in input/state stable systems,” IEEE Trans. Automat. Control, vol. 40, no. 8, pp. 1476–1478, Aug. 1995. [9] D. Angeli, E. Sontag, and Y. Wang, “A characterization of integral input- to-state stability,” IEEE Trans. Automat. Control, vol. 45, no. 6, pp. 1082–1097, Jun. 2000. [10] B. Jayawardhana, A. R. Teel, and E. P. Ryan, “iISS gain of dissipative systems,” in Proc. 46th IEEE Conf. Decision and Control, New Orleans, LA, 2007, pp. 3835–3840. [11] B. Jayawardhana and G. Weiss, “Convergence of the state of a passive nonlinear plant with l input,” in Proc. ECC 2007, Kos, Greece, Jul. 2007, CD-ROM. [12] A. Hansen, P. Soerensen, F. Blaabjerg, and J. Becho, “Dynamic modeling of wind farm grid interaction,” Wind Eng., vol. 26, no. 4, pp. 191–208, 2002. [13] Z. Lubosny, Wind Turbine Operation in Electric Power Systems. Berlin, Germany: Springer-Verlag, 2003. [14] C. Wang and G. Weiss, “Integral input-to-state stability of the drivetrain of a wind turbine,” in Proc. 46th IEEE Conf. Decision and Control, New Orleans, LA, 2007, pp. 6100–6105. [15] K. Johnson, L. Pao, M. Balas, and L. Fingersh, “Control of variablespeed wind turbines: Standard and adaptive techniques for maximizing energy capture,” IEEE Control Syst. Mag., vol. 26, no. 3, pp. 70–81, Jun. 2006. [16] T. Burton, D. Sharpe, N. Jenkins, and E. Bossanyi, Wind Energy Handbook. Chichester, U.K.: Wiley, 2001. [17] S. Heier, Wind Energy Conversion Systems. Chichester, U.K.: Wiley, 1998.
Abstract—We present an approach for feedback design which is based on recent developments in analytic interpolation with a degree constraint. Performance is cast as an interpolation problem with bounded analytic functions. Minimizers of a certain weighted-entropy functional provide interpolants having degree less than the number of constraints. The choice of weight parameterizes all such bounded degree solutions. However, the relationship between the weights and the shape of corresponding transfer functions is not direct. Thus, in this paper we develop a formalism that guides weight selection. Index Terms—Control synthesis, weighted entropy-like functionals.
I. INTRODUCTION Modern robust control design focuses on shaping the frequency response of closed-loop transfer functions. Performance is cast as a weighted optimization problem where weights relate to desired frequency responses [1], [2]. A drawback of standard H -based methodologies is that they result in a degree inflation for the controller and the feedback system beyond what is necessary for achieving performance. This paper is about a new formalism based on recent developments in analytic interpolation with a degree constraint [3]–[6]. Here, interpolants are obtained as minimizers of a weighted entropy-like functional and the choice of weight affects the shape of the optimal closedloop operator (interpolant). Although this approach allows some handle on the degree of interpolants, the relation between the weighting function and the shape of the corresponding interpolant is not direct. Thus, in this work, building on earlier studies by Nagamune, Blomqvist, and others (see e.g., [7]–[12]), we present an approach to address this issue. We formulate a quasi- convex optimization problem for weight selection based on a desired shape for the closed-loop response. We deal with sensitivity shaping of single-input/single-output systems and demonstrate the efficacy of the new methodology with illustrative examples.
1
II. EXTREMA OF WEIGHTED ENTROPY FUNCTIONALS Given a nominal scalar plant model P (z ), in discrete-time, internal stability of the closed-loop system with a suitable control C (z ) can be expressed via interpolation conditions on the sensitivity function 01 [13]. The conditions are as follows: first S (z ) = (1 + P (z )C (z )) c , and S (z ) must be analytic in the complement of the open unit disk then S (zi )
=
0
when zi is a pole of P in 1 when zi is a root of P in
c c
(1)
for i = 0; 1; . . . ; n, i.e., the number of interpolation conditions is assumed to be n + 1. Multiple poles and zeros induce interpolation on the Manuscript received May 2, 2007; revised July 30, 2007. Current version published September 24, 2008. This work was supported in part by the National Science Foundation (NSF) and AFOSR. Recommended by Associate Editor G. Chesi. The authors are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2008.929874 0018-9286/$25.00 © 2008 IEEE
