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Stable H∞ Flow Controller Design
Stable H∞ Flow Controller Design ¨ ˙ and Altu˘ g Iftar Hakkı Ula¸s Unal Department of Electrical and Electronics Engineering Anadolu University, 26470 Eski¸sehir, Turkey [email protected] [email protected] Abstract: Stable H∞ controller design, for a flow control problem which involves multiple time-delays, is considered. An algorithm, which produces a stable controller apart from the integral action, is developed utilizing a rational approximation of finite-impulse response filters. An example is also presented to show the performance of the stable H∞ flow controller obtained by the proposed approach. This controller robustly stabilizes the system and achieves all design requirements, whereas the H∞ -optimal controller, which is unstable, fails to achieve the same c IFAC 2009 requirements.Copyright ° 1. INTRODUCTION There exists numerous controller design methods for timedelay systems (Niculescu (2001)). An H∞ controller design for single-input single-output (SISO) systems with a single ¨ time-delay was formulated by Toker and Ozbay (1995). Meinsma and Zwart (2000) proposed an H∞ controller design approach for multi-input multi-output (MIMO) systems with a single time-delay using J-spectral factorizations. Later, an H∞ -optimal controller design approach for MIMO systems with multiple time-delays was proposed by Meinsma and Mirkin (2005) using the same factorization method in the chain-scattering framework. In this approach, the problem is splitted into a nested sequence of subproblems, called adobe problems, where each involve a single time-delay. Then the controller is obtained by solving these subproblems. In controller design, the first requirement is to guarantee the internal stability of the closed-loop system. In general, this may be achieved by an unstable controller. However, the stability of the controller is often desired, since unstable controllers are highly sensitive to non-linear effects, sensor/actuator faults, and numerical errors. Therefore, stability of the stabilizing controller may be introduced as an additional requirement. Stable H∞ controller design problem for finite dimensional systems has been considered ¨ for a long time (see Zeren and Ozbay (1999) and references therein). Stable H∞ controller design for SISO time-delay ¨ systems was given by G¨ um¨ u¸ssoy and Ozbay (2008). Re¨ ˙ cently, Unal and Iftar (2008a) proposed an algorithm to design a stable H∞ controller for MIMO systems with multiple time-delays. In the present paper, we consider the stable H∞ flow controller design for data-communication networks, which is an example of multiple time-delay systems. The flow controllers are designed to minimize the traffic congestion. However, existence of uncertain time-varying time-delays challenges controller design. Using the controller design ¨ approach of Toker and Ozbay (1995), a rate-based H∞ flow controller, which is robust to all uncertain timevarying time-delays in the different channels, was designed
by Quet et al. (2002). However, since the approach of ¨ Toker and Ozbay (1995) was limited to SISO systems, the proposed approach in Quet et al. (2002) was suboptimal in the H∞ sense. Later, an H∞ -optimal flow controller ¨ design approach was given by Unal et al. (2006) using the approach of Meinsma and Mirkin (2005). However, for ¨ technical reasons, in Unal et al. (2006) it was assumed that the uncertain part of the time-delays are always ¨ non-negative. This assumption was later removed in Unal ¨ ˙ et al. (2009) using the result of Unal and Iftar (2008b). ¨ Stable H∞ flow controller design was considered by Unal ˙ and Iftar (2008a), using the small-gain theorem. However, since the small-gain theorem provides only a sufficient condition, this approach may be conservative in general. In this study, we propose an alternative approach by considering a finite-dimensional approximation of the infinitedimensional part of the controller. 1.1 Notation For a matrix M , M T denotes its transpose. In denotes the n × n dimensional identity matrix. 0 denotes the zero matrix with appropriate·dimensions. 1n is the 1 × n ¸ A B denotes the transfer dimensional matrix of all 1’s. C D function matrix (TFM) C(sI − A)−1 B + D. k · k∞ and k · k2 respectively denote the H∞ norm and the L2 induced norm. A TFM is said to be stable if it is in H∞ , in addition, if its inverse exists in H∞ , it is said to be bistable. A constant square matrix is said to be stable if all its eigenvalues have negative real parts. A TFM Q ∈ H∞ is said to be contractive if kQk∞ < 1. For positive integers k and · ¸ l, Jk,l := blockdiag(Ik , −Il ). For a Σ11 Σ12 TFM Σ = , where Σ12 and Σ21 are respectively Σ21 Σ22 k × k and l × l dimensional and Σ21 is invertible, G = CHAIN (Σ) denotes the chain-scattering representation (Kimura (1996)) of Σ, where ¸ · ¸ · G11 G12 Σ12 − Σ11 Σ−1 Σ22 Σ11 Σ−1 21 21 . G =: = G21 G22 −Σ−1 Σ−1 21 Σ22 21
For k × l dimensional K, HM (G, K) denotes the homographic transformation (Kimura (1996)), which is defined as −1 Q := HM (G, K) = (G11 K + G12 ) (G21 K + G22 ) . 2. PROBLEM STATEMENT
1/s -
h
r =
r1h .. .
z1
-
rnh
2.1 Network Model
w1
-
+6
-
A data-communication network with n sources feeding a single bottleneck node is considered. The flow controller, which is to be designed, is implemented at the bottleneck node. The controller calculates a rate command for each source node to adjust the rate of data that will be sent to the bottleneck node in order to regulate the queue length at the bottleneck node. The dynamics of the queue length are given as (Quet et al. (2002)): n X
rib (t) − c(t)
(1)
i=1
where q(t) is the queue length at the bottleneck node at time t, rib (t) is the rate of data received by the bottleneck node at time t from the ith source, i = 1, . . . , n, and c(t) is the outgoing rate of data from the bottleneck node at time t, which equals to the capacity of the outgoing link assuming that q(t) is non-zero. The total amount of data received at the bottleneck node from the ith source node by time t can be written as (Quet et al. (2002)): t−τif (t) Z Zt ris (ϕ)dϕ, t − τif (t) ≥ 0 , rib (ϕ)dϕ = (2) 0 0 0, t − τif (t) < 0 where ris (t) = ri (t − τib (t)) is the rate of data sent from the ith source node at time t and ri (t) is the rate command for the ith source node issued by the controller at time t. By taking the derivative of both sides of (2), the rate of data received by the bottleneck node at time t, rib (t), is obtained as (Quet et al. (2002)): ½ (1 − δ˙if (t))ri (t − τi (t)), t − τif (t) ≥ 0 b ri (t) = , (3) 0, t − τif (t) < 0 where τi (t) = τib (t) + τif (t) is the round-trip time-delay, where τib (t) = hbi + δib (t) is the backward time-delay at time t, which is the time required for the rate command to reach the ith source. Here, hbi is the nominal timeinvariant known backward time-delay, and δib (t) is the time-varying backward time-delay uncertainty, τif (t) = hfi + δif (t) is the forward time-delay at time t, which is the time required for the data sent from the ith source to reach the bottleneck node. Here, hfi is the nominal time-invariant known forward timedelay, and δif (t) is the time-varying forward timedelay uncertainty.
