Using H2-control performance metrics for the optimal actuator location ...

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Using H2-control performance metrics for the optimal actuator location of distributed parameter systems Kirsten Morris, Michael A. Demetriou, Steven D. Yang

Abstract—This paper is concerned with the use of H2 -control performance metrics in finding optimal actuator locations for a class of infinite-dimensional systems. Conditions that guarantee convergence of the optimal actuator location for the finitedimensional representation of the infinite-dimensional system to the optimal actuator location of the original infinite-dimensional system are provided. Minimizing an H2 -performance index, parameterized by the possible actuator locations and disturbance functions, leads to optimal locations that in addition to enhanced performance, also provide robustness with respect to the spatial distribution of disturbances. Numerical studies for a cantilever beam and a diffusion problem are used to demonstrate the effects of the spatial distribution of disturbances on the optimal actuator location. The effect of the diffusivity parameter on the best actuator locations is also considered. Index Terms—Distributed parameter systems; optimal actuator location; H2 -control; spatial disturbances.

I. I NTRODUCTION In many control problems modeled by partial differential equations, the location, size, distribution and number of actuating and sensing devices constitute a challenging design problem. Examples of engineering systems that fall under this category include control of flexible structures and acoustic noise reduction [1]. Since the performance of the controlled system is intimately coupled to the actuator and sensor locations e.g. [2], [3], the importance of actuator/sensor location now becomes part of the controller design problem. This problem has been considered by many researchers; see for instance [4], [5], [6], [7], [8], [9], [10] and the review articles [11], [12], [13], [14]. One approach to actuator location optimization is to consider open-loop measures. In this approach, the location is typically chosen so as to enhance the controllability properties of the actuator. For example, this can be done by minimizing the controllability gramian Φ. One metric of the gramian is the quantity kz∗0 Φz0 k which represents the minimum energy required to reach state z0 . Related functions of the gramian involve the coercivity bound, the spectral radius and singular values, [15], [16], [17], [18], [19]. Relevant extensions consider optimization of the actuator authority over specific eigenfrequencies via the use of spatial norms [20], [21]. K. Morris is with the University of Waterloo, Department of Applied Mathematics, Waterloo, ON, N2L 3G1, CANADA, Email: [email protected]. The author gratefully acknowledges financial support from AFOSR, grant FA9550-10-1-0530. M. A. Demetriou is with Worcester Polytechnic Institute, Department of Mechanical Engineering, Worcester, MA 01609-2280, USA, [email protected]. The author gratefully acknowledges financial support from AFOSR, grant FA9550-12-1-0114. S. D. Yang is with the University of Waterloo, Department of Electrical and Computer Engineering, Waterloo, ON, N2L 3G1, CANADA, Email:

[email protected]

However, since the final system is affected by the controller design, it is sensible to choose the actuator location using the same criterion as used to design the controller. In [5], [7], [8] for example the actuator locations are chosen to minimize the linear-quadratic (LQ) cost. In [22] conditions for wellposedness of the LQ-optimal actuator problem were obtained and a numerical scheme for finding the global minimum is described in [23]. In many systems, the objective is to reduce the response to a disturbance. Two popular approaches to reducing the response to a disturbance are H∞ , where the disturbance frequency (or time) content is unknown, and H2 , where a particular disturbance signal is considered. In [24] H∞ -optimal actuator location is considered. Conditions for well-posedness of the problem, the use of approximations and also a numerical algorithm for calculating optimal locations are given there. In this paper the situation where disturbance is assumed to have a known frequency content is considered and this leads to the well-known H2 -controller design problem. The finitedimensional results on existence and characterization of an optimal controller are extended to infinite-dimensional systems. Furthermore, controller design and actuator location are integrated. Another level of optimization is also considered. The additional optimization considers the effects of the spatial component of the disturbances on optimal actuator location. The H2 -control formulation considered here incorporates the disturbances in the expression for the optimal value of the performance index, and this is subsequently employed to obtain optimal actuator locations that also address the effects of the spatial distribution of disturbances. In [25] open and closed-loop strategies for disturbance rejection in the context of a flexible beam were discussed. A modal approximation for a structure was used in [26] and the actuators (and sensors) were chosen to minimize an approximate H2 -norm in open loop. Modal reduced order models based on the open loop H2 -norm were also used in [27], but the closed loop H2 -performance was used to select actuator locations. Here the problem is formulated rigorously, both theoretically and in the context of using approximations. We also investigate the effect of various factors on H2 -optimal actuator location using several examples. Since the effect of using a reduced-order model on optimal actuator location selection is not known, the numerical scheme described in [23] is used to calculate the optimal actuator locations. This means that high-order models can be used in the actuator location/controller design step. Earlier attempts to include the effects of the spatial disturbance distribution on actuator locations considered different candidate disturbance distributions and subsequently solved the associated H2 -control problem for optimal actuator loca-

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tion, [28], [29]. These works concluded that if the disturbance distribution is known, then it should be incorporated into the performance-based actuator optimization problem. Equations for the control that minimize the H2 -norm for a fixed actuator location for infinite-dimensional systems are provided in section II. This is an extension of the analogous result for finite-dimensional systems. The optimal control is found by solving an operator algebraic Riccati equation (ARE) for an operator Π. A method to find the “worst” distribution of disturbances and to subsequently use it in the H2 -control performance for actuator optimization is presented. Such an actuator optimization provides both a performance enhancement and “spatial robustness”, as the optimal actuator is calculated for the worst possible disturbance distribution. In practice, an infinite-dimensional ARE cannot be solved and the control is calculated using an approximation. Use of approximations leads to solving a finite-dimensional ARE for a finite-rank approximation Πn . Conditions under which the optimal actuator locations for a LQ-cost calculated using approximations converge to the exact optimal location were derived in [22]. These results are used here to obtain conditions for the use of approximations in calculating H2 -optimal actuator locations. In practice certain components of the state of a spatially distributed process may be more important, or a certain spatial region may be of more interest than other spatial regions within the spatial domain. Consideration of the aspects of the state to be controlled should be included in the formulation of the controller design problem. This can be done by choosing the operator associated with the controller cost function to weight the aspects of the state that are of interest. In view of the above requirements, the effects of different choices of state weight on actuator location and controlled performance are of interest. These issues are explored using several examples: a flexible beam and diffusion on a two-dimensional region. A preliminary version of some of these results was presented in [30].

Consider systems described by the evolution equation z(0) = z0

(1)

where the operator A with domain D(A) generates a strongly continuous semigroup S(t) on a Hilbert space H , and B2 ∈ L (U, H ) where L (U, H ) indicates bounded linear operators from a separable Hilbert space U to H . Suppose that there are M actuators with locations that can be varied over some compact set Ω ⊂ Rn . Parametrize the actuator locations by r and indicate the dependence of the input operator B2 on the locations by the notation B2 (r). Letting ΩM indicate vectors of length M with each component in Ω, r ∈ ΩM . The linear-quadratic (LQ) controller design objective is to find a control u(t) to minimize the cost functional Z ∞

J(u, z0 ) =

hCz(t),Cz(t)i + hu(t), Ru(t)idt,

A∗ Π(r) + Π(r)A − Π(r)B2 (r)R−1 B∗2 (r)Π(r) +C∗C = 0 (3) (where this equation is understood to hold on D(A)) has a positive semi-definite solution Π(r) ∈ L (H , H ) and the optimal cost is hΠ(r)z0 , z0 i [31, sect. 6.2]. For a particular initial condition z0 , the optimal actuator problem is inf hz0 , Π(r)z0 i.

r∈ΩM

The objective function to be minimized by the best actuator location can be defined as minimizing kΠ(r)k (to minimize the response to the worst choice of initial condition) or 1 1 trace(V 2 Π(r)V 2 ) (if the initial condition is random, with zero mean and variance V [32]). These problems were analyzed in [22] and conditions for well-posedness of the problem, as well as conditions under which an approximation can be used to determine the best actuator location were obtained. However, in this work, minimizing the influence of disturbances on the performance is the major concern. The temporal component of the disturbance d(t) is fixed and it is assumed to lie in a finite-dimensional space V . The system being controlled is a distributed parameter system z˙(t) = Az + B1 d(t) + B2 (r)u(t),

