Robust Control of Infinite Dimensional Systems: Theory and - - PDF document

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Robust Control of Infinite Dimensional Systems: Theory and Applications

Hitay ¨ Ozbay

Abstract— The purpose of this paper is to complement the tutorial review given at the MTNS2006 on the skew Toeplitz approach to H∞ control of a class of infinite dimensional

• systems. Numerical computations of optimal and suboptimal

controllers are demonstrated in Matlab on different examples. Applications to infinite dimensional flexible beam models and systems with time delays are considered. We also focus on robust stability and performance problems for data flow control in computer communication networks, and give an application example from aerodynamic flow control: suppres- sion of cavity flow oscillations. Keywords— Robust Control, H∞-Control, Infinite Dimen- sional Systems, Time Delays, Congestion Control in Com- puter Networks, Cavity Flow Oscillations

• I. INTRODUCTION

This paper complements the tutorial review given by the author at the MTNS2006 on the skew Toeplitz approach to H∞ control of a class of infinite dimensional systems, [45]. It is not intended to be a survey of the vast subject

• f robust control of infinite dimensional systems. We will

just mention some related results in passing. For further reading see the papers and books listed in the references section, and their references. It is now well-known that robust controllers, under the presence of unstructured L∞-norm bounded perturbations in the plant transfer matrix, can be obtained from an H∞

• ptimization, [157]. Many different approaches have been

developed for H∞ control of finite dimensional systems, and computational tools are now widely available, [32], [34], [47], [53], [55], [81], [91], [118], [130], [147], [161], [162]. Robust control under ℓ1 optimality, and other types

• f uncertainty, are also studied widely, see [9], [12], [28],

[140] and their references. For time delay systems (an important class of infinite dimensional systems), H∞ con- trollers started to appear in the literature in the mid 1980s, [41], [46], [163]. Over the last 20 years there has been significant progress in the extension of these first results to larger classes of infinite dimensional systems, see e.g. [5], [24], [25], [31], [43], [52], [56], [57], [69], [73], [76], [83], [90], [103], [107], [108], [112], [113], [122], [123], [128], [136], [143]. One of the methods used in the computation

• f H∞ controllers for infinite dimensional systems is

This work was supported in part by the European Commission (contract

• no. MIRG-CT-2004-006666) and by T ¨

UB˙ ITAK (grant no. EEEAG- 105E065). H. ¨ Ozbay is with Dept.

• f

Electrical and Electronics Eng., Bilkent University, Bilkent, Ankara TR-06800, Turkey, on leave from

• Dept. of Electrical and Computer Eng., The Ohio State Univer-

sity, Columbus, OH 43210, U.S.A., hitay@bilkent.edu.tr,

• zbay@ece.osu.edu

the “skew Toeplitz” approach, [45]. The importance of skew Toeplitz operators in H∞ control has been noticed by Foias, Tannenbaum and their collaborators, and this term first appears in [11]. In this approach H∞ optimal and suboptimal controllers are directly computed without approximating the plant. For infinite dimensional systems robust controllers can also be obtained by approximating the plant and then using standard techniques developed for the control of finite dimensional systems, by keeping track of the original approximation error, see e.g. [6], [23], [74], [84], [85], [100], [101], [114], [126] and their references. See [22] for an early review of H∞ control of distributed parameter systems in general, [146] for state-space approach to H∞ control of such systems, and [96], [149] for reviews

