An algorithm for LQ optimal actuator location

In distributed parameter systems such as structures, heating processes and active noise control, there is generally choice in the placement of control hardware such as the sensors and actuators. Proper placement of sensors and actuators is essential in designing an active control system. Poor placement of sensors and actuators leads to poor performance of the controlled system. For simulation and controller design, distributed parameter systems are approximated, for instance by finite elements in calculations. For complex systems in several space variables a large number of elements is required for accuracy. As the number of finite elements increases, calculation of the optimal actuator placement can be computationally costly. Researchers have used a number of optimization criteria and also techniques to find the optimal location of actuators. Various optimization criteria exist in the literature. Many authors optimized actuator locations based on maximization of observability and controllability in active control [23, 34, 35, 40]. A number of researchers focused on minimizing the linear quadratic regulator cost [8, 20, 21, 30]. In [11], the authors located piezoelectric actuators to maximize the harvested strain energy in piezoceramic materials. The objective in [26] was to minimize the vibration amplitude. In [4], actuators and sensors are located in buildings based on two objective functions, the number of sensors and actuators and the interstory drifts. In [38] the optimization criteria was to maximize the closed-loop damping ratio. These criteria are well documented in the survey papers [14, 17, 33]. A numerical challenge is that the resulting optimization problem is generally non-convex.

Most researchers have used evolutionary optimization techniques. In [2, 3, 9, 13, 20, 36], the authors used genetic algorithms. Some studies have applied particle swarm methods [10, 27, 37]. In [24, 25] invasive weed optimization technique is applied, which is a new numerical stochastic technique and is inspired from the colonizing behaviour of weeds. One drawback of these methods is that they do not use gradient information and so convergence can be very slow. This is particularly a problem for control systems, such as structures where a large number of state variables are needed for accuracy, since genetic algorithms have a dramatic increase in computation time as the number of design variables increase. This is due largely to the fact that the gradient of the cost is not used in the optimization. Classical gradient-based techniques, on the other hand, have the advantage of accurate computation. However, they may be computationally expensive, particularly since multiple initial conditions are typically needed to ensure that a global optimum is found [16, 5, 18]. Consequently, a fast, yet accurate optimization method applicable to systems with a large number of degrees of freedom is needed.

In this study, we consider a linear quadratic cost which is a very popular controller design approach for multi-input systems. Since the optimal linear quadratic cost is dependent on the initial conditions, the initial condition to be considered should also be chosen. The response to the worst initial condition (in terms of effect on the cost) is minimized. Since the solution to the optimal linear quadratic problem with a particular actuator location is found by solving an algebraic Riccati equation (ARE) for a matrix, this means that the goal is to minimize the norm of a matrix P over possible actuator locations [28]. In [15] it was shown that by mapping actuator locations into zero–one vectors and projecting the solution of the Riccati equation, P, to a space of design parameters, the cost function becomes convex in the new topology. Geromel also suggested an algorithm to solve the optimal actuator location problem; however, details of implementation were not provided. Developing Geromel's scheme into an algorithm applicable for multiple actuators and high-order systems is the purpose of this paper. The algorithm is described in section 3.

This method is applied to optimal placement of piezoelectric actuator patches in vibration control of beam and plate structures in section 4. The results are compared with a genetic algorithm. The developed algorithm is faster and more accurate than a genetic algorithm. In addition, to verify the presented optimization method, experimental verification of the algorithm using a beam is presented in section 5. It is shown that optimally located actuators are much more effective in suppressing initial disturbances than actuators placed at other locations.