(2)

0

where R ∈ L (U,U) is a self-adjoint positive definite operator weighting the control, C ∈ L (H ,Y ) where Y is a separable

z(0) = z0

(4)

with a specific disturbance d, discussed below. The object is to choose the control u to minimize kyk2 =

II. P ROBLEM F ORMULATION z˙(t) = Az(t) + B2 u(t),

Hilbert space weights the state, and z(t) is determined by (1). Each set of actuator locations r ∈ ΩM defines an optimal control problem (2), indicated by J r (u, z0 ). Definition 1: The pair (A, B) is stabilizable if there exists K ∈ L (U, H ) such that A − BK generates an exponentially stable semigroup. Definition 2: The pair (A,C) is detectable if there exists F ∈ L (H ,Y ) such that A − FC generates an exponentially stable semigroup. Assume the family (A, B2 (r),C) is both stabilizable and detectable, for each r ∈ ΩM . Then for each r ∈ ΩM the location-parameterized Algebraic Riccati Equation (ARE)

Z ∞

ky(t)k2 dt,

y(t) = Cz(t) + Eu(t)

(5)

0

where B1 ∈ L (W, H ), C ∈ L (H ,Y ), E ∈ L (U,U) and W is a separable Hilbert space. We make the standard assumption that E ∗ E is invertible so that the control cost is non-singular. To simplify the formulae, it will be assumed throughout this paper that E ∗C = 0,

E ∗ E = I.

The cost (5) is then identical to the linear quadratic cost (2) with R = E ∗ E = I. The standard problem is to find the control law u so that kyk2 is minimized; where B1 is known as well as the actuator locations. The difference between this problem and the LQ problem is that in LQ the aim is to reduce the response to the initial condition z(0) with disturbance d = 0 while now the objective is to reduce the response to the disturbance d and set z(0) = 0. For any separable Hilbert spaces U,Y an inner product can be defined for operators M, N ∈ L (U,Y ) as hM, Ni = trace(M ∗ N) = ∑hNek , Mek i

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where {ek } is any orthonormal basis for U. It can be shown that this definition is independent of the choice of basis. The corresponding norm is known as the Hilbert-Schmidt norm, and the space of all operators for which this norm is finite is the space of Hilbert-Schmidt operators. The control system z˙(t) = Az(t) + Bu(t)

to the operator Lyapunov equations

kGk22

with transfer function G will often be abbreviated (A, B,C, E) or written   A B . G= C E

provided that G( jω) is a Hilbert-Schmidt operator. It is assumed throughout this paper that B1 and B2 are compact and that at least one of the following assumptions is satisfied: • B1 is a Hilbert-Schmidt operator, • C is a Hilbert-Schmidt operator, or • both B1 and C are trace class. This will guarantee that the controlled system is a HilbertSchmidt operator, and also that the optimal actuator problem is well-posed. These assumptions can be weakened for certain classes of systems. However, in practice, there are generally a finite number of disturbances, control signals, and outputs. Since every finite-rank operator is Hilbert-Schmidt operator, typically both B1 are B2 are Hilbert-Schmidt operators. Since the norm (5) is equal to the H2 -norm of the Laplace transform of y, this is known as an H2 -controller design problem [33]. The primary relevance of the H2 -norm is that kGk2 is the spectral power density, or expected power, of the output of a system subject to unit variance white noise (a signal with uniform spectral density). Alternatively, let {ek } be an orthonormal basis for U, and denote the output with input δ(t)ek by yk . Then kGk22 = ∑k kyk k22 where kyk k2 indicates the L2 -norm of the output yk . (See, for example, [34, chap. 3].) The case of other disturbances d is handled by absorbing the frequency content of d into the system description (4) so that the objective is to find u minimize the H2 -norm of the transfer function y(s) = C(sI − A)−1 (B1 + B2 u(s)) + Eu(s). It will be assumed here that this normalization has been done. Theorem 1: Consider the system (A, B,C, 0) with transfer function G where A is the infinitesimal generator of an exponentially stable C0 -semigroup and either B and C are both trace class or one of B or C is a Hilbert-Schmidt operator. The H2 -norm of the system is kGk22 = trace(B∗ Lo B) = trace(CLcC∗ ) where Lo and Lc are the observability and controllability Gramians respectively [35]. The Gramians are the solutions

= 0,

(6)

ALc + Lc A∗ + BB∗

= 0.

(7)

Proof: Letting ek indicate an orthonormal basis for U, and noting that each term in the sum below is non-negative,

y(t) = Cz(t) + Eu(t)

and its transfer function is G(s) = C(sI − A)−1 B + E. The H2 norm of G is rZ ∞ 1 kGk2 = trace(G( jω)∗ G( jω))dω, 2π 0

A∗ Lo + Lo A +C∗C

Z ∞

= Z0

∞

∑hCS(t)Bek , CS(t)Bek iH dt

= 0

=

trace(B∗ S(t)∗C∗CS(t)B) dt

k

∑hBek , k

Z ∞ 0

S(t)∗C∗CS(t)dt Bek iH

= trace(B∗ Lo B) where

Z ∞

Lo =

S(t)∗C∗CS(t)dt.

0

Since Lo solves (6) [31, Defn 4.1.20,Thm. 4.1.23] the first statement follows. The similar fact that kGk22 = trace(CLcC∗ ) follows by using duality.  If there is only a single control then B is defined by Bu = bu for some b ∈ H , and therefore kGk22

= hb, Lo bi.

Consider now the problem of choosing the control to minimize the H2 -norm. The following theorem is an extension of the analogous result for finite-dimensional systems [36], although the assumption in [36] that the control is a feedback control is not used here. Theorem 2: Consider the linear system (4) with cost (5) and assume that (A, B2 ) is stabilizable. The H2 -optimal control is the state feedback u(t) = −B∗2 (r)Π(r)z(t) with Π(r) the solution to (3), which yields optimal cost trace(B∗1 Π(r)B1 ) andpthe optimal norm of the closed loop transfer function trace(B∗1 Π(r)B1 ). If there is a single disturbance, so is B1 d can be written b1 d for some b1 ∈ H , the optimal cost is p hb1 , Π(r)b1 i and the optimal norm of the closed loop is hb1 , Π(r)b1 i. Proof: The dependence of B2 and Π on r will not be indicated. The problem is to minimize the H2 -norm of y over all u ∈ L2 (0, ∞;U). Let Π be the solution to the algebraic Riccati equation (3) and define the corresponding feedback K = −B∗2 Π. Redefine the controller signal as u = v + Kz. Since K is a stabilizing feedback, if u ∈ L2 (0, ∞;U) is a stabilizing control then v ∈ L2 (0, ∞;U) and vice versa. The equivalent problem of minimizing kyk2 over all v ∈ L2 (0, ∞;U) is considered. Defining AK = A + B2 K, CK = C + EK, this leads to z˙(t) = AK z(t) + B1 d(t) + B2 v(t) y(t) = CK z(t) + Ev(t). Defining the state-space systems    AK AK B1 , U= GK = CK 0 CK

B2 E

 ,

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The H2 -optimal actuator location problem is thus to find the actuator location rˆ that minimizes

the transfer function y = GK +Uv. ˜ = I for the finite-dimensional case in It is shown that UU [33, Thm. 14.3] and the extension to infinite-dimensions is an exercise in [31, Ex. 7.27]. For completeness, the details are given here. First, note that   −A∗K −CK∗ ˜ . U= B∗2 E∗ ˜ has realization The series connection UU       −A∗K −CK∗ CK −CK∗ E . ˜ = 0 AK B2 UU  ∗  ∗ B2 E CK I