• f state-space and operator theoretic approaches to H∞

control of time delay systems. For time delay systems, and some other classes of distributed parameter systems, H∞ controllers can also be derived from a game theoretic approach, [10], [132]. For the most recent results on H∞ control of systems with input-output delays, see [94], [160] and their references. Repetitive control design, [66], under certain performance and robustness conditions, can be posed as a robust control problem for systems with time delays, [61], [117], [134], [150]. Sampled-data controller design, with certain types of optimality conditions result in an H∞ control problem for time delay systems, see [8], [20], [65] for further references. Robust stability of time delay systems (within, and outside, the framework of H∞ control) is widely studied, [19], [33], [49], [50], [58], [62], [68], [72], [78], [80], [82], [86], [105], [106], [98], [115], [116], [138], [139], [145]. Several issues related to robust control of fractional delay systems has been considered in [14]. Stability robustness against small time delays have been considered for various types of plants, see e.g. [17], [87], [93], [99] and their references. Flexible structure models which include internal time delays, are considered in [63], [64]. For spatially invariant distributed parameter systems, [7], H∞ optimal controllers are obtained from a parameterized family of finite dimensional problems; see also [27]. Robust control of infinite dimensional systems is also covered in the book [26]. In Section 2 we review key results from operator theory and show how they are used in H∞ control. In Section 3 numerical computations of optimal and suboptimal con- trollers are demonstrated based on the formulae derived in [59], [141]. Plants considered in Section 3 are systems with time delays, [39], [60], and an infinite dimensional flexible

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beam model, [84], [85]. Applications are considered in Section 4: H∞ control design for data flow control in computer communication networks, [119], [120], and an example from aerodynamic flow control (suppression of cavity flow oscillations), [156]. Finally, we make some closing remarks in Section 5.

• II. H∞ CONTROL PROBLEMS

In this section we review some important results from

• perator theory and give a summary of how they are used

in finding optimal and suboptimal H∞ controllers. For simplicity of the presentation, all systems considered are single input single output unless otherwise stated. We will deal with the standard feedback system F(C, P) shown in Figure 1, where C is the controller to be designed and P is the plant to be controlled. P

c

u

C

v − + + + y u e r

• Fig. 1.

Feedback System F(C, P )

We say that a linear time invariant system whose input

• utput behavior is characterized by the transfer function

H(s) is said to be stable if H ∈ H∞ (see [164] for a technical discussion on the transfer functions of infinite dimensional systems, here we assume that the transfer function is the “quotient of the Laplace transform of the output and input, with initial condition zero”). The feedback system F(C, P) is stable if and only if S = (1 + PC)−1, PS and CS are in H∞. Here S is the sensitivity function. The set of all controllers stabilizing this feedback system for a given plant P is denoted by C(P). Most important features of the H∞ control problems are captured by the weighted sensitivity minimization problem, which is to find γo = inf

C∈C(P ) W(1 + PC)−1∞

(1) and the corresponding optimal controller Co ∈ C(P), for a given plant P and a weight W; typically W, W −1 ∈ H∞. There are several approaches to this problem depending on the structure of the problem data. First, let us assume that W is infinite dimensional (an irrational transfer function) and P is finite dimensional (a rational transfer function). We can solve this problem using Nevanlinna-Pick interpolation, as follows. For simplicity of the exposition assume that z1, . . . , znz are the zeros and p1, . . . , pnp are the poles of P in C+, and P has no poles

• r zeros on the imaginary axis. In this case, C ∈ C(P)

is equivalent to having the sensitivity function S = (1 + PC)−1 in H∞, with S(zi) = 1 and S(pj) = 0. Then, γo is the smallest γ > 0 such that there exists a function F ∈ H∞ such that F∞ ≤ 1 and F(zi) = γ−1W(zi), F(pj) = 0 for i = 1, . . . , nz, j = 1, . . . , np. This is the Nevanlinna- Pick interpolation problem, and can be solved from the problem data W and P, see [44], [45], [79], [159]. Once γo and the corresponding optimal F is computed, the resulting controller is C = (γ−1W − F)/PF. Note that in this case the problem solution depends on W(zi), and W(s) can be irrational. In summary, when the plant is finite dimensional and the weight is infinite dimensional, the weighted sensitivity minimization problem can be solved using Nevanlinna-Pick interpolation. Clearly, we cannot use this approach when the plant P is infinite dimensional (with infinitely many C+ zeros or poles). For such plants we will assume that W is finite dimensional and use the characterization of C(P) given

• below. First, assume that P can be written as P(s) =

N(s)/D(s) with N, D ∈ H∞ such that there exist X, Y ∈ H∞ satisfying X(s)N(s) + Y (s)D(s) = 1. Then the set of all controllers stabilizing F(C, P) is (see e.g. [3], [131], [155] and their references) C(P) = X + DQ Y − NQ : Q ∈ H∞, Y − NQ = 0

• .