r
q
W1 −
Λu
q(t) ˙ =
−? + - j
Po
∆
c
K
?+ j
qd
Fig. 1. Overall system (Ata¸slar (2004)) The nominal round-trip time-delay for the ith channel of the system is hi = hbi + hfi , and the time-varying roundtrip time-delay uncertainty is δi (t) = δib (t) + δif (t). It is assumed that the uncertainties are bounded as follows: f |δi (t)| < δ + , |δ˙i (t)| < βi , |δ˙i (t)| < β f (4) i
i
for some bounds δi+ > 0 and 0 < βif ≤ βi < 1. It is further assumed that, δi (t) is such that τi (t) ≥ 0 at all times. In a real application, there also exist some hard constraints, such as non-negativity constraints and upper bounds on the queue length and data rates. These non-linear constraints are ignored in the controller design. However, they are taken into account while doing simulations in Section 4. 2.2 Control Problem The flow controller, which is to be designed for the above described system, should robustly stabilize the system against all uncertain time-varying time-delays which satisfy (4). In addition to robust stability requirement, to avoid traffic congestion at the bottleneck node and share the network capacity fairly among the source nodes, the following performance requirements must also be considered (Quet et al. (2002)): tracking requirement: limt→∞ q(t) = qd , where qd > 0 is the desired queue length, weighted fairness requirement: limt→∞ ri (t) = αi c∞ , where αi > 0,Pi = 1, . . . , n, are the fairness weights, n which satisfy i=1 αi = 1, under the assumption that limt→∞ c(t) =: c∞ exists. Now, we can describe the overall system as shown in Fig. 1, where Po (s) := 1s 1n is the nominal plant except the time-delays, K is the controller to be designed, r := [r1 · · · rn ]¡T , ¢ Λu (s) := diag e−h1 s , . . . , e−hn s represents the nominal time-delays, which are taken outside the plant and ordered as h1 ≥ h2 ≥ . . . ≥ hn ≥ 0, in order to apply the approach of Meinsma and Mirkin (2005), ∆ is a possibly non-causal and linear time-varying block, which represents the uncertainties in the time-delays with k∆k ¤ £ 2 < 1, and W1 (s) := W 1 (s)· · · · W n (s) ¸, where, for i = 1, . . . , n, √ β +β f W i (s) := 2 √i i δi+ . s
1−βi
The structure of ∆ and the derivation of W1 can be found ¨ ˙ in Unal and Iftar (2008b).
z1 -
-
w1
P0
z
d
∆ ? W1
? W2
+ ? + i -
+ − ? i
Pb
yb - W3
w
u Λu
- e1
- K b
6
¨ Fig. 3. Equivalent four-block problem (Unal et al. (2009)) e2
W4
Λu u
K r
y
Fig. 2. System for the mixed sensitivity minimization problem (Ata¸slar (2004)) 2.3 H∞ -optimal Controller Design To solve the control problem stated above, i.e., to design a controller which robustly stabilizes the actual system and achieves all performance requirements, a mixed sensitivity minimization problem was set up in Ata¸slar (2004), as shown in Fig. 2. Here, W2 (s) = 1s , W3 (s) = σs1 , and α2 −1 0 0 1 α α3 0 −1 0 σ2 α1 , W4 (s) = . . .. s .. .. . . α. . n 0 0 −1 α1 where σ1 > 0 and σ2 > 0 are design parameters. Furthermore, d := q˙d − c, y := qd − q, e1 is introduced to achieve tracking requirement, and e2 is introduced to achieve the weighted fairness requirement. The flow control problem can now be posed as to determine a controller K which internally stabilizes the system in Fig. 2 with ∆ = 0 and minimizes the H∞ norm of the closed-loop TFM from ¤T ¤T £ £ to z := z1T eT1 eT2 . w := w1T dT To find an H∞ -optimal flow controller that achieves above objectives, an H∞ coprime factorization, Po (s) = f−1 (s)N e (s) is necessary, where N e and M f are chosen as M 1 s f e N (s) = s+² 1n and M (s) = s+² for an arbitrary ² > 0 (see ¨ Unal et al. (2009)). Utilizing this factorization, the system in Fig. 2 can also be considered as a 4-block problem shown f−1 y and in Fig. 3, where yˆ := M b f(s) = s K(s) . (5) K(s) := K(s)M s+² ¨ As shown in Unal et al. (2009), Pb satisfies the standard H∞ problem assumptions (Zhou et al. (1996)), hence the b latter problem can be defined as designing a controller K which minimizes the H∞ norm of the closed-loop TFM b for a given from w to z in Fig. 3, which is Fl (Pb, Λu K), γ > 0. For a satisfactorily large given sensitivity level, γ, a conb which satisfies kFl (Pb, Λu K)k b ∞ < γ, can be troller K, obtained by applying the approach proposed by Meinsma ¨ and Mirkin (2005), as shown in Unal et al. (2009). The corresponding controller, K, can then be found from (5) as b K(s) = s+² s K(s). The structure of the controller is shown
F2
r 6 F1
r
HM (G−1 Λ , QΛ )
qd − q − ?+ h y
? κ
s+ s
¨ ˙ Fig. 4. Structure of the controller K (Unal and Iftar (2008a)) in Fig. 4, where κ := q Pγ n
is a constant, F1 and 2 (δi+ ) F2 consist of time-delays and finite-impulse response (FIR) filters, GΛ is a bistable finite dimensional TFM, and QΛ is ¨ an arbitrary contractive TFM (see Unal et al. (2009) for details). 2
2
i=1
The optimal controller, Kopt , can be found by an itera¨ tive procedure on γ as described in Unal et al. (2009). The minimum γ, for which there exists a solution to b ∞ < γ, is denoted by γ opt . Then, Kopt (s) = kFl (Pb, Λu K)k s+² b b b b s Kopt (s), where Kopt satisfies kFl (P , Λu Kopt )k∞ < opt γ . Any designed controller K, including Kopt , obtained as above is unstable due to the integral term (see Fig. 4) which is necessary to satisfy the tracking requirement. However, apart from the integral part of the controller, i.e., b only the part from y¯ to r in Fig. 4 (which differs from K by a constant), may or may not be stable. When this part b is unstable, due to nonlinearities in the (equivalently K) system (i.e., the hard constraints), an unstable behaviour may be observed, at least for certain actual time-delays and/or initial conditions (see Section 4). In order to avoid such undesirable behaviour, a controller which is stable apart from the integral part may be required. 2.4 Stable Controller Design Consider the structure of the controller showm in Fig. 4. ¨ Here F1 and F2 are both stable (see Unal et al. (2009)). Hence, the TFM from y¯ to r is stable if and only if the TFM from y¯ to r¯ is stable. Utilizing the small-gain theorem (Sandberg (1964), Zames (1966)) and the fact that F2 is stable, the TFM from y¯ to r¯ is stable if a contractive QΛ 1 . An can be found such that kHM (G−1 Λ , QΛ )k∞ < kF2 k∞ ¨ ˙ algorithm to find such a QΛ is given in Unal and Iftar (2008a). This algorithm increases the sensitivity level γ, starting from γ opt , until such a solution can be found. However, note that, since the small-gain is not a necessary condition, the approach may unnecessarily increase γ, resulting in a conservative solution.