= z1 − Πz2 = z2

inf sup hb1 , Π(r)b1 i

(8)

where Π solves (3) with R = E ∗ E = I. Noting that (3) can be rewritten as A∗K Π + ΠAK +CK∗ CK = 0 and using the simplifying assumption E ∗ [C E] = [0 I], implies that (with respect to the realization [w1 , w2 ])       0 −A∗K 0 . ˜ = B2 0 AK UU  ∗  B2 0 I ˜ Thus, for all s, U(s)U(s) = I. Since AK generates an exponentially stable C0 -semigroup, GK ∈ H2 and U ∈ H∞ which implies that Uv ∈ H2 . This implies that y ∈ H2 . The H2 -norm of a function in H2 equals the L2 -norm of that function along the imaginary axis, if these boundary values are well-defined, as is the case here. Thus, kyk22

lim kB2 (s) − B2 (r)k = 0. s→r

If (A, B2 (r)) is stabilizable for all r ∈ ΩM and (A,C) is detectable, then there exists an optimal actuator location rˆ ∈ ΩM such that trace(B∗1 Π(ˆr)B1 ) = inf trace(B∗1 Π(r)B1 ). r∈ΩM

(9)

If B1 d = b1 d for some unknown b1 ∈ H then there is rˆ ∈ ΩM , φ ∈ H such that hφ, Π(ˆr)φi = kΠ(ˆr)k = inf sup hb1 , Π(r)b1 i r∈ΩM b1 ∈H

(10)

and this cost is achieved when b1 is the eigenfunction corresponding to the largest eigenvalue of Π(ˆr). Proof: The assumptions imply that Π(r) is a continuous function of r in the operator norm [22, Thm. 2.6]. Since ΩM is a compact set, the infimum is achieved by some rˆ ∈ ΩM . If B1 d = b1 d, b1 ∈H

˜ K iL2 = kGK k2L2 + kUvk2L2 + 2Rehv, UG 2 2 ˜ K iL2 . = kGK kL + kvkL + 2Rehv, UG 2

where Π(r) indicates the solution to the ARE with actuator location r. Theorem 3: Assume that for any r ∈ ΩM ,

sup hb1 , Π(r)b1 i = λmax (Π(r))

= kGK +Uvk2L2

˜ K is A realization for UG   −A∗K ˜ K = 0 UG  ∗ B2

Consider a single disturbance so that B1 d = b1 d for some b1 ∈ H . If the spatial distribution of the disturbance, b1 , is not known then the objective is to find the actuator location that minimizes the H2 -cost over possible disturbance distributions: The problem becomes that of choosing the actuator location r to minimize the closed loop response to the worst spatial disturbance distribution; that is r∈ΩM b1 ∈H

Indicating the original state of the above system by [z1 , z2 ], define a new state by w1 w2

trace(B∗1 Π(r)B1 ).

2

−CK∗ CK AK  ∗ E CK

 

0 B1

  .

0

Applying the same transformation (8) as was used to simplify ˜ the realization of UU,       −A∗K 0 −ΠB1 . ˜ K = 0 AK B1 UG  ∗  B2 0 0 ˜ K ∈ H2 ⊥ . Since v ∈ H2 , this implies that Thus, UG kyk2 = kGK k2L2 + kvk2L2 . The minimum H2 -norm is achieved when v = 0, that is the control signal u = Kz. The remaining statements follow from Theorem 1. 

= kΠ(r)k. Statement (10) then follows with φ equal to the eigenvector corresponding to the largest eigenvalue of Π(ˆr).  Thus, if B2 (r) is a continuous function of r, both the optimal actuator location problem with a fixed disturbance location, and the problem where the disturbance is unknown, lead to well-posed optimization problems. Note that if the spatial distribution of the disturbance b1 is unknown, the problem is to minimize kΠ(r)k over the actuator location r. III. C ALCULATION OF O PTIMAL ACTUATOR L OCATION In practice, the control is calculated using an approximation to the solution (3). Let {Hn } be a family of finite-dimensional subspaces of H and Pn the orthogonal projection of H onto Hn . The space Hn is equipped with the norm inherited from H . Consider a sequence of operators An ∈ L (Hn , Hn ) and define B2n (r) = Pn B2 (r), B1n = Pn B1 . This leads to a sequence of approximations to the system (4) z˙(t) = An z(t) + B1n d(t) + B2n (r)u(t), z(0) = zn0 = Pn z0 . (11)

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Defining Cn = C|Hn , the cost (5) becomes Z ∞ 0

Also, the sequence of approximating actuator locations rˆm has a convergent subsequence. Any convergent subsequence has the property that

kCn z(t) + Eu(t)k2 dt.

If (An , Bn ) is stabilizable and (An ,Cn ) is detectable, then the minimum cost is trace(B∗1n Πn B1n ), where Πn is the unique non-negative solution to the algebraic Riccati equation A∗n Πn + Πn An − Πn B2n R−1 B∗2n Πn +Cn∗Cn = 0,

(12)

on the finite-dimensional space Hn . (For simplicity of notation the dependence of B2 , B2n and Πn on r will not always be indicated in this section.) Assumptions that guarantee that Πn converges to Π in some sense are required. Let Sn (t) indicate the semigroup generated by An . The following set of assumptions is standard. We assume that for each z ∈ H , u ∈ U, y ∈ Y , and r ∈ ΩM , (A1 ) (i) kSn (t)Pn z − S(t)zk → 0, (ii) kSn∗ (t)Pn z − S∗ (t)zk → 0 uniformly in t on bounded intervals of t. (A2 ) (i) kB2n u − B2 uk → 0 and kCn Pn z −Czk → 0. (ii) kCn∗ y −C∗ yk → 0 and kB∗2n Pn z − B∗2 zk → 0. (A3) (i) The family of pairs (An , B2n ) is uniformly exponentially stabilizable, i.e., there exists a uniformly bounded sequence of operators Kn ∈ L (Hn ,U) such that (An −B2n Kn )t Pn z ≤ M1 e−ω1 t |z| e for some positive constants M1 ≥ 1 and ω1 . (ii) The family of pairs (An ,Cn ) is uniformly exponentially detectable, that is, there exists a uniformly bounded sequence of operators Fn ∈ L (Y, Hn ) such that (An −FnCn )t Pn ≤ M2 e−ω2 t , t ≥ 0, e for some positive constants M2 ≥ 1 and ω2 . Although not all approximation schemes satisfy assumptions (A1)-(A3), most common approximations, such as linear splines for the diffusion equation and cubic splines for damped beam vibrations do, provided that the original system is stabilizable and detectable; see the review article [37]. Assumptions (A1)-(A3) guarantee not only that the solutions to the approximating ARE’s converge strongly, but that the resulting feedback controls determine uniformly stable semigroups. Define µ = infr∈ΩM hB1 , Π(r)B1 i = hB1 , Π(ˆr)B1 i where rˆ is an optimal actuator location and define µn , rˆn similarly. The following theorem shows that strong convergence of Πn → Π implies convergence of the cost: µn → µ. Convergence of the optimal actuator location and controllers is also implied.That is, performance arbitrarily close to optimal can be achieved with the approximating actuator locations and controllers. Theorem 4: Consider the control system (4) and assume that for any r ∈ ΩM , lim kB2 (s) − B2 (r)k = 0. s→r

Assume also that (A1)-(A3) are satisfied for each (An , B2n (r),Cn ). Then the approximating optimal costs converge to the exact optimal cost, that is inf trace(B∗1 Π(r)B1 ) = lim inf trace(B∗1n Πn (r)B1n ).

r∈ΩM

n→∞ r∈ΩM

µ = lim trace(B∗1 Π(ˆrm )B1 ); m→∞

(13)

and the corresponding controllers converge. Proof: Let ek indicate an orthogonal basis for W. µn

