A Probabilistic Approach for Robust Input Shapers Design for Precise ...

A Probabilistic Approach for Robust Input Shapers Design for Precise Point-to-Point Control Puneet Singla∗, Tarunraj Singh† and Umamaheswara Konda‡ A probabilistic approach which exploits the domain and distribution of the uncertain model parameters has been developed for the design of robust input shapers. A Polynomial Chaos based expansion approach is used to approximate uncertain system states and cost functions in terms of finite-dimensional series expansion in the stochastic space. Residual energy of the system is used as cost function to design robust input shapers for precise restto-rest maneuvers. An optimization problem which minimizes any moment or combination of moments of the distribution function of the residual energy is formulated. Numerical examples are used to illustrate the benefit of using the Polynomial Chaos based probabilistic approach for the determination of robust Input Shapers for uncertain linear systems. The solution of Polynomial Chaos based approach is compared to the minimax optimization based robust input shaper design approach which emulates a Monte Carlo process.

I.

Introduction

Precise point-to-point control is of interest in a variety of applications including, scanning probe microscopes, hard disk drives, gantry cranes, flexible arm robots etc. All of these systems are characterized by under-damped modes which are excited by the actuators. Precise position control mandates that the energy in the vibratory modes be dissipated by the end of the maneuver. These are challenging demands on the controller when the model parameters are known precisely. However, uncertainties in model parameters are ubiquitous and these uncertainties manifest themselves as a deterioration in the performance of the controller. Input Shaping is a technique which shapes the reference command to the system so as to eliminate or minimize the residual energy.1 The first solution to desensitize the Input Shaper to modeling uncertainties was to impose constraints on the sensitivity of the residual energy to model parameter error to zero at the end of the maneuver. The resulting solution is referred to as the ZVD Input Shaper.2 The ZVD Input Shaper improves the performance of the pre-filter in the proximity of the nominal model of the system. To exploit knowledge of the domain of uncertainty, the EI Input Shaper3 and the minimax Input Shaper4 were proposed which are worst case designs, i.e., they minimize the worst performance of the system over the domain of uncertainty. The minimax Input Shaper requires sampling of the uncertain space which results in a computationally expensive design as the dimension of the uncertain parameter space increases. ∗ Assistant Professor, Senior AIAA, AAS Member, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-14260, Email: [email protected]. † Professor, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-14260, Email: [email protected]. ‡ Graduate student, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-14260, Email: [email protected].

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A technique to incorporate the distribution of uncertainty in the Input Shaper design process was proposed by Chang et al.5 where the expected value of the residual vibration was minimized. The minimax Input Shaper4 incorporated the probability distribution function as a weighting scheme to differentially weigh the plants that are sampled in the domain of uncertainty. Tenne and Singh6 proposed the use of unscented transformation to map the Gaussian distributed uncertain parameters into the residual energy space. The unscented transformation force fits the distribution of the residual energy to a Gaussian. The sum of the mean and deviation of the residual energy distribution was minimized which resulted in a robust input shaper. Clearly, this approach cannot represent non-Gaussian distributions which limits its capability. This paper formulates an optimization problem exploiting the strengths of Polynomial Chaos to determine representative parameters (moments or cumulants) of the probability density function of the function of states of a linear dynamical system whose model parameters are random variables. The Polynomial Chaos expansion provides a computationally efficient approach compared to Monte Carlo simulation for the estimation of moments or cumulants of the function of uncertain state variables. This paper is organized as follows: Following the introductory section, a brief overview of robust input shaper design is presented. This is followed by the development of the equations which represent the dynamics of the coefficients of the Polynomial Chaos series. The specific problem considered is a spring-mass system with an uncertain coefficient of friction. Numerical simulations are used to compare the results of minimax optimization based robust Input Shaper design with those of the Polynomial Chaos based design.

II.