Let us now assume that the plant has finitely many unstable modes, i.e. D(s) can be taken as a rational function. In this case X(s) should be chosen in such a way that Y (s) = 1 − X(s)N(s) D(s) ∈ H∞. Clearly, using Lagrange interpolation one can find a ratio- nal X(s) satisfying the above requirement. Note also that stabilizing controllers are in the form C = (X + DQ) 1 − N(X + DQ) (2) and they can be implemented as shown in Figure 2.

X+DQ Controller

c

u e + + N

• Fig. 2.

Stabilizing Controller

In particular, when the plant is stable we can choose N = P, D = 1 and X = 0. This leads to γo = inf

Q∈H∞ W(1 − PQ)∞

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The next crucial step is to perform inner outer factorization

• f the plant:

P = MnNo where Mn is inner and No is outer. Defining Q1 = WNoQ (we assume that WNo is invertible in H∞; otherwise, see the discussion in [40], [48] on the absorption of the outer factor), we have γo = inf

Q1∈H∞ W − MnQ1∞.

(3) The problem (3) can be put into the framework of the Nehari problem, [104], and that gives γo = Γ where Γ is the Hankel operator whose symbol is M ∗

nW ∈

L∞. We will assume that the norm is achieved on the discrete spectrum, i.e., Γ > Γe, the essential norm (see [45], [158] for the computation of the essential norm). In this case γo and the corresponding optimal Q1 ∈ H∞, and hence the optimal controller C ∈ C(P), can be

• btained by computing the largest singular value, and

the corresponding singular vector of Γ, see also [2] for the characterization of all suboptimal solutions of this

• problem. For more details see [45].

The problem (3) can also be solved using Sarason’s Theorem, [127], or the Commutant Lifting Theorem, [44], [135], as follows. Let us map the right half plane, C+, to the unit disc, D, via the conformal map z = ϕ(s) = s−1

s+1,

s = ϕ−1(z) = 1+z

1−z. Define functions w(z) = W(ϕ−1(z))

and mn(z) = Mn(ϕ−1(z)). For the inner function m := mn define H(m) = H2(D) ⊖ mH2(D). When m is rational (finite Blaschke product), the space H(m) is finite dimensional, otherwise it is infinite dimensional. Let S be the unit shift operator on ℓ2, i.e. it can be seen as the multiplication by z on H2(D). Then the compressed shift

• perator T is defined as T = ΠH(m)S|H(m), where ΠH(m)

denotes the orthogonal projection onto H(m). Now the solution of (3) is γo = w(T). Since w(z) is rational, it can be written as w(z) = b(z)/a(z). Under the assumption γo > w(T)e, (the norm is strictly greater than the essential norm), γo is the largest γ for which there exists a non-zero f ∈ H(m) such that 0 =

• b(T)∗b(T) − γ2a(T)∗a(T)
• f =: Aγf .

(4) The operator Aγ is called a skew Toeplitz operator, and γo is the largest γ which makes Aγ singular; the optimal Q1 ∈ H∞, and hence the optimal controller C ∈ C(P), are determined from the corresponding f ∈ H(m) as w − mnqopt

1

= b(T)f a(T)f . (5) A closer examination of (4) leads to a finite set of linear equations for the existence of a non-zero f ∈ H(m), even when H(m) is infinite dimensional. This set of linear equations determine γo and the corresponding f. Then the

• ptimal controller is obtained from (5) using this f. For

complete details and further references see [45]. The weighted sensitivity minimization is known as a

• ne-block H∞ control problem. An extension of this

problem, which also takes into account robust stability

• f the feedback system, [32], is the mixed sensitivity
• ptimization: find

γo = inf

C∈C(P )

• W1(1 + PC)−1

W2PC(1 + PC)−1

• ∞

(6) and the corresponding optimal controller. Again, using the parameterization (2), we transform the problem (6) into a problem of finding γo = inf