3. PROPOSED DESIGN APPROACH
F2app
In this section, we propose an alternative approach to the stable controller design problem stated in Subsection 2.4. This approach is based on the approximation of the F2 block, which consists of FIR filters and timedelays, by a finite-dimensional TFM. For this purpose we use the approximation method of Zhong (2006), which is summarized in the following subsection. This method yields a stable rational approximation of a FIR filter and ensures that the H∞ -norm of the approximation error converges to zero when the approximation step N approaches infinity.
ye
r P
yF2 −? y¯ h+
uq
yq -
QΛ
Fig. 5. Representation with rational approximation of F2 r
y
Gp yq
3.1 Rational Approximation of FIR filters
uq - QΛ
Consider the following FIR filter described by the TFM Z(s) = (I − e−(sI−Az )hz )(sI − Az )−1 Bz .
(6)
Let us define as in Zhong (2006) −1 hz ZN ´ ³ hz e−Az N + I , ΦN := e−Az ζ dζ
3.2 Stable H∞ Flow Controller Design (7)
0
where N is the number of approximation steps. Then, let us define hz
hz
ΓN (s) := (e N (sI−Az ) − I)(e N (sI−Az ) + I)−1 ΦN .
(8)
ΓN has the following property, called the limiting property, limN →∞ ΓN (s) = sI − Az . Utilizing ΦN and ΓN , e−(sI−Az )hz can be written as e−(sI−Az )hz = (ΦN − ΓN (s))N (ΦN + ΓN (s))−N . Therefore, Z in (6) can be written as Z(s) = (I − (ΦN − ΓN (s))N (ΦN + ΓN (s))−N ) ×(sI − Az )−1 Bz .
(9)
Utilizing the limiting property of ΓN , ΓN (s) ≈ (sI − Az ), Z can be approximated by ZN given below ZN (s) = (I − (ΦN − sI + Az )N (sI − Az + ΦN )−N ) ×(sI − Az )−1 Bz =
N −1 X
ΠkN (s)ΞN (s)Bz ,
Fig. 6. Equivalent representation with rational approximation of F2 .
(10)
k=0
where ΠN (s) := (ΦN − sI + Az )(sI − Az + ΦN )−1 and ΞN (s) = 2(sI − Az + ΦN )−1 . The approximation guarantees that limN →∞ kEN k∞ = 0, where EN := Z − ZN (Zhong (2006)). The stability of ZN depends on N . Using the numerical calculations, the lower bound for N to satisfy the stability of ZN is given in Zhong (2006) by the following theorem. Theorem 1 (Zhong (2006)): ZN given in (10) is stable ˜ with for any N > N ¼ » h ˜ · max |λi (Az )| , N= i 2.8 where λi (·) denotes the ith eigenvalue of · and d·e is the ceiling function.
Utilizing the above described approximation method for ∞ FIR filters, a stable H can be designed as ¸ · flow controller eF eF B A 2 be a rational approximafollows. Let F2app = e 2 C F2 0 tion of F2 in Fig. 4 obtained by the approach summarized in Subsection 3.1. Then, replacing F2 by F2app , the system from y¯ to r¯ can be represented as shown in Fig. 5, where · ¸ A − B2 C2 B2 B1 Σ11 Σ12 C1 0 In , (11) Σ := := Σ21 Σ22 1 0 −C2
· ¸ A B1 B2 G11 G12 where C1 In 0 := := G−1 Λ . The system G21 G22 C2 0 1 in Fig. 5 can equivalently be represented as in Fig. 6, where Ap Bp1 Bp2 Gp := Fu (Σ, F2app ) := Cp1 Dp11 Dp12 Cp2 Dp21 Dp22 eF −B2 B1 A − B2 C2 B2 C 2 B eF eF C1 eF 0 B A 2 . 2 2 := (12) C1 0 0 In eF 1 −C2 −C 0 2 In (12), since Dp21 is invertible, Gp has a chain-scattering representation, G∆ = CHAIN (Gp ), and the system shown in Fig. 6 can be represented as in Fig. 7, where ·
G∆11 G∆12 G∆ =: G∆21 G∆22 :=
¸
b A b =: C1 b2 C
b1 B In 0
b2 B 0 1
Ap − Bp1 Cp2 Bp2 Bp1 Cp1 Dp11 0 , 0 Dp21 −Cp2
(13)
which can be shown to be bistable. The problem can now be defined as to find a contractive QΛ such that the closed-loop TFM from y¯ to r¯ in Fig. 7, which is S := HM (G∆ , QΛ ), is stable. From (13), S can be written
u q
r G∆ = CHAIN (Gp )
QΛ yq
-
y
6
Fig. 7. Chain-Scattering Representation as S = λ(G∆11 QΛ + G∆12 )(λG∆21 QΛ + λG∆22 )−1 , for any λ > 0, which does not affect the stability, however could reduce the conservativeness in the controller design (Lee and Soh (2005)). For any contractive QΛ , λ(G∆11 QΛ + G∆12 ) is stable. Furthermore, (λG∆21 QΛ + λG∆22 )−1 (and thus S) is stable if kλ(G∆21 QΛ + G∆22 ) − 1k∞ < 1. The problem of finding a contractive QΛ which satisfies kλ(G∆21 QΛ + G∆22 ) − 1k∞ < 1 can be defined as a two block problem as shown by Lee and Soh (2002): kHM (T, QΛ )k∞ < 1 , (14) where ·
¸ ·
λG∆21 In 0 ¸ · AT BT . =: CT D T
T =
¸
b A λG∆22 − 1 λC = b2 0 0 1 0
b1 B b2 B 0 λ−1 In 0 0 1 (15)
Finding a contractive QΛ which satisfies (14) will be solved ¯ J)-lossless ˆ via (J, factorization of T , where J¯ := Jn+1,1 ˆ and J := Jn,1 (Kimura (1996)). The necessary condition ¯ J)-lossless ˆ for the (J, factorization of T is the existence ¯ T = E T JE ˆ T = of a nonsingular E such that DTT JD T T ¸ · In 0 . This is satisfied for 0 < λ < 2. Moreover, 0 λ(λ − 2) for simplicity, the nonsingular ET can be selected as; ¸ · In p 0 . ET = λ(2 − λ) 0 Theorem 2 (Kimura (1996)): For a given realization of T in (15), the two block problem given in (14) can be solved if there exists a solution X ≥ 0 for 0 < λ < 2 satisfying ¯ T = 0 , (16) ¯ T )−1 RT + CTT JC XAT + ATT X − R(DTT JD ¯ T +XBT , such that AF := AT +BT FT where R := CTT JD T ¯ T )−1 RT . In that case, the is stable, where FT := −(DTT JD contractive QΛ can be written as b , QΛ = HM (Φ−1 , Γ) (17) T
b is any contractive parameter and where Γ ¸ · AT + BT FT BT ET−1 . Φ−1 = T FT ET−1 Therefore, a controller which solves the problem of Subsection 2.2 and which is stable apart from the integral action can be obtained by the following algorithm. Algorithm 1: 1. Find the optimal sensitivity level γ opt (see Subsection 2.3) and let γ = γ opt . 2. Find F1 , F2 , and GΛ (see Subsection 2.3) for the current sensitivity level γ.