=

inf trace((Pn B1 )∗ Πn (r)Pn B1 )

r∈ΩM

≤ trace((Pn B1 )∗ Πn (ˆr)Pn B1 ) =

∑hΠn (ˆr)Pn B1 ek , Pn B1 ek i = ∑hB1 ek , Πn (ˆr)B1 ek i k

=

k

∑hB1 ek , (Πn (ˆr) − Π(ˆr))B1 ek i + ∑hB1 ek , Π(ˆr)B1 ek i k

k

= trace (B∗1 ( Πn (ˆr) − Π(ˆr) )B1 ) + µ. The assumptions imply that Πn (ˆr) converges strongly to Π(ˆr) [38, Thm. 6.9],[39, Thm. 2.1, Cor. 2.2] and since B1 is a compact operator, limn→∞ trace(B∗1 (Πn (ˆr) − Π(ˆr))B1 ) = 0 and so lim sup µn ≤ µ. It remains only to show lim inf µn ≥ µ. To this end, choose a subsequence µm → lim inf µn , with corresponding actuator locations rˆm . Since rˆm ⊂ ΩM , it has a convergent subsequence, also denoted rˆm , with limit r ∈ ΩM . Now, kB2m (ˆrm ) − B2 (r)k = kPm B2 (ˆrm ) − B2 (r)k ≤ kPm B2 (ˆrm ) − Pm B2 (r)k + kPm B2 (r) − B2 (r)k ≤ kPm kkB2 (ˆrm ) − B2 (r)k + kPm B2 (r) − B2 (r)k. Thus, kB2m (ˆrm ) − B2 (r)k → 0. By assumption (A3i) there is a uniformly bounded sequence Km (r) ∈ L (H ,U) such that Am − B2m (r)Km (r) generate semigroups bounded by Me−ωt for some M > 0, ω > 0. For some ε < ω/M, choose N large enough that kB2m (ˆrm ) − B2m (r)k < ε for m > N. Then for all m > N, Am − B2m (ˆrm )Km (r) generates a C0 -semigroup with bound Me(−ω+Mε)t . Consider the sequence of systems (Am , B2m (ˆrm ),Cm ), as approximations to (A, B2 (r),C). Applying [38, Thm. 6.9],[39, Thm. 2.1, Cor. 2.2], Πm (ˆrm ) converges strongly to Π(r) and since B1 is a compact operator, kΠm (ˆrm )B1 − Π(r)B1 k → 0. Thus, lim inf µn = lim µm m→∞

= lim trace((Pm B1 )∗ Πm (ˆrm )Pm B1 ) m→∞

= lim trace (B∗1 ( Πm (ˆrm ) − Π(r) )B1 )

(14)

m→∞

+trace(B∗1 Π(r)B1 ) = trace(B∗1 Π(r)B1 ) ≥ µ. It follows that lim inf µn ≥ µ and so lim µn = µ as required. We now show (13). Since {ˆrn } is contained in a compact set, it has a convergent subsequence rˆm → r¯, r¯ ∈ ΩM . Now, trace(B∗1 Π(ˆrm )B1 ) = trace(B∗1 (Π(ˆrm ) − Π(¯r))B1 ) + trace(B∗1 (Π(¯r) − Πm (ˆrm ))B1 ) + trace(B∗1 Πm (ˆrm )B1 ). (15) The first term converges to zero because Π(r) is a continuous function of r [22, Thm. 2.6]. The second term converges to zero by an argument identical to that used in the first

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part of this proof. Since the third term is µm it follows that limm→∞ trace(B∗1 Π(ˆrm )B1 ) = µ . The corresponding approximating optimal controllers are Km = −B2m (ˆrm )Π(ˆrm ) and the optimal controller is K = −B2 (¯r)Π(¯r). From the arguments above, kB2m ((ˆrm ) − B2 (¯r)k → 0, Πm (ˆrm ) converges strongly to Π(r) and so kKm − Kk → 0.  Now consider the situation where there is a single, but unknown disturbance so that the actuator location should minimize inf kΠ(r)k.

point. The control is a force centered on the point r with width ∆. Including Kelvin-Voigt and air damping with parameters cd I and cv respectively leads to the partial differential equation (PDE)

Theorem 5: Consider the control system (4). Assume that for any r ∈ ΩM ,

w(0,t) = 0, w0 (0,t) = 0, w00 (L,t) = 0, w000 (L,t) = 0.

r

lim kB2 (s) − B2 (r)k = 0, s→r

and that either (i) C is a compact operator or (ii) A generates an analytic semigroup and (αI − A)−1 is compact. Assume also that (A1)-(A3) are satisfied for each (An , B2n (r),Cn ). Consider the problem of minimizing the H2 -cost over all possible disturbance functions inf sup hb1 , Π(r)b1 i.

r∈ΩM b1 ∈H

Let rˆ be the optimal actuator location for (A, B2 (r),C) with optimal cost µ and define similarly µn , rˆn . It follows that • µ = limn→∞ µn , • the sequence of approximating actuator locations rˆm has a convergent subsequence and any convergent subsequence has the property that µ = limm→∞ kΠ(ˆrm )k; and also the corresponding controllers converge. Proof: The assumptions imply that kΠn (r) − Π(r)k → 0 for each r [22, Thm. 3.3,Thm. 3.4]. The optimal cost is µ = infr∈ΩM kΠ(r)k with a similar expression for µn and [22, Thm. 3.5] implies that µ = limn→∞ µn . Since {ˆrn } is contained in a compact set, it has a convergent subsequence rˆm → r¯, r¯ ∈ ΩM . Now, kΠ(ˆrm )k ≤ kΠ(ˆrm ) − Π(¯r)k + kΠ(¯r) − Πm (ˆrm )k + kΠm (ˆrm )k. The first term converges to zero because Π(r) is a continuous function of r [22, Thm. 2.6]; the second to zero by the argument used in the proof of Theorem 4 and the third term is µm . Thus, limm→∞ kΠ(ˆrm )k ≤ µ and so limm→∞ kΠ(ˆrm )k = µ. Controller convergence follows as in the previous theorem.  Compactness of C, in addition to assumptions (A1)-(A3) and continuity of B2 (r) ensures that Πn (r) converges to Π(r) in norm for each r [22, Thm. 3.3]. If A generates an analytic semigroup, uniform convergence of Πn (r) can be obtained without compactness of the state weight C [40, Thm. 4.1.4.1]; see also [22, Thm. 3.4]. IV. E XAMPLE : F LEXIBLE BEAM Consider an Euler-Bernoulli model for a thin beam of length L fixed at one end and free at the other. (This is known as a cantilevered beam.) Let w(ξ,t) denote the deflection of the beam from its rigid body motion at time t and position ξ. The deflection is controlled by applying a force u(t) at a spatial

ρ

∂w ∂4 w ∂5 w ∂2 w + cv + EI 4 = br u(t) + b1 d(t), + cd I 2 4 ∂t ∂t ∂t∂ξ ∂ξ

for t ≥ 0, 0 < ξ < L, where, letting ∆ indicate the width of the actuator and r its location, br (ξ) = 1/∆ for |r − ξ| < ∆2 and br (ξ) = 0, otherwise. The spatial part of the disturbance is some b1 ∈ L2 (0, L). The boundary conditions are

Beam parameters were set to ρ = 0.093 Kg/m, cd I = 6.49 × 10−5 s N m2 , cv = 0.0013 s N m2 EI = 0.491 N m2 , L = 0.4573 m and ∆ = 0.01L for the computer simulations. Letting V = {w ∈ H 2 (0, L); w(0) = 0, w0 (0) = 0},

H = L2 (0, L),

where H 2 (0, L) = {φ ∈ L2 (0, L) : φ0 , φ00 ∈ L2 (0, L)}, and V has the norm inherited from H 2 (0, L), define the stiffness and damping operators K ∈ L (V,V 0 ), D ∈ L (V,V 0 ) by hKφ, ψi = hDφ, ψi =