Q∈H∞

• W1(1 − N(X + DQ))

W2N(X + DQ)

• ∞

and the corresponding optimal Q ∈ H∞. After a series

• f inner-outer factorizations, [47], the above problem is

further reduced to finding smallest γ such that there exists Q1 ∈ H∞ satisfying

• W − MQ1

G

• ∞

≤ γ (7) where W, G, M, Q1 are determined from the problem data W1, W2, N, D. In particular Q1 is determined from Q by an invertible relation, M is inner infinite dimensional, and G is finite dimensional. If the plant is stable N = P and D = 1, then W is finite dimensional as well. Otherwise, it has a special structure W = Wo + M1 Wo where Wo, Wo are finite dimensional and M1 is the infinite dimensional part of M, i.e., M = M1M2 with M2 being finite dimensional (assuming that the plant has finitely many unstable modes). Next step is to do a spectral factorization: F ∗

γ Fγ = γ2 − G∗G.

That transforms (7) into a problem of finding smallest γ such that WF −1

γ

− MQ2∞ ≤ 1 (8) where Q2 = Q1F −1

γ . Clearly, now the problem (8) is in the

form (3), and the approach outlined earlier is applicable. The problem (7) is a two-block H∞ control problem, and as we have seen above, it can be reduced to a one-block problem by a spectral factorization. One step extension of the two-block problem is the four-block problem. That also can be reduced to a one- block problem by a series of spectral factorizations, see e.g. [47]. The key observation we make in this context is that when the weights are finite dimensional, and the inner-

• uter factorizations of the plant have already been made

(see below for examples), we only need to do spectral factorizations for finite dimensional systems, see [77], [45] for further details. In general, there are several technical difficulties in performing spectral factorizations for infinite dimensional systems, [151]. The problem of robust stabilization in the gap metric can be posed as a special case of the mixed sensitivity

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minimization, (6), with special weights W1 and W2. This problem has been studied for various classes of infinite dimensional systems, see e.g. [35], [52], [142] and their

• references. The uncertainty model in this case is coprime

factor perturbations. Briefly, if the uncertain plant is given as P∆ = N∆ D∆ = N + ∆N D + ∆D N, D, ∆N, ∆D ∈ H∞ with N ∗N + D∗D = 1

• ∆N

∆D

• ∞ < δ

then a controller C ∈ C(P), where P = N/D, is also in C(P∆) if and only if it satisfies

• D−1(1 + PC)−1
• 1

C

• ∞

≤ 1 δ . (9) For certain classes of infinite dimensional systems this type

• f uncertainty modeling is very helpful in finding finite

dimensional controllers. See for example [114] for robust controller design for a thin airfoil where the D∆ term contains an irrational transfer function (the Theodorsen’s function) which is approximated by a second order rational function, [110]. It turns out that, see e.g. [51], [54], [148], minimizing the left hand side of (9) over all C ∈ C(P) (for the largest allowable δ), is equivalent to finding the norm of a Hankel

• perator whose symbol is [D∗ N ∗]. Clearly, (9) is a special

case of the two block problem, (6), (with W1 = D−1

• , the

inverse of the outer part of D, and W2 = N −1

• ), and it too

reduces to a one block problem. In summary, several different H∞ control problems can be reduced to the generic form (3), which is solved via (5). Then the optimal controller is computed using the optimal Q ∈ H∞ in (2).

• III. OPTIMAL SOLUTION OF THE MIXED

SENSITIVITY MINIMIZATION PROBLEM In this section we present a closed form solution for the mixed sensitivity problem (6), taken from [141].

• A. Examples of Plants Considered

The plants considered in [141] have the coprime factor- ization in the form P(s) = Mn(s)No(s) Md(s) (10) where Mn is inner, No is outer and Md is finite dimen- sional and inner. The formula given here is also valid for certain different classes of infinite dimensional plants (including plants with infinitely many C+ poles) with some modifications, see [59], [60]. Examples of infinite dimensional plants in the form (10) are as follows.