Consider a linear time-invariant system,

$$\begin{eqnarray} &&\dot {x}(t)=A x(t)+B u(t),\tqs x(0)={x}_{0}. \end{eqnarray} \tag{ 1 }$$

The control u(t) is chosen to minimize a quadratic performance index

$$\begin{eqnarray} &&J=\frac{1}{2}\int \nolimits _{0}^{\infty }(x(t)^{\prime}Q x(t)+u(t)^{\prime}R u(t))\hspace{0.167em} \mathrm{d}t, \end{eqnarray} \tag{ 2 }$$

where Q and R are positive semi-definite and positive definite weighting matrices, respectively. For stabilizable (A,B) and detectable (Q^1/2,A), the optimal cost is [29]

$$\begin{eqnarray} &&{J}_{\mathrm{opt}}=\frac{1}{2}{x}_{0}^{\prime}P{x}_{0}, \end{eqnarray} \tag{ 3 }$$

where P is the unique positive semi-definite solution of the algebraic Riccati equation (ARE)

$$\begin{eqnarray} &&{A}^{\prime}P+P A-P B{R}^{-1}{B}^{\prime}P+Q=0. \end{eqnarray} \tag{ 4 }$$

The optimal control is

$$\begin{eqnarray} &&u(t)=-{R}^{-1}{B}^{\prime}P x(t). \end{eqnarray} \tag{ 5 }$$

The optimal cost (3) depends on the initial condition. This dependence can be handled in several ways, depending on the application. Most commonly, the initial condition that has the worst effect on the cost or else a random initial condition is considered. In this study, we minimize the actuator location by considering the worst initial condition. The cost is [28]

$$\begin{eqnarray} &&\mathop{\max }\limits _{\Vert {x}_{0}\Vert =1}x_{0}^{\prime}P{x}_{0}=\Vert P\Vert , \end{eqnarray} \tag{ 6 }$$

where, letting λ_i indicate the eigenvalues of P,||P|| = max_iλ_i.

Now consider the situation where there are a number of possible actuator locations. The control operator B in (1) is dependent on the actuator locations. This yields a cost ||P|| that varies with actuator location. This cost function is in general a non-convex function of the actuator locations. However, in [15] the problem is reformulated into a convex optimization problem. The formulation relies on considering a discrete set of N possible actuator locations. In some situations, the number of possible actuator locations is finite due to engineering constraints. In other cases, the region of possible actuator needs to be discretized. Since each actuator occupies a non-zero amount of space, this does not present a practical constraint on possible actuator locations.

Suppose that there are M_N actuators and N possible actuator locations. Define a set of possible control operators B_j by considering a single actuator at the jth location. Define similarly the control weight R_j by considering one actuator at the jth location. Following the approach in [15], let π be a vector of N logical elements where the jth entry has a 1 when an actuator exists in that location and a value of 0 otherwise. Note that π has exactly M_N non-zero elements so

$$\begin{eqnarray} &&\sum _{j=1}^{N}{\pi }_{j}={M}_{N}. \end{eqnarray} \tag{ 7 }$$

Each such vector π defines a possible set of actuator locations.

The joint actuator placement and control problem is

$$\begin{eqnarray} &&\fl \mathop{\min }\limits _{\pi ,u}J=\mathop{\min }\limits _{\pi ,u}\frac{1}{2}\int \nolimits _{0}^{\infty }\left \{ {x}^{\prime}Q x+\sum _{j=1}^{N}{u}_{j}^{\prime}{\pi }_{j}{R}_{j}{u}_{j}\right \} \mathrm{d}t\nonumber\\ &&\text{s.t.}\hspace{0.167em} \dot {x}=A x+\sum _{j=1}^{N}{\pi }_{j}{B}_{j}{u}_{j}. \end{eqnarray} \tag{ 8 }$$

For each possible set of actuator locations, π, there is a different cost, obtained by solving the ARE (4) with the appropriate definition of the current B and R, and so the objective function is

$$\begin{eqnarray} &&\sigma (\pi )=\Vert P(\pi )\Vert . \end{eqnarray} \tag{ 9 }$$

The optimization problem is

$$\begin{eqnarray} &&\fl \min \{ \sigma (\pi );\pi \in \Phi \} ,\text{ subject to}\nonumber\\ &&\Phi =\left \{ \pi \in {\mathbb{R}}^{N}\text{ s.t. }{\pi }_{j}\in \{ 0,1\} ;\sum _{j=1}^{N}{\pi }_{j}={M}_{N}\right \} , \end{eqnarray} \tag{ 10 }$$

where M_N indicates the number of actuators.