e from Theorem 1, choose an upper bound 3. Compute N e , and let N = N e. Nmax > N 4. Let N = N + 1. 5. Find F2app for the current N (see Subsection 3.1). 6. If there exists a solution X ≥ 0 to the Riccati equation (16), go to step 8. Otherwise, if N < Nmax go to step 4 else continue with step 7 7. Increase γ by a small amount and go to step 2. 8. If (1 + F2 (s)H(s))−1 is unstable (can be checked by the Nyquist criterion), where H(s) = HM (G−1 Λ , QΛ ) and QΛ is obtained from (17), then go to step 4 if N < Nmax or go to step 7 otherwise. If (1 + F2 (s)H(s))−1 is stable, the desired controller is then given by (see Fig. 4) K(s) = F1 (s)H(s)
κ(s + ²) . s(1 + F2 (s)H(s))
(18)
4. SIMULATION RESULTS We performed a number of simulations in order to demonstrate the performance of the stable H∞ flow controller obtained by the proposed approach and to compare it to the performance of the unstable H∞ -optimal controller. Due to space limitations, we present only one case here for a network of two sources feeding one bottleneck node. The simulations are performed using MATLAB Simulink, where the non-linear effects (hard constraints) are also taken into account. The nominal time-delays are assumed to be h1 = 5 tu and h2 = 2 tu, where tu stands for time unit. Other design parameters are taken as δ1+ = 1.0, δ2+ = 0.5, β1 = 0.3, β2 = 0.1, β1f = β2f = 0.1, α1 = 32 , α2 = 13 , and σ1 = σ2 = 0.25. The desired queue length, qd , is taken as 30 packets and the buffer size (maximum queue length) is taken as 60 packets. Moreover, the capacity of the outgoing link is taken as 90 packets/tu and the rate limits for the sources are taken as 150 packets/tu. Timedomain simulations of both the H∞ -optimal controller and the proposed controller are given. For the optimal controller, we take QΛ = 0 and for the proposed controller, we b = 0. In Fig. 8–9, q (whose scale is on the right) is the take Γ queue length and rsi (whose scale is on the left) is the actual flow rate at source i, for i = 1, 2. The actual time-delays for the simulations are taken as τ1b (t) = 4.5 + 0.5 sin( 2π 50 t), f 2π 2π b τ1 (t) = 2.6 + 0.1 sin( 100 t), τ2 (t) = 1.7 + 0.1 sin( 50 t), and 2π t). τ2f (t) = 1.1 + 0.1 sin( 100 Applying the approach of Subsection 2.3, the optimal sensitivity level is obtained as γ opt = 3.9110. The response of the H∞ -optimal flow controller is shown in Fig. 8, where an unstable behaviour is observed. The stable H∞ flow controller, designed using Algorithm 1, uses approximation ˜ = 1 is obtained from Theorem 1) step N = 2 (N and the obtained sensitivity level is γ = 5.2370. The response of this controller is shown in Fig. 9. As shown in the figure, the proposed controller robustly stabilizes the actual system and also satisfies the tracking and weighted fairness requirements apart from low frequency oscillations at steady state. These oscillations are due to the timevarying forward time-delays, and can not be avoided unless they are known in advance (see Quet et al. (2002)).
90
150
75
q 120
60
90
45
rs
rs
1
2
60
30
30
15
0
0
50
100
150
200
Time in tu
250
300
350
Queue length in packets
Flow rates at sources in packets/tu
180
0 400
Fig. 8. Response of the optimal controller 40
q 90
30
rs1
60
20
30
10
Queue length in packets
Flow rates at sources in packets/tu
120
s
r2
0
0
50
100
150
200
Time in tu
250
300
350
0 400
Fig. 9. Response of the proposed controller 5. CONCLUSION Stable H∞ flow controller design for systems which involve multiple time-delays has been considered. As opposed to ¨ ˙ the approach presented in Unal and Iftar (2008a), which was based on the small-gain theorem, a finite-dimensional approximation of the infinite-dimensional part of the controller is utilized. Note that, however, this approximation is used only to find an appropriate QΛ which produces a stable stabilizing controller. The controller which is implemented is as shown in Fig. 4, which uses actual (infinitedimensional) F2 . Although a specific flow control problem is chosen for demonstration purposes, the proposed approach, summarized by Algorithm 1, can be applied to any control problem, for which the structure of the controller turns out to be as shown in Fig. 4. REFERENCES B. Ata¸slar. Robust flow control for data communication networks, July 2004. (In Turkish). ¨ S. G¨ um¨ u¸ssoy and H. Ozbay. Stable H∞ controller design for time-delay systems. International Journal of Control, 81:546–556, 2008.