Z L Z0 0

L

EIφ00 (ξ)ψ00 (ξ) dξ, cD Iφ00 (ξ)ψ00 (ξ) dξ + cv

Z L

φ(ξ)ψ(ξ) dξ. 0

For each φ ∈ V, Kφ defines an element of V 0 . If φ ∈ H 4 (0, L) then integrating by parts can be used to identify Kφ with an element of L2 (0, L). Similar behaviour holds for D. Defining Be2 (r)u(t) = br u(t), Be1 d(t) = b1 d(t), the PDE can be written w(t) ¨ + Dw(t) ˙ + Kw(t) = Be2 (r)u(t) + Be1 d(t). To write this in a first-order state space formulation define the state-space H = V × H with state z(t) = (w(·,t), ∂t∂ w(·,t)) and define the operators " # " # " # 0 0 0 I B1 = , B2 (r) = , A= −K −D Be1 Be2 (r) with D(A) = {(φ, ψ) ∈ V ×V : Kφ+Dψ ∈ L2 (0, L), φ00 (L) = φ000 (L) = 0}. It is well-known that A with domain D(A) generates an exponentially stable analytic semigroup on H . A state-space formulation of the above PDE problem is z˙(t) = Az(t) + B1 d(t) + B2 (r)u(t). There is only one control so the control weight is chosen to be 1; that is   0 E= . 1 We will consider state weights of the form.   Co C= , 0

(16)

7

where Co ∈ L (V,Y ). An obvious choice is Co = I. If Co = I, due to the norm on the state-space H , 2

ky(t)kH

=

(|w(t)|V2

=



w(t) w(t) ˙ u(t)



K

 0 0

5 b (ξ)

4 3

3

2

2

0

  w(t)  0 0   ˙ . I 0   w(t) 0 1 u(t)

1 0

0.2

0.4

0.6

0.8

1

5

M w(t) ¨ + Dw(t) ˙ + Kw(t) = B2 (r)u(t) + B1 d(t),

0

0.2

0.4

0.6

0.8

1

b14(ξ)

4

3

3

2

2

1

If the controlled output is the tip displacement, scaled, then defining CL z(t) = w(L,t), the state weight Co = CL in (16). Since CL acts on V ⊂ H 2 (0, L), Sobolev’s Inequality implies that it is a bounded operator. Since the semigroup generated by A is exponentially stable, and B2 (r) has finite rank, the assumptions of Theorem 3 are satisfied. Thus, the cost kΠ(r)k depends continuously on the actuator location and there exists an optimal actuator location for the problem of an unknown disturbance location and also for a fixed disturbance location. Since a closed form solution to the partial differential equation problem is not available, the optimal actuator location must be calculated using an approximation. We use a Galerkin method with cubic splines φi (ξ) as the basis functions [41] and corresponding approximation w(t, ξ) ≈ ∑ni=1 wi (t)φi (ξ). This leads to the system of equations [1]

0

5 b13(ξ)

4

0

b12(ξ)

4

11

1

+ |w(t)| ˙ 2H ) + u(t)2 

5

1 0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

0.6

0.8

1

0.4 0.6 0.8 Spatial variable ξ/L

1

4 1.5 2 1

0 −2 −4

0

0.2

0.4 0.6 0.8 Spatial variable ξ/L

b16(ξ)

0.5

b15(ξ) 1

0

0

0.2

Fig. 1. Normalized spatial disturbance distributions. The mark ◦ denotes the maximum value of each b1 j (ξ), j = 1, . . . , 6.

β1 = 1.87510407, s1 = 0.7341, β2 = 4.69409113, s2 = 1.0185, h  2 i 1 L b11 (ξ) = √2πσ exp − 21 ξ−µ , , µ = L4 , σ = 12 σ h  2 i 1 b12 (ξ) = √2πσ exp − 21 ξ−µ , µ = L2 , σ = L8 , σ h  2 i L 1 exp − 21 ξ−µ , µ = 3L b13 (ξ) = √2πσ σ 4 , σ = 12 , b14 (ξ) = cosh(β1 ξ) − cos(β1 ξ) − s1 (sinh(β1 ξ) − sin(β1 ξ)) , b15 (ξ) = cosh(β2 ξ) − cos(β2 ξ) − s2 (sinh(β2 ξ) − sin(β2 ξ)) ,

T



for w = w1 (t), . . . wn (t) , where the mass, damping, stiffness, control and disturbance influence matrices are

Z L

[B1 ]i =

Z L0

Mi j = Di j =

Z 0L 0

1 ∆ Z

b1 (ξ)φi (ξ) dξ, [B2 (r)]i =

Z r+∆/2 r−∆/2

L

ρ(ξ)φi (ξ)φ j (ξ) dξ, Ki j = 00

00

0

φi (ξ) dξ, 00

00

EI(ξ)φi (ξ)φ j (ξ) dξ,

cd Iφi (ξ)φ j (ξ) + cv φi (ξ)φ j (ξ) dξ,

i, j = 1, . . . , n.

This approximation scheme satisfies the assumptions of [22, Thm. 3.3] and so the approximating Riccati operators converge uniformly. Theorems 4 and 5 imply that use of the approximations in simulations will lead to predicted optimal cost and actuator locations that converge to the true optimal cost and location for both fixed and unknown disturbance locations respectively.

A. Simulation results The optimal actuator location was calculated for six different disturbance functions b1 , all normalized so that kb1 kL2 = 1. The first three are Gaussian distributions centered at different parts of the spatial domain. The next two are the first and second modes e.g. [42], [43], [44] of the beam. Finally, b16 (ξ) is a constant function tapered at the left endpoint. The various functions are shown in Figure 1, and are defined as, letting

b16 (ξ) = 1. The response of the beam was approximated using 80 cubic splines. Tables I and II show the optimal actuator location for each disturbance b1 j , for Co = I and Co = CL respectively. Figure 3 depicts the H2 -norm of the closed-loop transfer function from each disturbance to the entire state, denoted by Tzd (s; ξ), for each disturbance b1 j for both Co = I and Co = CL . The location of the actuators that lead to minimum cost hb1 , Π(r)b1 i are marked by ◦ and shown in Tables I, II. The best actuator location is not always at the point where b1 j (ξ) is maximized and that the best actuator location for a given b1 j depends on the state weight Co . The optimal actuator locations depend strongly on the spatial distribution of disturbances if Co is a multiple of the identity operator. This dependence is weaker if Co = CL . In this case, the optimal actuator locations are influenced by the weight on the tip displacement; the optimal actuator location is between the maximum of b1 j and beam tip. Next, the optimal control problem with Co = CL was solved with the actuator placed at points optimal for various disturbance functions: r = 0.5091L, 0.9950L and r = 0.4475L. The spatial distributions representing the worst spatial distributions (corresponding to the largest eigenvector of the Riccati matrix Π(r)) are shown in Figure 2. If each of these worst spatial distributions were regarded a priori as known, then the actuator location optimization problem using the metric hb1 , Π(r)b1 i would lead to the above three actuator locations as optimal. Figure 4 depicts the norm kΠ(ξ)k for the two cases of the weight Co . If the weight is the entire state, Co = I, the cost is

8

distribution b11 (ξ) b12 (ξ) b13 (ξ) b14 (ξ) b15 (ξ) b16 (ξ)

0.5 b1(ξ) corresponding to ξ=0.5091L

0 −0.5 −1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

max of b1 j (ξ) 0.25L 0.50L 0.75L 1L 0.471L [0,L]

opt. location 0.2650L 0.4960L 0.7491L 0.7519L 0.4701L 0.6546L

TABLE I S PATIAL DISTRIBUTION OF DISTURBANCES AND CORRESPONDING OPTIMAL ACTUATOR LOCATION WITH Co = I

0.5 0 −0.5 b1(ξ) corresponding to ξ=0.9950L

−1 −1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5

distribution b11 (ξ) b12 (ξ) b13 (ξ) b14 (ξ) b15 (ξ) b16 (ξ)

b1(ξ) corresponding to ξ=0.4475L

0 −0.5 −1

0

0.1

0.2

0.3

0.4 0.5 0.6 Spatial variable ξ/L

0.7

0.8

0.9

1

Fig. 2. Spatial distributions corresponding to largest eigenvector of Π(r).