Theorem  Define the convex set Φ_c = {π∈R^N s.t. π_j ≥ 0}. The function σ(π):Φ_c → R is a convex function. For π₀∈Φ_c the following defines a subgradient μ of σ(π₀)

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \mu ({\pi }_{0})=[{\mu }_{1}({\pi }_{0})\cdots {\mu }_{N}({\pi }_{0})],\\ \displaystyle {\mu }_{j}({\pi }_{0})=\mathrm{Tr}\{ {L}_{j}S({\pi }_{0})\} \\ \displaystyle {L}_{j}={B}_{j}{R}_{j}^{-1}{B}_{j}^{\prime}\tqs j=1,\ldots ,N\\ \displaystyle S({\pi }_{0})=-\frac{1}{2}P({\pi }_{0})\theta ({\pi }_{0})P({\pi }_{0}) \end{array} \end{eqnarray} \tag{ 11 }$$

where, letting z indicate the normalized eigenvector associated with the maximum eigenvalue of P(π₀),θ(π₀) is the solution of the Lyapunov equation

$$\begin{eqnarray} &&\fl (A-B K({\pi }_{0}))\theta ({\pi }_{0})+\theta ({\pi }_{0})(A-B K({\pi }_{0}))^{\prime}+z{z}^{\prime}=0. \end{eqnarray} \tag{ 12 }$$

Since the problem is now reformulated as a convex problem, any gradient-based optimization method will converge to a global minimum.

Since σ(π) is a convex function of π,

$$\begin{eqnarray} &&\fl \sigma (\pi )-\sigma ({\pi }_{0})\geq \langle \mu ({\pi }_{0}),\pi -{\pi }_{0}\rangle \tqs \text{for all }\pi \in \Phi \end{eqnarray} \tag{ 13 }$$

where μ is defined in (11) and 〈μ(π₀),π − π₀〉 denotes the usual inner product of μ(π₀) and π − π₀. To optimize the actuator locations, we should look for the smallest σ(π) that satisfies (13). In [15], the following optimization scheme is proposed. Equation (13) is rewritten as

$$\begin{eqnarray} &&\sigma (\pi )\geq \sigma ({\pi }_{0})+\langle \mu ({\pi }_{0}),\pi -{\pi }_{0}\rangle . \end{eqnarray} \tag{ 14 }$$

To solve the optimization, σ(π) in (14) can be replaced with θ that can take any real value and the optimization problem is written as

$$\begin{eqnarray} &&\mathop{\min }\limits _{\pi }\theta \hspace{0.167em} \text{s.t. }~\theta \geq \sigma ({\pi }_{0})+\langle \mu ({\pi }_{0}),\pi -{\pi }_{0}\rangle . \end{eqnarray} \tag{ 15 }$$

Letting π* indicate the optimizer of (15), θ(π*) is not restricted to be equal to σ(π*). Consequently, (15) is called a relaxation of the original optimization problem [32]. Since (15) is a linear optimization problem the solution falls on the boundary of the inequality constraint and

$$\begin{eqnarray} &&{\theta }^{\ast }=\mathop{\min }\limits _{\pi }\sigma ({\pi }_{0})+\langle \mu ({\pi }_{0}),\pi -{\pi }_{0}\rangle . \end{eqnarray} \tag{ 16 }$$

If the solution of this relaxed problem, π*, has an objective value θ* which equals σ(π*) then

$$\begin{eqnarray} &&\sigma ({\pi }^{\ast })=\sigma ({\pi }_{0})+\langle \mu ({\pi }_{0}),{\pi }^{\ast }-{\pi }_{0}\rangle . \end{eqnarray} \tag{ 17 }$$