H. Kimura. Chain-Scattering Approach to H∞ Control. Birkhauser, Boston, 1996. P. H. Lee and Y. C. Soh. Synthesis of stable H ∞ controller via the chain scattering framework. Systems and Control Letters, 47:121–127, 2002. P. H. Lee and Y. C. Soh. Synthesis of simultaneous stabilizing H ∞ controller. International Journal of Control, 78:1437–1446, 2005. G. Meinsma and L. Mirkin. H∞ control of systems with multiple I/O delays via decomposition to adobe problems. IEEE Transactions on Automatic Control, 50:199–211, 2005. G. Meinsma and H. Zwart. On H∞ control for dead-time systems. IEEE Transactions on Automatic Control, 45: 272–285, 2000. S. I. Niculescu. Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, No. 269. Springer-Verlag, London, 2001. ˙ ¨ P. Quet, B. Ata¸slar, A. Iftar, H. Ozbay, S. Kalyanaraman, and T. Kang. Rate-based flow controllers for communication networks in the presence of uncertain timevarying multiple time-delays. Automatica, 38:917–928, 2002. I. W. Sandberg. On the L2 -boundedness of solutions of nonlinear functional equations. Bell Syst. Tech. J., 43: 1581–1599, 1964. ¨ O. Toker and H. Ozbay. H∞ optimal and suboptimal controllers for infinite dimensional SISO plants. IEEE Transactions on Automatic Control, 40:751–755, 1995. ¨ ˙ H. U. Unal and A. Iftar. Stable H ∞ controller design for systems with multiple time-delays: The case of datacommunication networks. In Proceedings of the 17th World Congress of IFAC, Seoul, Korea., July 2008a. ¨ ˙ H. U. Unal and A. Iftar. Utilization of non-causal uncertainty blocks in the robust flow controller design problem for data-communication networks. In Proceedings of the 17th IEEE International Conference on Control Applications, San Antonio, TX, U.S.A., September 2008b. ¨ ˙ ¨ H. U. Unal, B. Ata¸slar-Ayyıldız, A. Iftar, and H. Ozbay. Robust controller design for multiple time-delay systems: The case of data-communication networks. In Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, July 2006. ¨ ˙ ¨ H. U. Unal, B. Ata¸slar-Ayyıldız, A. Iftar, and H. Ozbay. Robust flow control in data-communication networks with multiple time-delays. 2009. Submitted for publication. G. Zames. On the input-output stability of time-varying nonlinear feeedback systems Part I: Conditions derived using concepts of loop gain, conicity, and positivity. IEEE Transactions on Automatic Control, AC–11:228– 238, 1966. ¨ M. Zeren and H. Ozbay. On the synthesis of stable H ∞ controllers. IEEE Transactions on Automatic Control, 44:431–435, 1999. Q. C. Zhong. Robust Control of Time-Delay Systems. Springer-Verlag, London, 2006. K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, Englewood Cliffs, 1996.
by Quet et al. (2002). However, since the approach of ¨ Toker and Ozbay (1995) was limited to SISO systems, the proposed approach in Quet et al. (2002) was suboptimal in the H∞ sense. Later, an H∞ -optimal flow controller ¨ design approach was given by Unal et al. (2006) using the approach of Meinsma and Mirkin (2005). However, for ¨ technical reasons, in Unal et al. (2006) it was assumed that the uncertain part of the time-delays are always ¨ non-negative. This assumption was later removed in Unal ¨ ˙ et al. (2009) using the result of Unal and Iftar (2008b). ¨ Stable H∞ flow controller design was considered by Unal ˙ and Iftar (2008a), using the small-gain theorem. However, since the small-gain theorem provides only a sufficient condition, this approach may be conservative in general. In this study, we propose an alternative approach by considering a finite-dimensional approximation of the infinitedimensional part of the controller. 1.1 Notation For a matrix M , M T denotes its transpose. In denotes the n × n dimensional identity matrix. 0 denotes the zero matrix with appropriate·dimensions. 1n is the 1 × n ¸ A B denotes the transfer dimensional matrix of all 1’s. C D function matrix (TFM) C(sI − A)−1 B + D. k · k∞ and k · k2 respectively denote the H∞ norm and the L2 induced norm. A TFM is said to be stable if it is in H∞ , in addition, if its inverse exists in H∞ , it is said to be bistable. A constant square matrix is said to be stable if all its eigenvalues have negative real parts. A TFM Q ∈ H∞ is said to be contractive if kQk∞ < 1. For positive integers k and · ¸ l, Jk,l := blockdiag(Ik , −Il ). For a Σ11 Σ12 TFM Σ = , where Σ12 and Σ21 are respectively Σ21 Σ22 k × k and l × l dimensional and Σ21 is invertible, G = CHAIN (Σ) denotes the chain-scattering representation (Kimura (1996)) of Σ, where ¸ · ¸ · G11 G12 Σ12 − Σ11 Σ−1 Σ22 Σ11 Σ−1 21 21 . G =: = G21 G22 −Σ−1 Σ−1 21 Σ22 21
For k × l dimensional K, HM (G, K) denotes the homographic transformation (Kimura (1996)), which is defined as −1 Q := HM (G, K) = (G11 K + G12 ) (G21 K + G22 ) . 2. PROBLEM STATEMENT
1/s -
h
r =
r1h .. .
z1
-
rnh
2.1 Network Model
w1
-
+6
-
A data-communication network with n sources feeding a single bottleneck node is considered. The flow controller, which is to be designed, is implemented at the bottleneck node. The controller calculates a rate command for each source node to adjust the rate of data that will be sent to the bottleneck node in order to regulate the queue length at the bottleneck node. The dynamics of the queue length are given as (Quet et al. (2002)): n X
rib (t) − c(t)
(1)
i=1
where q(t) is the queue length at the bottleneck node at time t, rib (t) is the rate of data received by the bottleneck node at time t from the ith source, i = 1, . . . , n, and c(t) is the outgoing rate of data from the bottleneck node at time t, which equals to the capacity of the outgoing link assuming that q(t) is non-zero. The total amount of data received at the bottleneck node from the ith source node by time t can be written as (Quet et al. (2002)): t−τif (t) Z Zt ris (ϕ)dϕ, t − τif (t) ≥ 0 , rib (ϕ)dϕ = (2) 0 0 0, t − τif (t) < 0 where ris (t) = ri (t − τib (t)) is the rate of data sent from the ith source node at time t and ri (t) is the rate command for the ith source node issued by the controller at time t. By taking the derivative of both sides of (2), the rate of data received by the bottleneck node at time t, rib (t), is obtained as (Quet et al. (2002)): ½ (1 − δ˙if (t))ri (t − τi (t)), t − τif (t) ≥ 0 b ri (t) = , (3) 0, t − τif (t) < 0 where τi (t) = τib (t) + τif (t) is the round-trip time-delay, where τib (t) = hbi + δib (t) is the backward time-delay at time t, which is the time required for the rate command to reach the ith source. Here, hbi is the nominal timeinvariant known backward time-delay, and δib (t) is the time-varying backward time-delay uncertainty, τif (t) = hfi + δif (t) is the forward time-delay at time t, which is the time required for the data sent from the ith source to reach the bottleneck node. Here, hfi is the nominal time-invariant known forward timedelay, and δif (t) is the time-varying forward timedelay uncertainty.