||Tzd(s;ξ)||2 vs ξ

max of b1 j (ξ) 0.25L 0.50L 0.75L 1L 0.471L [0,L]

opt. location 0.5091L 0.9950L 0.9950L 0.9950L 0.4475L 0.9950L

TABLE II S PATIAL DISTRIBUTION OF DISTURBANCES AND CORRESPONDING OPTIMAL ACTUATOR LOCATION WITH Co = CL

||Tzd(s;ξ)||2 vs ξ

0.03

0.2 Normalized optimal cost ||Π(ξ)|| 1.001

0.02 0.1 0.01 0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.1

0.2

0

0.2

0.4

0.6

0.8

1

1

0.999

0.998

0

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

0.6

0.8

1 0.997

0.04

0.4

0.02

0.2

0

0

0.2

0.4 0.6 0.8 Spatial variable ξ/L

1

0

0.996 optimal at ξ=0.0115 L

0

0.2

0.4 0.6 0.8 Spatial variable ξ/L

1 0.995

(a) H2 -norm of ξ-parameterized Tzd (s; ξ) when Co = I ||Tzd(s;ξ)||2 vs ξ

−5

4

x 10

x 10

0.1

0.2

0.3

0.4 0.5 0.6 Spatial variable ξ/L

0.7

0.8

0.9

1

(a) Normalized optimal cost vs ξ/L with C = 103 I.

||Tzd(s;ξ)||2 vs ξ

−4

6

0

Normalized optimal cost ||Π(ξ)|| 1

4 2 0.9

2 0

0

0.2

0.4

0.6

0.8

1

0

−3

1

0

0.2

0.4

0.6

0.8

1

−3

x 10

1.5

0.8

0.7

x 10

1

0.6

0.5

0.5

0.5

0

0

0.2

0.4

0.6

0.8

1

0

−5

3

0

0.2

0.4

0.6

0.8

1

0.4

−3

x 10

1

x 10

0.3

2

0.2

0.5 1 0.1

0

0

0.2

0.4 0.6 0.8 Spatial variable ξ/L

1

0

0

0.2

0.4 0.6 0.8 Spatial variable ξ/L

(b) H2 -norm of ξ-parameterized Tzd (s; ξ) when Co = CL

optimal at ξ=0.9950L

1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 Spatial variable ξ/L

0.7

0.8

0.9

(b) Normalized optimal cost vs ξ/L with C = CL .

Fig. 3. Optimal costs vs ξ/L . Fig. 4. Optimal cost vs ξ/L for different output operators.

1

9

Evolution of beam energy norm 10 non−optimal optimal 9

8

7

6

5

4

3

2

1

0

0

0.1

0.2

0.3 Time (sec)

0.4

0.5

0.6

(a) L2 state norm of the closed loop system.

Fig. 6. Region Ω for the diffusion problem: with origin referenced at bottom left corner, a 4 × 4 units square where a circle of radius 0.4 units centered at (3, 1) is removed.

Evolution of beam tip displacement 0.5 non−optimal optimal 0.4

0.3

on a two-dimensional irregular shape Ω shown in Figure 6. Assume Dirichlet boundary conditions at the edges ∂Ω. Let z(x, y,t) denote the temperature at the point (x, y) ∈ Ω at time t. The temperature is controlled by applying a heat source u(t) on a square-shaped actuator patch. The disturbance v(t) is distributed spatially as d(x, y)v(t). This leads to the model

0.2

0.1

0

−0.1

−0.2

∂z ∂t (x, y,t)

= ∇ · (κ(x, y)∇z(x, y,t)) + b(x, y)u(t) + d(x, y)v(t), = 0, (x, y) ∈ ∂Ω, (18) where, letting (rx , ry , ε) indicate a square centered at (rx , ry ) 1 with side length 2ε, b(x, y) = 2ε , if (x, y) ∈ (rx , ry , ε) and b(x, y) = 0 otherwise. The disturbance d(x, y) is assumed to 1 be of a similar form as the actuator, that is d(x, y) = 10 2ε if (x, y) ∈ (dx , dy , ε) and d(x, y) = 0 otherwise. The state-space formulation of (18) on H = L2 (Ω) is

−0.3

z(x, y,t)

−0.4

−0.5

0

0.1

0.2

0.3 Time (sec)

0.4

0.5

0.6

(b) Beam tip displacement of the closed loop system. Fig. 5. Evolution of state L2 norm and beam tip displacement.

almost independent of the actuator location. If the weight is the tip position CL , the optimal cost is weakly dependent on actuator location, as long as r > 0.2L. The minimum occurs at 0.995L. Finally, Figure 5 shows the time evolution of the L2 state norm and beam tip displacement of the closed loop system. The system was simulated in the time interval [0, 0.6]s using a 4th order Runge-Kutta scheme (ode23s in the Matlab ode library), with disturbance 300b15 (ξ). The optimal case has the actuator placed at r = 0.4701L, and the non-optimal case used an actuator placed at r = 0.7162L. As illustrated in Figure 3(a), this corresponds to the worst actuator location for the disturbance b15 (ξ). The temporal component of the disturbance was a normal distribution with zero mean and unit variance in both the optimal and non-optimal cases. The performance is far superior with the actuator optimally placed than with the actuator placed at a non-optimal location. V. E XAMPLE : D IFFUSION Consider the heat diffusion problem with variable diffusivity coefficient 2 −(2−y)2

κ(x, y) = 3(3 − x)2 e−(2−x)

+ 0.01

(17)

z˙(t) = Az(t) + B(r)u(t) + Dv(t), where Ah := ∇ · (κ∇h), B(r)u := b(x, y)u, Dv := d(x, y)v, with D(A) = {h ∈ H 2 (Ω) | h = 0 on ∂Ω}. A finite element method (FEM) [45] with linear splines {φi } as the basis for the finite-dimensional subspace is used for approximating (18). A FEM used for approximating (18) using the standard Galerkin approximation yields finite-dimensional approximating systems. The system (18) is both stabilizable and detectable and so are the approximating problems; see [46]. Two performance costs are compared: (1) cost based on the entire state z ∈ L2 (Ω) leads to, defining C1 z := z,     C1 0 y1 (t) = z(t) + u(t), 0 1 and (2) cost based on average temperature over the whole doR R main z(x, y) dΩ, which leads to, defining C2 z := z(x, y) dΩ, Ω

Ω

    C 0 y2 (t) = 2 z(t) + u(t). 0 1

10

Theorems 4 and 5 imply that use of the approximations in simulations will lead to predicted optimal cost and actuator locations that converge to the true optimal cost and location for both fixed and unknown disturbance locations respectively. Since there is a single control signal, the control weight is chosen to be 1. In the approximation, 479 elements were used and the half-width of the actuator is ε = 0.2. The cost functions for each case using full state cost y1 (t) have a clear global minimum. As an example, the cost function for the case with disturbance at (3, 3) is shown in Figure 7.

4 5 3.5 4.5

D1 & A1

3

4 3.5

2.5

A4 3

D4

2

A3 D3

1.5

2.5 2 1.5

1

1 0.5

800

D2 & A2

0.5

700

0

600

0

0.5

1

1.5

2

2.5

3

3.5

4

cost

500

Fig. 8. Disturbance location and resulting optimal actuator location using state measurement y1 (t).

400 300 200

4 3

100 0

2

1 2

Similar qualitative behaviour was obtained with the other 3 disturbance locations discussed above.

1 3 4

0

y

x 1100

Fig. 7. Cost function with disturbance at (3, 3) using state measurement y1 (t).

1000 900

With this modified measurement and disturbance D4, the optimal actuator location is found to be at the disturbance location (1, 2). This suggests that if the state weighting is large enough, the optimal actuator location will coincide with the disturbance location. However, the weighting on the state for this to occur is affected by the diffusivity. Now, consider average temperature measurement y2 (t). The cost function with the disturbance at (3, 3) is shown in Figure 9. Note that there is no global minimum and there are many points with cost value close to the minimum cost.