• 1. A stable first order system with transport delay:

P1(s) = e−hs τps + 1, h > 0, τp > 0. Md(s) = 1, Mn(s) = e−hs, No(s) = 1 τps + 1.

• 2. An unstable system with transport delay:

P2(s) = e−hs 1 s − a, h > 0, a > 0. Md(s) = s − a s + a, Mn(s) = e−hs, No(s) = 1 s + a.

• 3. An unstable system with internal time delays:

P3(s) = s + 3 + 2(s − 1)e−0.4s s2 + se−0.2s + 5e−0.5s , which can be re-written as P3(s) = PN(s) PD(s), where PN(s) = s + 3 + 2(s − 1)e−0.4s (s + 1)2 , PD(s) = s2 + se−0.2s + 5e−0.5s (s + 1)2 . It can be shown that PD(s) has only two zeros in C+, these are the unstable poles of the plant, p1 and p2. On the other hand, PN(s) has infinitely many zeros in C+. Inner-outer factorization of this plant can be done, [60], by finding p1, p2, and the single C+ zero, z1, of PN(s) = 2(s + 1) + (s − 3)e−0.4s (s + 1)2 . At this point we should mention that there are several tools for finding the zeros of a quasi-polynomial, see e.g. [36], [37], [38], [124] for the Matlab-based program DDE- BIFTOOL, and also [29], [88], [109] for other techniques and further references on this topic. Now, the zeros of T (s) in C+ are computed as p1,2 ≈ 0.467±j1.889, and the zero

• f PN(s) in C+ is z1 ≈ 0.247. We define

Md(s) = (s − p1)(s − p2) (s + p1)(s + p2), M1(s) = s − z1 s + z1 . Then, the plant P3 can be written in the form (10) with Md as above and Mn(s) = M1(s) s + 3 + 2(s − 1)e−0.4s 2(s + 1) + (s − 3)e−0.4s , No(s) = PN(s) M1(s) Md(s) PD(s) .

• 4. A flexible beam: Consider an Euler-Bernoulli beam

having free ends with Kelvin-Voigt damping. Assume that the beam length and all other parameters are normalized to unity, except the damping constant ε > 0, [85]. Denote the deflection of the beam at time t and location x along the elastic axis of the beam by w(x, t). Suppose that a transverse force, −u(t), is applied at one end of the beam, e.g. at x = 1. See Figure 3.

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x=1 force input u(t) w(0,t) x=0

• Fig. 3.

Beam with free ends.

The beam dynamics are as follows, [16], [84], ∂2w ∂t2 + ε ∂5w ∂x4∂t + ∂4w ∂x4 = 0 (11) with boundary conditions ∂2w ∂x2 (0, t) + ε ∂3w ∂x2∂t(0, t) = 0, ∂2w ∂x2 (1, t) + ε ∂3w ∂x2∂t(1, t) = 0, ∂3w ∂x3 (0, t) + ε ∂4w ∂x3∂t(0, t) = 0, ∂3w ∂x3 (1, t) + ε ∂4w ∂x3∂t(1, t) = u(t). Let the output of the system be yo(t) :=

∂2 ∂t2 w(0, t) and

consider a low-pass characteristics for the sensor dynam- ics: H(s) = e−hs/(1 +τs), with h ≥ 0 and τ > 0. Define the output available for feedback as Y (s) = H(s)Yo(s). Note that w(0, t) is the deflection at the opposite end of the beam as the applied force −u(t), i.e., even if the sensing delay is zero the plant is non-minimum phase due to non- collocated actuator and sensor. We will ignore the actuator dynamics here. Transfer function of the plant (including the sensor dynamics) can be derived as in [84]: P4(s) := Y (s)

U(s),

P4(s) = s2(sinh β − sin β) e−hs β3(cos β cosh β − 1)(1 + εs)(1 + τs), (12) where β4 =

−s2 (1+εs).