r
q
W1 −
Λu
q(t) ˙ =
−? + - j
Po
∆
c
K
?+ j
qd
Fig. 1. Overall system (Ata¸slar (2004)) The nominal round-trip time-delay for the ith channel of the system is hi = hbi + hfi , and the time-varying roundtrip time-delay uncertainty is δi (t) = δib (t) + δif (t). It is assumed that the uncertainties are bounded as follows: f |δi (t)| < δ + , |δ˙i (t)| < βi , |δ˙i (t)| < β f (4) i
i
for some bounds δi+ > 0 and 0 < βif ≤ βi < 1. It is further assumed that, δi (t) is such that τi (t) ≥ 0 at all times. In a real application, there also exist some hard constraints, such as non-negativity constraints and upper bounds on the queue length and data rates. These non-linear constraints are ignored in the controller design. However, they are taken into account while doing simulations in Section 4. 2.2 Control Problem The flow controller, which is to be designed for the above described system, should robustly stabilize the system against all uncertain time-varying time-delays which satisfy (4). In addition to robust stability requirement, to avoid traffic congestion at the bottleneck node and share the network capacity fairly among the source nodes, the following performance requirements must also be considered (Quet et al. (2002)): tracking requirement: limt→∞ q(t) = qd , where qd > 0 is the desired queue length, weighted fairness requirement: limt→∞ ri (t) = αi c∞ , where αi > 0,Pi = 1, . . . , n, are the fairness weights, n which satisfy i=1 αi = 1, under the assumption that limt→∞ c(t) =: c∞ exists. Now, we can describe the overall system as shown in Fig. 1, where Po (s) := 1s 1n is the nominal plant except the time-delays, K is the controller to be designed, r := [r1 · · · rn ]¡T , ¢ Λu (s) := diag e−h1 s , . . . , e−hn s represents the nominal time-delays, which are taken outside the plant and ordered as h1 ≥ h2 ≥ . . . ≥ hn ≥ 0, in order to apply the approach of Meinsma and Mirkin (2005), ∆ is a possibly non-causal and linear time-varying block, which represents the uncertainties in the time-delays with k∆k ¤ £ 2 < 1, and W1 (s) := W 1 (s)· · · · W n (s) ¸, where, for i = 1, . . . , n, √ β +β f W i (s) := 2 √i i δi+ . s
1−βi
The structure of ∆ and the derivation of W1 can be found ¨ ˙ in Unal and Iftar (2008b).
z1 -
-
w1
P0
z
d
∆ ? W1
? W2
+ ? + i -
+ − ? i
Pb
yb - W3
w
u Λu
- e1
- K b
6
¨ Fig. 3. Equivalent four-block problem (Unal et al. (2009)) e2
W4
Λu u
K r
y
Fig. 2. System for the mixed sensitivity minimization problem (Ata¸slar (2004)) 2.3 H∞ -optimal Controller Design To solve the control problem stated above, i.e., to design a controller which robustly stabilizes the actual system and achieves all performance requirements, a mixed sensitivity minimization problem was set up in Ata¸slar (2004), as shown in Fig. 2. Here, W2 (s) = 1s , W3 (s) = σs1 , and α2 −1 0 0 1 α α3 0 −1 0 σ2 α1 , W4 (s) = . . .. s .. .. . . α. . n 0 0 −1 α1 where σ1 > 0 and σ2 > 0 are design parameters. Furthermore, d := q˙d − c, y := qd − q, e1 is introduced to achieve tracking requirement, and e2 is introduced to achieve the weighted fairness requirement. The flow control problem can now be posed as to determine a controller K which internally stabilizes the system in Fig. 2 with ∆ = 0 and minimizes the H∞ norm of the closed-loop TFM from ¤T ¤T £ £ to z := z1T eT1 eT2 . w := w1T dT To find an H∞ -optimal flow controller that achieves above objectives, an H∞ coprime factorization, Po (s) = f−1 (s)N e (s) is necessary, where N e and M f are chosen as M 1 s f e N (s) = s+² 1n and M (s) = s+² for an arbitrary ² > 0 (see ¨ Unal et al. (2009)). Utilizing this factorization, the system in Fig. 2 can also be considered as a 4-block problem shown f−1 y and in Fig. 3, where yˆ := M b f(s) = s K(s) . (5) K(s) := K(s)M s+² ¨ As shown in Unal et al. (2009), Pb satisfies the standard H∞ problem assumptions (Zhou et al. (1996)), hence the b latter problem can be defined as designing a controller K which minimizes the H∞ norm of the closed-loop TFM b for a given from w to z in Fig. 3, which is Fl (Pb, Λu K), γ > 0. For a satisfactorily large given sensitivity level, γ, a conb which satisfies kFl (Pb, Λu K)k b ∞ < γ, can be troller K, obtained by applying the approach proposed by Meinsma ¨ and Mirkin (2005), as shown in Unal et al. (2009). The corresponding controller, K, can then be found from (5) as b K(s) = s+² s K(s). The structure of the controller is shown
F2
r 6 F1
r
HM (G−1 Λ , QΛ )
qd − q − ?+ h y
? κ
s+ s
¨ ˙ Fig. 4. Structure of the controller K (Unal and Iftar (2008a)) in Fig. 4, where κ := q Pγ n
is a constant, F1 and 2 (δi+ ) F2 consist of time-delays and finite-impulse response (FIR) filters, GΛ is a bistable finite dimensional TFM, and QΛ is ¨ an arbitrary contractive TFM (see Unal et al. (2009) for details). 2
2
i=1
The optimal controller, Kopt , can be found by an itera¨ tive procedure on γ as described in Unal et al. (2009). The minimum γ, for which there exists a solution to b ∞ < γ, is denoted by γ opt . Then, Kopt (s) = kFl (Pb, Λu K)k s+² b b b b s Kopt (s), where Kopt satisfies kFl (P , Λu Kopt )k∞ < opt γ . Any designed controller K, including Kopt , obtained as above is unstable due to the integral term (see Fig. 4) which is necessary to satisfy the tracking requirement. However, apart from the integral part of the controller, i.e., b only the part from y¯ to r in Fig. 4 (which differs from K by a constant), may or may not be stable. When this part b is unstable, due to nonlinearities in the (equivalently K) system (i.e., the hard constraints), an unstable behaviour may be observed, at least for certain actual time-delays and/or initial conditions (see Section 4). In order to avoid such undesirable behaviour, a controller which is stable apart from the integral part may be required. 2.4 Stable Controller Design Consider the structure of the controller showm in Fig. 4. ¨ Here F1 and F2 are both stable (see Unal et al. (2009)). Hence, the TFM from y¯ to r is stable if and only if the TFM from y¯ to r¯ is stable. Utilizing the small-gain theorem (Sandberg (1964), Zames (1966)) and the fact that F2 is stable, the TFM from y¯ to r¯ is stable if a contractive QΛ 1 . An can be found such that kHM (G−1 Λ , QΛ )k∞ < kF2 k∞ ¨ ˙ algorithm to find such a QΛ is given in Unal and Iftar (2008a). This algorithm increases the sensitivity level γ, starting from γ opt , until such a solution can be found. However, note that, since the small-gain is not a necessary condition, the approach may unnecessarily increase γ, resulting in a conservative solution.