Since σ is a convex function of π, for all π∈Φ,π ≠ π*,

$$\begin{eqnarray} &&\sigma (\pi )\gt \sigma ({\pi }_{0})+\langle \mu ({\pi }_{0}),{\pi }^{\ast }-{\pi }_{0}\rangle \end{eqnarray} \tag{ 18 }$$

and so for all π ≠ π*

$$\begin{eqnarray} &&\sigma (\pi )\gt \sigma ({\pi }^{\ast }) \end{eqnarray} \tag{ 19 }$$

which means that σ(π*) is the global minimum for the actuator location problem. If σ(π*) ≠ θ*, then define σ₁ = σ(π*) and μ₁ = μ(π*), and add another constraint

$$\begin{eqnarray} &&\theta \geq \sigma ({\pi }_{1})+\langle \mu ({\pi }_{1}),\pi -{\pi }_{1}\rangle \end{eqnarray} \tag{ 20 }$$

to (15). Continuing this way, a series of linear optimization problems

$$\begin{eqnarray} &&\fl \mathop{\min }\limits _{\pi }\theta \hspace{0.167em} \text{s.t. }\theta \geq \sigma ({\pi }_{i})+\langle \mu ({\pi }_{i}),\pi -{\pi }_{i}\rangle \nonumber\\ &&i=1,\ldots ,n \end{eqnarray} \tag{ 21 }$$

is obtained. Since the optimization problem is linear, θ* at each iteration is on one of the constraint boundaries. From (13),

$$\begin{eqnarray} &&\sigma (\pi )\geq {\sigma }_{i}+\langle {\mu }_{i},\pi -{\pi }_{i}\rangle \tqs \text{for all }\pi \in \Phi . \end{eqnarray} \tag{ 22 }$$

Therefore, in (15)

$$\begin{eqnarray} &&\sigma (\pi )\geq {\theta }^{\ast }\tqs \text{for all }\pi \in \Phi . \end{eqnarray} \tag{ 23 }$$

When θ* = σ(π*), then π* is the global solution of this problem. As the number of constraints for this problem increases, its feasibility area gets smaller, and it will finally converge to the optimal actuator locations.

This algorithm can be summarized as follows.

• (A)  
  Assume π₀∈Φ and calculate σ(π₀) and μ(π₀). Set k = 0 and choose a sufficiently small value ε > 0.
• (B)  
  Choose an initial location for actuators π₀∈Φ and calculate σ(π₀) and μ(π₀).
• (C)  
  Equation (13) can be relaxed as

  $$\begin{eqnarray} &&\fl \mathop{\min }\limits _{\pi \in \Phi }\theta \hspace{0.167em} \text{s.t. }\theta \geq \sigma ({\pi }_{i})+\langle \mu ({\pi }_{i}),\pi -{\pi }_{i}\rangle \nonumber\\ &&i=1,\ldots ,k. \end{eqnarray} \tag{ 24 }$$
• (D)  
  Using π_k+1, calculate σ_k+1. If (σ_k+1 − θ_k+1) ≤ ε, terminate. If not, calculate μ_k+1 and return to step (C).

For a single actuator, the relaxed problem (24) in step (C) can be simply solved, since the linear constraint in this problem is a scalar equation. However, for multiple actuators, solution of the relaxed master problem (24) is challenging, especially for large N and M_N.

The relaxed master problem (24) in step (C) can be written

$$\begin{eqnarray} &&\fl \mathop{\min }\limits _{\pi \in \Phi }\mathop{\max }\limits _{i}\sigma ({\pi }_{i})+\langle \mu ({\pi }_{i}),\pi -{\pi }_{i}\rangle \tqs i=1,\ldots ,k, \end{eqnarray} \tag{ 25 }$$

which minimizes the largest value of a set of functions. Thus, (24) can be written as

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl \mathop{\min }\limits _{\pi \in \Phi }\mathop{\max }\limits _{i}\sigma ({\pi }_{i})-\langle \mu ({\pi }_{i}),{\pi }_{i}\rangle +\langle \mu ({\pi }_{i}),\pi \rangle \\ \displaystyle i=1,\ldots ,k\\ \displaystyle \fl \text{s.t. }\Phi =\left \{ \pi \in {\mathbb{R}}^{\mathbb{N}}\text{ s.t. }{\pi }_{j}\in \{ 0,1\} ;\sum _{j=1}^{N}{\pi }_{j}={M}_{N}\right \} . \end{array} \end{eqnarray} \tag{ 26 }$$