800 cost

Figure 8 shows the disturbance location and resulting optimal actuator location for four cases using state measurement y1 (t), plotted over a graph of the diffusivity coefficient on the domain Ω. As seen in the figure, when the disturbance is located at (3, 3) (D1 in figure) or at (1.4, 0.4) (D2 in figure), the optimal actuator location (A1 and A2, respectively) coincides with the disturbance location. With disturbance at (2, 1.6) (D3 in figure), the optimal actuator location (A3) is close to the disturbance location, at (2.2, 1.8). With disturbance at (1, 2) (D4 in figure), the optimal actuator location (A4) is further from the disturbance location, at (2, 2.4). The results suggest that the optimal actuator location is the disturbance location when the disturbance is located in a region of low diffusivity, but not so when located in a region of higher diffusivity. However, consider a modified measurement where the weighting on the output matrix C1 is increased,     10C1 0 y˜1 (t) = z(t) + u(t). (19) 0 1

700 600 500 400 300 4 3

4 3

2 2 1 y

1 0

0

x

Fig. 9. Cost function with disturbance at (3, 3) using average temperature measurement y2 (t).

Figure 10 shows the disturbance location and resulting optimal actuator location for the same four disturbance locations as discussed above, plotted over a graph of the diffusivity coefficient. As seen in the figure, the optimal actuator location is found to be (3.6, 3.6) (A1 in figure) when the disturbance is at (3, 3) (D1 in figure), and (3.6, 0.4) for the other three cases. Using a modified measurement     10C2 0 y˜2 (t) = z(t) + u(t), (20) 0 1 the optimal actuator location becomes (3.6, 0.4) for all cases. To investigate further the role of the diffusivity in actuator location, consider a model that is identical except that the diffusivity is changed to 1 κ(x, y) = ((x − 2)2 + (y − 3)2 ) + 0.01. 3

(21)

11

4

4 5

A1 3.5

4

D3 A3

3.5 4.5

3.5

D1

3

4

D1 & A1

3

3 3.5 2.5

2.5 2.5

3

D4

2

2 2.5

2

D3 1.5

2

1.5 1.5

1.5 1

1 1 1

0.5

D2

A2 & A3 & A4 0.5

0

A2 D2

0.5

0.5

0 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 10. Disturbance location and resulting optimal actuator location using average temperature measurement y2 (t).

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 11. Disturbance location and resulting optimal actuator location using state measurement y1 (t) for a different diffusivity function (21). 4

Figure 11 shows the disturbance location and resulting optimal actuator location for three new cases using state measurement y1 (t), plotted over a graph of the diffusivity coefficient (21) on the domain Ω. It is again seen that the optimal actuator location coincides with the disturbance location when the disturbance is located in a region of low diffusivity (D1 and A1 in figure). For disturbances at higher diffusivity (D2 and D3), the optimal actuator location (A2 and A3, respectively) is still close to the disturbance location. For average temperature measurement y2 (t) with diffusivity (21), the disturbance and resulting optimal actuator location is shown using the same three cases in Figure 12. The optimal actuator location does not coincide with the disturbance location, with the exception of A1 which is found to be close to D1. The results from this figure as well as Figure 10 appear to suggest that for average temperature measurement, the actuator favours locations in lower diffusivity regions. As a final example, consider constant diffusivity κ(x, y) = 2 over the domain. For full state measurement y1 (t), with disturbance at (1, 3), the optimal actuator location coincides with the disturbance at (1, 3), and with disturbance at (2, 1) the optimal actuator location again is very close to the disturbance, (1.8, 1). For average temperature measurement y2 (t) and disturbance at (1, 3) or (2, 1), the optimal actuator location is found to be (1.6, 2.2) in both cases. VI. C ONCLUSIONS The effects of disturbances, and in particular their spatial distribution, on the choice of optimal actuator locations for a class of infinite dimensional system was examined. Including the effect of a disturbance with known frequency content leads to an optimal H2 -control problem. The solution to the problem is shown to be the solution of an algebraic Riccati equation, as for the finite-dimensional case. Earlier efforts on the use of control performance metrics, predominantly linear quadratic ones, for the optimization of actuator locations were extended

4

D3 3.5

3.5

D1

3

A1 3 2.5 2.5 2 2

A3 A2

1.5

1.5

1 1

0.5

0.5

D2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 12. Disturbance location and resulting optimal actuator location using average temperature measurement y2 (t) for a different diffusivity function (21).

to include the spatial distribution of disturbances. This was made feasible via the use of H2 -control performance metrics for actuator location optimization. Such a performance metric accounted for the spatial distribution of disturbances directly, thus allowing for the actuator location optimization that also provided a robustness against disturbances. In practice, a finite-dimensional approximation is needed to compute the controller, and actuator location. Sufficient conditions under which the approximating controller and optimal locations converge to the exact controller and optimal location are given. The double optimization problem of finding the worst disturbance distribution and the associated optimal actuator location was also examined. When the worst spatial distribution is unknown, the cost function is the norm of the Riccati operator. The worst disturbance turns out to be the eigenfunction of the

12

Riccati operator corresponding to the largest eigenvalue. Thus, the optimal actuator location was identical to that found using a linear-quadratic cost [22]. Various examples showed the effects of different spatial disturbances on the optimal actuator location. It is clear that knowledge of the spatial distribution of disturbances should be used in the locating actuators as in general this affects the actuator location and hence system performance. Furthermore, other system parameters such as diffusivity also have an effect on the best actuator location. The dual problem of sensor placement and the associated state estimation problem, as well as the joint sensor-actuator location optimization are currently being examined. R EFERENCES [1] H. T. Banks, R. C. Smith, and Y. Wang, Smart Material Structures: Modeling, Estimation and Control. Wiley, 1996. [2] F. Fahroo, “Optimal location of controls for an acoustic problem,” in Proc. of the 34th IEEE Conference on Decision and Control, vol. 4, December 1995, pp. 3765 –3766. [3] K. A. Morris, “Noise reduction achievable by point control,” ASME Journal on Dynamic Systems, Measurement and Control, vol. 120, no. 2, pp. 216–223, 1998. [4] F. Fahroo and M. A. Demetriou, “Optimal location of sensors and actuators for an active noise control problem,” in Proc. of the American Control Conference, vol. 3, June 1999, pp. 1717 –1721. [5] F. Fahroo and Y. Wang, “Optimal location of piezoceramic actuators for vibration suppression of a flexible structure,” in Proc. of the 36th IEEE Conference on Decision and Control, vol. 2, December 1997, pp. 1966 –1971. [6] M. A. Demetriou and F. Fahroo, “Optimal location of actuators for control of a 2-D structural acoustic model,” in Proc. of the 38th IEEE Conference on Decision and Control, vol. 5, December 1999, pp. 4290 –4295. [7] C. Antoniades and P. D. Christofides, “Integrated optimal actuator/sensor placement and robust control of uncertain transport-reaction processes,” Comp. & Chem. Eng., vol. 26, pp. 187–203, 2002. [8] F. Fahroo and M. A. Demetriou, “Optimal location of sensors and actuators for an active noise control problem,” Journal of Computational and Applied Mathematics, vol. 114, pp. 137–158, 2000. [9] F. Peng, A. Ng, and Y.-R. Hu, “Actuator placement optimization and adaptive vibration control of plate smart structures,” J. Intell. Mat’l Sys. & Struct., vol. 16, pp. 263–271, 2005. [10] M. A. Demetriou and F. Fahroo, “Multi-stage optimization of flexible structures,” in Proc. of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, July 24-28 2006. [11] M. I. Frecker, “Recent advances in optimization of smart structures and actuators,” J. Intell. Mat’l Sys. & Struct., vol. 14:4-5, pp. 207–216, 2003. [12] M. van de Wal and B. de Jager, “A review of methods for input/output selection,” Automatica, vol. 37, pp. 487–510, 2001. [13] V. A. Gupta, M. B. Sharma, and N. Thakur, “Optimization criteria for optimal placement of piezoelectric sensors and actuators on a smart structure: A technical review,” J. Intell. Mat’l Sys. & Struct., vol. 21, no. 12, pp. 1227–1243, 2010. [14] C. S. Kubrusly and H. Malebranche, “Sensors and controllers location in distributed systems-a survey,” Automatica, vol. 21, pp. 117–128, 1985. [15] M. A. Demetriou and J. Borggaard, “Optimization of an integrated actuator placement and robust control scheme for distributed parameter processes subject to worst-case spatial disturbance distribution,” in Proc. of the American Control Conference, vol. 3, June 4-6 2003, pp. 2114 – 2119. [16] W. Gawronski, Dynamics and Control of Structures, A Modal Approach. New York: Springer-Verlag, 1998. [17] J. L. Junkins and Y. Kim, Introduction to Dynamics and Control of Flexible Structures. Washington, DC: AIAA Education Series, 1993. [18] S. Omatu and J. H. Seinfeld, Distributed Parameter Systems: Theory and Applications. New York: Oxford University press, 1989. [19] R. E. Skelton, Dynamic Systems Control: Linear Systems Analysis and Synthesis. New York: John Wiley & Sons, Inc., 1988. [20] A. Armaou and M. A. Demetriou, “Optimal actuator/sensor placement for linear parabolic pdes using spatial norm,” Chemical Engineering Science, vol. 61, no. 22, pp. 7351 – 7367, 2006.