3. PROPOSED DESIGN APPROACH
F2app
In this section, we propose an alternative approach to the stable controller design problem stated in Subsection 2.4. This approach is based on the approximation of the F2 block, which consists of FIR filters and timedelays, by a finite-dimensional TFM. For this purpose we use the approximation method of Zhong (2006), which is summarized in the following subsection. This method yields a stable rational approximation of a FIR filter and ensures that the H∞ -norm of the approximation error converges to zero when the approximation step N approaches infinity.
ye
r P
yF2 −? y¯ h+
uq
yq -
QΛ
Fig. 5. Representation with rational approximation of F2 r
y
Gp yq
3.1 Rational Approximation of FIR filters
uq - QΛ
Consider the following FIR filter described by the TFM Z(s) = (I − e−(sI−Az )hz )(sI − Az )−1 Bz .
(6)
Let us define as in Zhong (2006) −1 hz ZN ´ ³ hz e−Az N + I , ΦN := e−Az ζ dζ
3.2 Stable H∞ Flow Controller Design (7)
0
where N is the number of approximation steps. Then, let us define hz
hz
ΓN (s) := (e N (sI−Az ) − I)(e N (sI−Az ) + I)−1 ΦN .
(8)
ΓN has the following property, called the limiting property, limN →∞ ΓN (s) = sI − Az . Utilizing ΦN and ΓN , e−(sI−Az )hz can be written as e−(sI−Az )hz = (ΦN − ΓN (s))N (ΦN + ΓN (s))−N . Therefore, Z in (6) can be written as Z(s) = (I − (ΦN − ΓN (s))N (ΦN + ΓN (s))−N ) ×(sI − Az )−1 Bz .
(9)
Utilizing the limiting property of ΓN , ΓN (s) ≈ (sI − Az ), Z can be approximated by ZN given below ZN (s) = (I − (ΦN − sI + Az )N (sI − Az + ΦN )−N ) ×(sI − Az )−1 Bz =
N −1 X
ΠkN (s)ΞN (s)Bz ,
Fig. 6. Equivalent representation with rational approximation of F2 .
(10)
k=0
where ΠN (s) := (ΦN − sI + Az )(sI − Az + ΦN )−1 and ΞN (s) = 2(sI − Az + ΦN )−1 . The approximation guarantees that limN →∞ kEN k∞ = 0, where EN := Z − ZN (Zhong (2006)). The stability of ZN depends on N . Using the numerical calculations, the lower bound for N to satisfy the stability of ZN is given in Zhong (2006) by the following theorem. Theorem 1 (Zhong (2006)): ZN given in (10) is stable ˜ with for any N > N ¼ » h ˜ · max |λi (Az )| , N= i 2.8 where λi (·) denotes the ith eigenvalue of · and d·e is the ceiling function.
Utilizing the above described approximation method for ∞ FIR filters, a stable H can be designed as ¸ · flow controller eF eF B A 2 be a rational approximafollows. Let F2app = e 2 C F2 0 tion of F2 in Fig. 4 obtained by the approach summarized in Subsection 3.1. Then, replacing F2 by F2app , the system from y¯ to r¯ can be represented as shown in Fig. 5, where · ¸ A − B2 C2 B2 B1 Σ11 Σ12 C1 0 In , (11) Σ := := Σ21 Σ22 1 0 −C2
· ¸ A B1 B2 G11 G12 where C1 In 0 := := G−1 Λ . The system G21 G22 C2 0 1 in Fig. 5 can equivalently be represented as in Fig. 6, where Ap Bp1 Bp2 Gp := Fu (Σ, F2app ) := Cp1 Dp11 Dp12 Cp2 Dp21 Dp22 eF −B2 B1 A − B2 C2 B2 C 2 B eF eF C1 eF 0 B A 2 . 2 2 := (12) C1 0 0 In eF 1 −C2 −C 0 2 In (12), since Dp21 is invertible, Gp has a chain-scattering representation, G∆ = CHAIN (Gp ), and the system shown in Fig. 6 can be represented as in Fig. 7, where ·
G∆11 G∆12 G∆ =: G∆21 G∆22 :=
¸
b A b =: C1 b2 C
b1 B In 0
b2 B 0 1
Ap − Bp1 Cp2 Bp2 Bp1 Cp1 Dp11 0 , 0 Dp21 −Cp2
(13)
which can be shown to be bistable. The problem can now be defined as to find a contractive QΛ such that the closed-loop TFM from y¯ to r¯ in Fig. 7, which is S := HM (G∆ , QΛ ), is stable. From (13), S can be written
u q
r G∆ = CHAIN (Gp )
QΛ yq
-
y
6
Fig. 7. Chain-Scattering Representation as S = λ(G∆11 QΛ + G∆12 )(λG∆21 QΛ + λG∆22 )−1 , for any λ > 0, which does not affect the stability, however could reduce the conservativeness in the controller design (Lee and Soh (2005)). For any contractive QΛ , λ(G∆11 QΛ + G∆12 ) is stable. Furthermore, (λG∆21 QΛ + λG∆22 )−1 (and thus S) is stable if kλ(G∆21 QΛ + G∆22 ) − 1k∞ < 1. The problem of finding a contractive QΛ which satisfies kλ(G∆21 QΛ + G∆22 ) − 1k∞ < 1 can be defined as a two block problem as shown by Lee and Soh (2002): kHM (T, QΛ )k∞ < 1 , (14) where ·
¸ ·
λG∆21 In 0 ¸ · AT BT . =: CT D T
T =
¸
b A λG∆22 − 1 λC = b2 0 0 1 0
b1 B b2 B 0 λ−1 In 0 0 1 (15)
Finding a contractive QΛ which satisfies (14) will be solved ¯ J)-lossless ˆ via (J, factorization of T , where J¯ := Jn+1,1 ˆ and J := Jn,1 (Kimura (1996)). The necessary condition ¯ J)-lossless ˆ for the (J, factorization of T is the existence ¯ T = E T JE ˆ T = of a nonsingular E such that DTT JD T T ¸ · In 0 . This is satisfied for 0 < λ < 2. Moreover, 0 λ(λ − 2) for simplicity, the nonsingular ET can be selected as; ¸ · In p 0 . ET = λ(2 − λ) 0 Theorem 2 (Kimura (1996)): For a given realization of T in (15), the two block problem given in (14) can be solved if there exists a solution X ≥ 0 for 0 < λ < 2 satisfying ¯ T = 0 , (16) ¯ T )−1 RT + CTT JC XAT + ATT X − R(DTT JD ¯ T +XBT , such that AF := AT +BT FT where R := CTT JD T ¯ T )−1 RT . In that case, the is stable, where FT := −(DTT JD contractive QΛ can be written as b , QΛ = HM (Φ−1 , Γ) (17) T
b is any contractive parameter and where Γ ¸ · AT + BT FT BT ET−1 . Φ−1 = T FT ET−1 Therefore, a controller which solves the problem of Subsection 2.2 and which is stable apart from the integral action can be obtained by the following algorithm. Algorithm 1: 1. Find the optimal sensitivity level γ opt (see Subsection 2.3) and let γ = γ opt . 2. Find F1 , F2 , and GΛ (see Subsection 2.3) for the current sensitivity level γ.