The variable π in this problem is a vector of binary components. Consequently, an integer optimization algorithm is required for the solution. The objective function in (26) is a linear function of π and can be easily calculated in a considerably short time. For such an objective function, a branch and bound optimization method is suitable. Details of branch and bound techniques can be found in [12]. Applying this method, at first all the integer boundaries on the π vector are removed. Consequently, (26) can be relaxed to

$$\begin{eqnarray} &&\fl \mathop{\min }\limits _{\pi \in \Phi }\mathop{\max }\limits _{i}\sigma ({\pi }_{i})-\langle \mu ({\pi }_{i}),{\pi }_{i}\rangle +\langle \mu ({\pi }_{i}),\pi \rangle \nonumber\\ &&i=1,\ldots ,k\nonumber\\ &&\fl \text{s.t.}\Phi =\left \{ \pi \in {\mathbb{R}}^{\mathbb{N}}\text{ s.t. }0\leq {\pi }_{j}\leq 1;\sum _{j=1}^{N}{\pi }_{j}={M}_{N}\right \} . \end{eqnarray} \tag{ 27 }$$

This relaxation of the original problem is then solved. If it does not result in a binary solution, an element of this vector is chosen and equality binary constraints are added on this element and the relaxed problem is solved once again. These added constraints form the branches of a binary tree. Each branch of this tree is called a candidate sub-problem. To avoid enumeration of all candidate sub-problems, fathoming tests should be applied.

The objective function in (27) is an affine function of π, and all the constraints in this problem are linear. Consequently, sequential quadratic programming can be used. In this study, the fminimax code from the MATLAB optimization toolbox is used.

Consider now the problem of optimizing the locations of piezoelectric patch actuators on structures. It is assumed that the actuators are perfectly bonded to the top surface of the structure, as shown in figure 1. The finite element approximation is used to model the base structures, and the actuators are assumed to have the same area as the finite elements.

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Letting σ indicate the stress tensor, f the body force vector, D the electric displacement vector, q the electric charge density, ρ the mass density and w the structural deflection, the electrostatic field in piezoelectric materials is modelled by

$$\begin{eqnarray*} &&\nabla \cdot \sigma +f=\rho \ddot{w},\tqs \nabla \cdot D+q=0. \end{eqnarray*}$$

The first equation is the elasticity equilibrium equation and the second equation is Gauss's law of electrostatics [22]. Moreover, the constitutive equations of piezoelectric materials can be written as

$$\begin{eqnarray} &&\sigma =C s+e \nabla \phi \end{eqnarray} \tag{ 28a }$$

$$\begin{eqnarray} &&D=e S-\epsilon \nabla \phi . \end{eqnarray} \tag{ 28b }$$

In this equation, s is the strain vector, ϕ, is the electric potential, C contains the material elasticity constants (dependent on modulus of elasticity and Poisson's ratio), is the piezoelectric constants matrix, e = dC contains the piezoelectric coupling constants, and d is the dielectric constant matrix.

The patches are assumed to be perfectly bonded to the base structure. For voltage driven actuators, the finite element equations take the form [41]

$$\begin{eqnarray} &&M\ddot{\zeta }(t)+{C}_{\mathrm{D}}\dot {\zeta }(t)+K \zeta (t)-{K}_{\zeta \phi }\phi (t)=F(t), \end{eqnarray} \tag{ 29 }$$

where ζ(t) is the generalized nodal displacement vector, M and K are the mass and stiffness matrices respectively and C_D is the damping matrix. A very small Rayleigh structural damping, C_D = 10⁻⁸K is assumed. Also K_ζϕ is the (symmetric) electro-mechanical coupling matrix, while F(t) is the vector of external forces. Details for finite element modelling of piezo-laminated smart structures and M,K and K_ζϕ can be found in [31]. The second-order system (29) is often rewritten in first-order state-space form by defining a state $[\zeta ,\;\dot {\zeta }]$. However, this leads to poorly conditioned matrices and often numerical problems result. To avoid these problems, the finite element equations are transferred into state-space form by defining the state variable

$$\begin{eqnarray} &&x=\left \{ \begin{array}{@{}c@{}} \displaystyle {K}^{0.5}\zeta \\ \displaystyle \dot {\zeta } \end{array}\right \} . \end{eqnarray} \tag{ 30 }$$