One can show that P4(s) can be expressed as infinite products of second order terms. These product repre- sentations display its poles and zeros. It also facilitate inner/outer factorizations which are essential for solving the H∞ optimization problems. P4(s) = 2e−hs τs + 1

∞

• n=1

gn(s) (13) where gn(s) =

• 1 + εs −

s2 4α4

n

• 1 + εs + s2

φ4

n

• for values of s where this infinite product converges. In

[84] it is shown that (13) converge everywhere in the closed right half plane and can be written as quotients of H∞ functions. We factor P4 = MnNo, where Mn(s) = e−hsB(s), No(s) = 2 (τs + 1)

∞

• n=1
• 1 + s
• ε2 +

1 α4

n +

s2 4α4

n

• 1 + εs + s2

φ4

n

• , (14)

B(s) =

∞

• n=1

  2α4

n

• ε +
• ε2 +

1 α4

n

• − s

2α4

n

• ε +
• ε2 +

1 α4

n

• + s

  . (15) It has been shown that [84], [85], No(s) ∈ H∞ and B(s) ∈ H∞ converge in the closed right half-plane. The zeros of P4 are at s = 2α4

n

• ε ±
• ε2 + 1

α4

n

• for

n = 1, 2, ... (16) where cos(αn) sinh(αn) = sin(αn) cosh(αn), for αn > 0. Also, P4 has a singularity at −1/ε, and poles at s = −φ4

n

2

• ε ±
• ε2 − 4

φ4

n

• for

n = 1, 2, ... where cos(φn) cosh(φn) = 1, for φn > 0.

• B. H∞ Optimal Controller

Assume that the weights W1 and W2 are rational and (W2No), (W2No)−1 ∈ H∞, then optimal H∞ controller for plant (10) can be written as, [141], Copt = Eγo(s)Md(s) N −1

• (s)Fγo(s)L(s)

1 + Mn(s)Fγo(s)L(s) (17) where Eγ(s) =

• W1(−s)W1(s)

γ2

− 1

• , and for the definition
• f the other terms, let the right half plane zeros of Eγ(s)

be βi, i = 1, . . . , n1, the right half plane poles of P(s) be αk, k = 1, . . . , ℓ and that of W1(−s) be ηi i = 1, . . . , n1. Then, Fγ(s) = Gγ(s) n1

i=1 s−ηi s+ηi where

G∗

γGγ =

• 1 −

W ∗

2 W2

γ2 − 1

• Eγ

−1 (18) and Gγ, G−1

γ

∈ H∞, and L(s) = L2(s)

L1(s) , L1 and L2 are

polynomials with degrees ≤ (n1 + ℓ − 1) and they are determined by the following interpolation conditions, = L1(βi) + Mn(βi)Fγ(βi)L2(βi), (19) = L2(−βi) + Mn(βi)Fγ(βi)L1(−βi), = L1(αk) + Mn(αk)Fγ(αk)L2(αk), = L2(−αk) + Mn(αk)Fγ(αk)L1(−αk) for i = 1, . . . , n1 and k = 1, . . . , ℓ. The optimal performance level, γo, is the largest γ value such that the spectral factorization (18) can be done and the interpolation conditions (19) are satisfied for some non-zero L1, L2. A Matlab-based computer program is available for computing γo and all the functions appearing

SLIDE 6

in (17); it can be downloaded from the author’s web site: www.ece.ohio-state.edu/˜ozbay search for HINFCON under research heading. At the MTNS06 basic features of this program will be demonstrated. With respect to the above controller formula, first point to note is that the spectral factorization in question is finite dimensional and the order of Fγ is at most the order of W1 plus the order of W2. Secondly, L(s) is characterized by its 2(n1 + ℓ) coefficients, and the interpolation conditions (19) can be re-written as Rγ Φ = 0 (20) where the entries of the 2(n1 +ℓ)×1 vector Φ contain the coefficients of L1 and L2, and the 2(n1 + ℓ) × 2(n1 + ℓ) matrix Rγ is constructed from Mn(s), F(s) and βi, αk, for i = 1, . . . , n1 and k = 1, . . . , ℓ. Clearly, explicit computation of Mn(s) is not necessary for the construction