[21] S. R. Moheimani, D. Halim, and A. J. Fleming, Spatial Control of Vibration. World Scientific, 2003. [22] K. A. Morris, “Linear quadratic optimal actuator location,” IEEE Tran. on Automatic Control, vol. 56, pp. 113–124, 2011. [23] N. Darivandi, K. A. Morris, and A. Khajepour, “An algorithm for LQoptimal actuator location,” Smart Materials and Structures, vol. 22, no. 3, p. 035001, 2013. [24] D. Kasinathan and K. A. Morris, “H∞ -optimal actuator location’,” IEEE Tran. Auto. Control, vol. 58, no. 10, pp. 2522–2535, 2013. [25] M. A. Demetriou, “Integrated optimal actuator/sensor placement and hybrid controller design of flexible structures under worst case spartiotemporal disturbance variations,” J. Intell. Mat’l Sys. & Struct., vol. 15, pp. 901–921, 2004. [26] P. Ambrosio, F. Resta, and F. Ripamonti, “An H2 -norm approach for the actuator and sensor placement in vibration control of a smart structure,” Smart Materials and Structures, vol. 21, no. 12, 2012. [27] W. Liu, Z. Hou, and M. A. Demetriou, “A computational scheme for the optimal sensor/actuator placement of flexible structures using spatial H2 measures,” Mechanical Systems and Signal Processing, vol. 20, no. 4, pp. 881 – 895, 2006. [28] M. A. Demetriou and J. Borggaard, “Optimization of a joint sensor placement and robust estimation scheme for distributed parameter processes subject to worst case spatial disturbance distributions,” in Proc. of the American Control Conference, vol. 3, June 30-July 2 2004, pp. 2239 –2244. [29] ——, “Design of worst spatial distribution of disturbances for a class of parabolic partial differential equations,” in Proc. of the American Control Conference, vol. 6, June 8-10 2005, pp. 3894 – 3899. [30] M. A. Demetriou and K. A. Morris, “Using H2 -control metrics for the optimal actuator location of infinite-dimensional systems,” in Proc. of the American Control Conference, 2010. [31] R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Berlin: Springer Verlag, 1995. [32] W. S. Levine and M. Athans, “On the determination of the optimal constant output feedback gains for linear multivariable systems,” IEEE Tran. Auto. Control, vol. AC-15, no. 1, pp. 44–48, 1970. [33] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1996. [34] P. C. Chandrasekharan, Robust Control. Academic Press, 1996. [35] M. Tucsnak and G. Weiss, Observation and control for operator semigroups, ser. Birkh¨auser Advanced Texts: Basler Lehrb¨ucher. [Birkh¨auser Advanced Texts: Basel Textbooks]. Basel: Birkh¨auser Verlag, 2009. [36] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. Francis, “State-space solutions to standard H2 and H∞ control problems,” IEEE Tran. Auto. Control, vol. 34, no. 8, pp. 831–847, 1989. [37] K. A. Morris, “Control of systems governed by partial differential equations,” in ed. W.S. Levine, The Control Handbook, Second Edition, Control System Advanced Methods. CRC Press, 2010. [38] H. T. Banks and K. Kunisch, “The linear regulator problem for parabolic systems,” SIAM J. Control and Optim., vol. 22(5), pp. 684–698, 1984. [39] K. Ito, “Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces,” in Distributed Parameter Systems, W. Schappacher, F. Kappel, and K. Kunisch, Eds. Springer-Verlag, 1987. [40] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press, 2000, vol. I. [41] P. M. Prenter, Splines and Variational Methods. New York: Wiley, 1975. [42] R. D. Blevins, Formulas for Natural Frequency and Mode Shape. Malabar, Fla: Krieger Publishing Company, 1979. [43] D. J. Inman, Vibration with Control, Measurement and Stability. Englewood Cliffs, N.J.: Prentice Hall, 1989. [44] A. Leissa, Vibration of Plates. The Acoustical Society of America through the American Institute of Physics, 1993. [45] J. Alberty, C. Carstensen, and S. Funken, “Remarks around 50 lines of Matlab: short finite element implementation,” Numerical Algorithms, vol. 20, pp. 117–137, 1999. [46] K. A. Morris, “Design of finite-dimensional controllers for infinitedimensional systems by approximation,” Jour. of Mathematical Systems, Estimation and Control, vol. 4, no. 1-30, 1994.

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Kirsten Morris ’ research interests are systems modelled by partial differential equations and also systems, such as smart materials, involving hysteresis. She has also written an undergraduate textbook ”Introduction to Feedback Control”, and was editor of the book ?Control of Flexible Structures?. Professor Morris is a member of the Applied Mathematics Department at the University of Waterloo and is cross-appointed to the Department of Mechanical & Mechatronics Engineering. From 2005-2008 she was Associate Dean for Graduate Studies & Research in the Faculty of Mathematics. She was an associate editor with the IEEE Transactions on Automatic Control during 2007-2012, SIAM Journal on Control & Optimization 2010-2012 and has been a member of the editorial board of the SIAM book series Advances in Design & Control since 2005. Prof. Morris has served on IEEE Control System Society Board of Governors since 2010 and was vice-president, membership, 2013-2014.

Michael A. Demetriou , Student Member 1989, Member 1993, Senior Member 2002. He is currently with the Department of Mechanical Engineering at Worcester Polytechnic Institute in Worcester, Massachusetts. He received his B.S. in Mechanical Engineering in 1987, his M.S. in Applied Mathematics in 1989 and his Ph.D. in Electrical EngineeringSystems in 1993, all from the University of Southern California. He spent three years as a visiting Assistant Professor with the Center for Research in Scientific Computation at NCSU. From 2004-2007 he served as an Associate Editor for the IEEE Transactions on Automatic Control and from 2009-2011 served as an Associate Editor for ASME Journal of Dynamic Systems, Measurement, and Control. He is currently finishing a five-year term as an Associate Editor for SIAM J. Control and Optimization. He established the IEEE-CSS Technical Committee on Distributed Parameter Systems in 2003 which he chaired till 2012. His current research interests include optimization and control of mobile sensor and actuator networks in spatially distributed systems, hybrid control of distributed parameter systems, adaptive estimation and control of infinite dimensional systems, fault detection, diagnosis, and control of distributed and lumped parameter systems with application to structural, acoustic-structure interaction systems and thermal manufacturing systems.

Steven Yang received the B.A.Sc. degree in computer engineering from the University of Waterloo, Waterloo, Canada in 2013. He has held research internships at the Department of Applied Mathematics, University of Waterloo, working in the areas of control theory and numerical computation for partial differential equations. He is currently working towards a master’s degree in mathematics at Queen’s University, Kingston, Canada, with a focus in geometric control theory.