The system dynamics (29) are then

$$\begin{eqnarray} &&\dot {x}(t)=A x(t)+B \phi (t), \end{eqnarray} \tag{ 31 }$$

where

$$\begin{eqnarray} A=\left [\begin{array}{@{}cc@{}} \displaystyle 0&\displaystyle {K}^{0.5}\\ \displaystyle -{M}^{-1}{K}^{0.5}&\displaystyle -{M}^{-1}{C}_{\mathrm{ D}} \end{array}\right ];\tqs \nonumber\\ &&B=\left [\begin{array}{@{}c@{}} \displaystyle 0\\ \displaystyle {M}^{-1}{K}_{\zeta \phi } \end{array}\right ]. \end{eqnarray} \tag{ 32 }$$

We considered patches 40 μm thick and made of PZT 5A. The base structures are made of steel. Material properties of steel and PZT 5A are given in table 1.

The optimal cost and location of actuators obtained from the finite element formulation should converge as the size of the finite elements decreases. For damped structures with a finite number of actuators where B in the partial differential formulation is bounded this is ensured for finite element approximations if Q is compact. For details, see [28]. When piezoelectric actuators are used, however, the B operator is only bounded in a space larger than the state-space [1]. In this study, since the aim is to control the position, which includes the first part of the state vector,

$$\begin{eqnarray} {Q}^{1/2}=\left [\begin{array}{@{}cc@{}} \displaystyle I&\displaystyle 0 \end{array}\right ] \end{eqnarray} \tag{ 33 }$$

which is compact on the state-space [28]. Numerical tests in [6] indicate convergence of the optimal cost and actuator locations for the problems studied in this paper.

4.1. Pinned beam

4.2. Plate

To model the plates in ANSYS, 'SHELL181' elements are used which are based on Reissner Mindlin plate theory. The equations of motion for Reissner Mindlin plates with attached piezoelectric patches can be found in [1]. Plate dimensions are 500 mm × 500 mm × 1 mm and it is divided into 100 finite elements. The structure is augmented with actuators on its top surface, and the actuators are 5 mm × 5 mm × 40 μm each, which have the same area as the finite elements.

First, the location of one actuator is optimized on a cantilever plate. The result is shown in figure 2. To verify the accuracy of the optimization code, we calculated the value of objective function in each element, and observed that in the optimized location, the objective function has its minimum value. Moreover, the actuator is located in two other elements, as shown in figure 3, and the plate is subjected to a vertical impulse at the tip. Figure 4 shows the vertical deformation at the same point with the three actuator locations. Clearly, when the actuator is optimally placed, the disturbances are suppressed much faster.

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Next, the optimal placement for 10 actuators is calculated on the same plate and figure 5 shows the result. The same problem is solved with GA.

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Applying GA, a random initial population is chosen and, as in the previous example, the optimization constraint is taken into account by assigning a very large number to the fitness value when the number of ones in binary genes of each chromosome goes higher than 10. Figure 6 shows the optimal location of actuators with this method. For both methods, the optimum objective values and the elapsed optimization time are compared in table 4. In addition to being faster in computation, the gradient-based method yields a smaller cost than the genetic algorithm.

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Table 4.  Comparison of GA and the introduced algorithm for the cantilevered plate.