• f Rγ; all we need is its values at βi’s and αk’s. See [42]

for more detailed discussion of this point and further ref-

• erences. Due to symmetry in the interpolation conditions,

L(s) term in the optimal controller satisfies |L(jω)| = 1, but it may or may not be stable. The final observation we make, probably the most important one from the point of view of “controller implementation,” is that the controller has internal unstable pole-zero cancellations: the C+ zeros of Eγo and Md are cancelled by the zeros of 1 + Mn(s)Fγo(s)L(s). Since Mn is infinite dimensional, e.g. time delay, exact cancellation is not always possible. For this reason, these cancellations should be studied in more detail and a new equivalent structure, which can be implemented in a stable manner, should be investigated, see [60] for a discussion of this problem within the framework

• f general time delay systems. In this context, the optimal

controller for the plant P3 given above will be presented at the MTNS2006; it will be an example taken from [60]. Time permitting, a discussion on optimal H∞ controllers for plants P2, [39], [111], and P4, [84], [85], will also be

• included. Here we illustrate the optimal H∞ controller for

P1, with the mixed sensitivity problem weights W1(s) = 1 + ǫs s + ǫ W2(s) = k(1 + ατps). In this case the plant is stable, so ℓ = 0, and W1 is first

• rder, so n1 = 1, and thus n1 + ℓ − 1 = 0, which means

that L(s) is just a constant, +1 or −1. We have only one linear equation to form, and that gives γo. When we let ǫ → 0, we obtain γo as the largest root of the equation

• 1 − (k/γ)2 − kατp/γ2 − sin(h/γ) = 0

in the interval

2h π

≤ γ < ∞. It can be shown that, [133], the optimal H∞ controller has an “internal model controller” structure Copt = Qopt 1 − P(s)Qopt Qopt = Qo 1 + FoQo where Qo(s) = k 1 + ατps γos Fo(s) = 1 1 + ατps γs(sin(h/γo) + γs cos(h/γ)) + e−hs 1 + (γs)2 When the time delay, h, is “relatively small,” we have Fo(s) ≈

1 1+ατps, and then the closed loop transfer function

from r to y, is T (s) ≈ e−hs 1 + τcls where τcl = αkτp/γ is the approximate closed loop time constant.

• IV. APPLICATIONS

In this section we present two applications of robust con- trol theory for infinite dimensional systems. Both problems involve plant models with time delays.

• A. Flow Control in Data Communication Networks

There are interesting control problems in communication networks, due to time varying time delays, [102]. To illustrate the main underlying difficulty, let us consider a generic data communication set-up shown in Figure 4. The senders adjust their packet sending rates based on the acknowledgment signals transmitted by the receivers.

Link Senders Receivers Network Congested

• Fig. 4.

Generic data communication network.

Using fluid flow approximation, the queue at the bottle- neck (congested) link evolves according to ˙ q(t) = rin(t) − c(t) where rin(t) is the incoming data packet flow rate and c(t) is the outgoing flow rate (available capacity of the link). As the packets pass through the network, they reach their destination with a forward time delay hf(t) = hp + q(t)

c(t),

where hp is the propagation delay in the forward path, and

q(t) c(t) is the queueing delay. Once the packets are

received, acknowledgments are sent back; assuming they pass through the same network, with a separate buffer, which is almost always empty, the backward delay is hb ≈ hp. So, we define the return-trip-time RTT (t) = h(t) = 2hp + q(t)

c(t). Typically, the input flow rate is

set by the controller as a function of the “congestion information” relayed to the controller, delayed by h(t). In other words, we may assume that rin(t) = rc(t−h(t)), where rc(t) is the assigned flow rate at time t (i.e. output

• f the controller). Thus, we have different plant models,

depending on how this congestion information (feedback)

SLIDE 7

is set. From the above discussion it is clear that queue evolution equation will contain a time delay which depends