e from Theorem 1, choose an upper bound 3. Compute N e , and let N = N e. Nmax > N 4. Let N = N + 1. 5. Find F2app for the current N (see Subsection 3.1). 6. If there exists a solution X ≥ 0 to the Riccati equation (16), go to step 8. Otherwise, if N < Nmax go to step 4 else continue with step 7 7. Increase γ by a small amount and go to step 2. 8. If (1 + F2 (s)H(s))−1 is unstable (can be checked by the Nyquist criterion), where H(s) = HM (G−1 Λ , QΛ ) and QΛ is obtained from (17), then go to step 4 if N < Nmax or go to step 7 otherwise. If (1 + F2 (s)H(s))−1 is stable, the desired controller is then given by (see Fig. 4) K(s) = F1 (s)H(s)
κ(s + ²) . s(1 + F2 (s)H(s))
(18)
4. SIMULATION RESULTS We performed a number of simulations in order to demonstrate the performance of the stable H∞ flow controller obtained by the proposed approach and to compare it to the performance of the unstable H∞ -optimal controller. Due to space limitations, we present only one case here for a network of two sources feeding one bottleneck node. The simulations are performed using MATLAB Simulink, where the non-linear effects (hard constraints) are also taken into account. The nominal time-delays are assumed to be h1 = 5 tu and h2 = 2 tu, where tu stands for time unit. Other design parameters are taken as δ1+ = 1.0, δ2+ = 0.5, β1 = 0.3, β2 = 0.1, β1f = β2f = 0.1, α1 = 32 , α2 = 13 , and σ1 = σ2 = 0.25. The desired queue length, qd , is taken as 30 packets and the buffer size (maximum queue length) is taken as 60 packets. Moreover, the capacity of the outgoing link is taken as 90 packets/tu and the rate limits for the sources are taken as 150 packets/tu. Timedomain simulations of both the H∞ -optimal controller and the proposed controller are given. For the optimal controller, we take QΛ = 0 and for the proposed controller, we b = 0. In Fig. 8–9, q (whose scale is on the right) is the take Γ queue length and rsi (whose scale is on the left) is the actual flow rate at source i, for i = 1, 2. The actual time-delays for the simulations are taken as τ1b (t) = 4.5 + 0.5 sin( 2π 50 t), f 2π 2π b τ1 (t) = 2.6 + 0.1 sin( 100 t), τ2 (t) = 1.7 + 0.1 sin( 50 t), and 2π t). τ2f (t) = 1.1 + 0.1 sin( 100 Applying the approach of Subsection 2.3, the optimal sensitivity level is obtained as γ opt = 3.9110. The response of the H∞ -optimal flow controller is shown in Fig. 8, where an unstable behaviour is observed. The stable H∞ flow controller, designed using Algorithm 1, uses approximation ˜ = 1 is obtained from Theorem 1) step N = 2 (N and the obtained sensitivity level is γ = 5.2370. The response of this controller is shown in Fig. 9. As shown in the figure, the proposed controller robustly stabilizes the actual system and also satisfies the tracking and weighted fairness requirements apart from low frequency oscillations at steady state. These oscillations are due to the timevarying forward time-delays, and can not be avoided unless they are known in advance (see Quet et al. (2002)).
90
150
75
q 120
60
90
45
rs
rs
1
2
60
30
30
15
0
0
50
100
150
200
Time in tu
250
300
350
Queue length in packets
Flow rates at sources in packets/tu
180
0 400
Fig. 8. Response of the optimal controller 40
q 90
30
rs1
60
20
30
10
Queue length in packets
Flow rates at sources in packets/tu
120
s
r2
0
0
50
100
150
200
Time in tu
250
300
350
0 400
Fig. 9. Response of the proposed controller 5. CONCLUSION Stable H∞ flow controller design for systems which involve multiple time-delays has been considered. As opposed to ¨ ˙ the approach presented in Unal and Iftar (2008a), which was based on the small-gain theorem, a finite-dimensional approximation of the infinite-dimensional part of the controller is utilized. Note that, however, this approximation is used only to find an appropriate QΛ which produces a stable stabilizing controller. The controller which is implemented is as shown in Fig. 4, which uses actual (infinitedimensional) F2 . Although a specific flow control problem is chosen for demonstration purposes, the proposed approach, summarized by Algorithm 1, can be applied to any control problem, for which the structure of the controller turns out to be as shown in Fig. 4. REFERENCES B. Ata¸slar. Robust flow control for data communication networks, July 2004. (In Turkish). ¨ S. G¨ um¨ u¸ssoy and H. Ozbay. Stable H∞ controller design for time-delay systems. International Journal of Control, 81:546–556, 2008.
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