	Objective value	Elapsed time (s)
Current method	1.5845	491.9577
Genetic algorithm	1.7483	4.4433 × 104

To verify the optimization results, the plate is subjected to a vertical impulse at its tip. The response at the free end of the plate is shown in figure 8 for the optimal placements from the genetic algorithm and the presented method. The responses from two non-optimal actuator placements (shown in figure 7) are also included in this figure. Figure 8 shows that with the optimal placement the vibrations are suppressed much faster than when actuators are placed at the non-optimal locations. Also, the locations obtained by the developed algorithm are slightly more effective in vibration suppression than those obtained from the genetic algorithm.

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In the next step, the boundary conditions of the cantilevered plate are changed to pinned conditions on the two opposite sides, instead of a fixed side as shown in figure 9. Consequently, the nodes on these two sides are free to rotate but restricted to move. Figure 9 shows the optimal location of these actuators on the plate. The same problem is also solved with GA and the results are compared in table 5. Figure 10 shows the optimal location of actuators achieved by GA. The same as previous examples, table 5 shows that the proposed method is much faster and yields a better actuator location than a genetic algorithm.

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Table 5.  Comparison of GA and the introduced algorithm for the plate with pinned end conditions.

	Objective value	Elapsed time (s)
Current method	2.4115	9.1625 × 103
Genetic algorithm	2.4433	1.553 × 105

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To study the actual effect of optimal actuator placement, experiments were performed on a cantilever beam. The experimental setup is shown in figure 12. A thin steel beam is hung from the top of an aluminum frame. The beam has fixed boundary condition at the top and is free at the bottom. The optimal location of two actuators on this cantilever beam is studied for vibration control. To study the optimal actuator location problem, four patches are attached to the beam surface with superglue, as shown in figure 13. This figure also shows the position number for each actuator. In each experiment only two of these actuators are activated to regulate it to the desired shape.

Table 6 shows the beam and piezoelectric patch dimensions. The beam material properties are shown in table 1. The actuators are PSI-5A4E piezo sheets from Piezosystems. These actuators are made of lead zirconate titanate and are Industry Type 5A or Navy Type II.

Keyence LK081 and LK031 non-contact laser sensors are applied to read the deformations and estimate the states. The experimental setup also includes an SA11 power amplifier for input voltages of actuators and a Sensory 626 data acquisition card. Figure 11 shows the block diagram for the experiment.

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To find the optimal location of two actuators on the beam, the beam is modelled with 10 'BEAM3' finite elements in ANSYS finite element software. Each finite element has the same dimensions as the actuator patches. The optimal location of two actuators on the beam are positions 1 and 2 shown in figure 13. Figure 14 shows the uncontrolled and controlled responses which are read by sensor 2. The controlled responses in this figure are for three combinations of actuators. The actuator numbers are shown in figure 13. Applying the optimization scheme, the optimal location of two actuators on this beam are at positions 1 and 2. In figure 14, the optimally located actuators suppress the effect of vibrations in a shorter amount of time.

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The beam starts vibrating from 13 mm initial displacement at its tip. Table 7 shows the suppression time for all the existing actuator locations in this problem. It is observed that when actuators are placed optimally, the disturbances are suppressed faster. For larger values of Q elements, smaller suppression times can be achieved.

In this paper an optimization algorithm to solve the LQ optimal actuator location problem was described. The objective function is reformulated so that the optimal actuator location becomes a convex optimization problem. Next, a subgradient-based integer minimax optimization using a branch and bound technique is applied to minimize the performance index. The proposed optimization method is used to find the optimal location of piezoelectric actuator patches for vibration control for several examples. Even for the case of a simple uniform beam and a single actuator, the optimal location is not the obvious one. Our method was shown to be more accurate and considerably faster than a genetic algorithm. Since the problem is convex, a global minimum is found, regardless of the choice of initial condition. Experiments performed in this study reveal that, even for a simple beam, optimal actuator placement leads to a more effective control system.

The problem of sensor placement is dual to that of actuator location and so the algorithm described in this paper can be used for optimal sensor placement.

Related work of our group has been concerned with obtaining an algorithm for H_∞ optimal actuator placement [19]. We are currently investigating to what extent different cost criteria lead to different optimal actuator locations, as well as the extension of the algorithm described in this paper to H₂-optimal control [7].