• n the queue itself. More precisely,

˙ q(t) = rc(t − h(t)) − c(t) , h(t) = 2hp + q(t) c(t). Analysis of these type of state dependent delay systems have been considered in the literature, see e.g. [21], [67], [92] and their references . But the controller design in this context is a difficult problem. So, typically, we assume that the delay has a nominal value, ho = 2hp + qo/co, where qo and co are the nominal values of the queue and the link capacity, respectively, and the variations of the delay is bounded. When the explicit queue information is fed back to the controller (as in ATM networks, see e.g. [13], [15], [129]) H∞-based robust controllers can be designed, [119], by using the uncertainty bound on the time delay and the time derivative of the delay. Nominal time delays in each source-destination link are assumed to be the same in [119]. This assumption is relaxed in [144], and an H∞ controller is designed using the approach proposed in [94]. In active queue management of TCP flows, feedback information is implicit: as data packets pass through the congested link, some percentage of them are marked according to the average queue level, and the sources adjust their rates according to the packet marks received, see [89] for more details and references. For this scenario there is a nonlinear time delay system model, [97], which has been linearized and used in various robust control schemes, [30], [70], [120], [154]. The linearized time delay model in this case can be taken in the form P(s) = K A(s) e−hos 1 + A(s) hos e−hos where A(s) = (Wo(hos)2+(Wo+1)hos+2)−1 and K, Wo are constants. Clearly, this model fits the H∞ controller design scheme discussed earlier. The weights are chosen in such a way that uncertainty in the time delay, number

• f users and link capacity are taken into account when the

mixed sensitivity optimization problem is set up. See [120] for full details.

• B. Control of Cavity Flow Oscillations

Control of cavity flow oscillations is an interesting aerodynamic flow control problem; see [152], [153] for a background on the subject and a list of references. The physical experimental set-up is as illustrated in Figure 5. The actuator is a two-dimensional synthetic-jet issuing from a slot embedded in the cavity leading edge. Pressure fluctuations at the bottom of the cavity are measured by Kulite dynamic pressure transducers. In general such flow problems are best modeled by Navier-Stokes equations. Unless there is a special geome- try in the structure and some assumptions on the flow are made, it is difficult to design controllers for these infinite dimensional nonlinear models (see [1], [4], [18], [71], [75],

pressure sensor flow cavity baseline flow control flow actuator Controller

• Fig. 5.

Cavity Flow System.

[137] for recent approaches to flow control problems and future directions). For the cavity problem at hand, a linear plant model in the form P(s) = KsH(s)Go(s)e−τss 1 − KrKsH(s)Go(s)e−τss has been proposed in [125], where Ks, Kr are constants Go(s) = ω2

• s2 + 2ζωos + ω2
• H(s) =

e−τas 1 − r(s)e−2τas r(s) = r 1 + s/ωr . Time delays τa and τs are related to the physical geometry and the flow. All other parameters r, ωr, ωo, ζ, Kr, Ks are adjusted in such a way that the open loop system response measured matches the output of this plant model. We define v(t) in Figure 1, standard feedback system, as the noise which generates the cavity oscillations when feedback control is absent (i.e. baseline flow only, actuator output is zero). The open loop response is Y (s) = P(s)V (s). Typically, energy of this signal is concentrated at particular frequencies (called Rossiter modes). The goal of feedback control is to spread the energy of the closed loop response, Y = P(1 + PC)−1V (s), in a wide frequency band. One can pose this problem as a weighted sensitivity minimiza- tion problem, [156], of finding the optimal controller in C(P) achieving inf

C∈C(P ) P(1 + PC)−1∞.

Note that in this problem P is infinite dimensional, and the weight W1 = P is infinite dimensional too. In [156] the weight W1 is approximated by another infinite dimensional system, for which the special structure of this weight and the plant makes the problem solvable by using the H∞ control design method discussed here. At the MTNS2006 we will briefly show these results from [156].

• V. CONCLUSIONS

This paper is an outline of the invited talk to be given at the MTNS, in July 2006, Kyoto, Japan. It also provides background on the topics to be discussed during the author’s talk. Not all details could be included here. References are given for the missing details. Several other

SLIDE 8

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