Input Shaping

Input-Shapers, also referred to as time-delay filters, are a simple and powerful approach for the shaping of reference input to eliminate or minimize residual motion of system undergoing transition from one set point to another. The system being controlled is assumed to be stable or marginally stable and could represent an open-loop or a closed-loop system. One of strengths of Input Shaper is their robustness to modeling uncertainties. The earliest solution to addressing the problem of sensitivity of the Input-Shaper to uncertainties in model parameter errors involved forcing the sensitivity of the residual energy to model errors to zero at the end of the maneuver. This works well for perturbations of the parameters about the nominal values. In case, one has knowledge of the domain of uncertainty and the distribution of the uncertain variable, this additional knowledge can be exploited in the design of a Input-Shaper by posing a minimax optimization problem. In this formulation, the maximum magnitude of the residual energy over the domain of uncertainty is minimized. The minimax optimization problem is computationally expensive and as the dimension of the uncertain space grows, the computational cost grows exponentially since the uncertain space has to be finely sampled.

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Let us consider a linear mechanical system of the following form:

˙ p) + K(p)x(t, p) = D(p)u(t) M¨ x(t, p) + C(p)x(t,

(1)

where x(t, p) ∈ Rn represents the generalized displacement coordinate wile u(t) ∈ Rm represents the deterministic system input vector. M, C(p) and K(p) are the mass, damping and stiffness matrices, respectively. We assume that the mass matrix is not uncertain and p ∈ Rr is a vector of uncertain system parameters which are function of random variable ξ with known probability distribution function (pdf) f (ξ), i.e., p = p(ξ). For example, the uncertain variable p could be represented as:

p ∼ U[a, b]

(2)

which implies that p is a uniformly distributed random variable and lies in the range

a ≤ p ≤ b.

(3)

p ∼ N [µ, Σ]

(4)

or p could be a Gaussian variable

where µ is the mean of the random variable p and Σ is the covariance matrix. p is not restricted to uniform or Gaussian distributions. However, in this paper, we will focus on these two distributions. The transfer function of the Input-shaper/Time-delay filter is parameterized as:

G(s) =

Z X

Ai e−sTi

(5)

i=0

where T0 is zero and Z is the number of delays in the pre-filter. Ai and Ti are the parameters that need to be determined to satisfy the objective of completing the state transition with minimal excursion from the desired final states in the presence of modeling uncertainties. For rest-to-rest maneuvers, the residual energy is a germane cost function. The residual energy at the final time TZ can be defined as:

V (TZ ) =

1 T 1 ˙ Z ) + (x(TZ ) − xf )T K(p)(x(TZ ) − xf ) x˙ (TZ )Mx(T 2 2

(6)

where xf is the vector of the desired final displacement. The first term of V (TZ ) is the kinetic energy in the system and the last term is the pseudo-potential energy which measures the energy resident in a hypothetical

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set of springs when x deviates from xf . If K(p) is not positive definite, the cost function V (TZ ) has to be augmented with a quadratic term to ensure that the cost function is positive definite. The resulting cost function is:

V (TZ ) =

1 T 1 1 ˙ Z ) + (x(TZ ) − xf )T K(p)(x(TZ ) − xf ) + (xr − xrf )2 x˙ (TZ )Mx(T 2 2 2

(7)

where xr refers to the rigid body displacement and xrf is the desired final position. The design of a robust Input-Shaper can be posed as the problem:

min max V (TZ )

Ai ,Ti

p

(8)

which is a minimax optimization problem and emulates a Monte Carlo process. This as indicated earlier is a computationally expensive problem when the dimension of p grows. To alleviate this problem, this paper endeavors to identify a probabilistic representation of uncertain residual energy V (TZ ) as a function of uncertain parameter vector p(ξ). To emulate the minimax optimization problem in the probability space, one would requires the distribution of V (TZ ) to have a small mean and small variance which would correspond to small residual energy and small spread over the range of the random variable. One can add higher moments such as skew and kurtosis and minimizing the skew and kurtosis is consistent with the goal of minimizing the worst cost over the domain of uncertainty. In the next section, we describe a probabilistic method based on Polynomial Chaos series expansion of an uncertain variable which exploits the domain and distribution of the uncertain model parameters to get exact moments or cumulants of residual energy.

III.

Polynomial Chaos

Polynomial chaos is a term originated by Norbert Wiener in 1938,7 to describe the members of the span of Hermite polynomial functionals of a Gaussian process. According to the Cameron-Martin Theorem,8 the Fourier-Hermite polynomial chaos expansion converges, in the L2 sense, to any arbitrary process with finite variance (which applies to most physical processes). This approach is combined with the finite element method to model uncertainty in dynamical systems.9 This has been generalized10 to efficiently use the orthogonal polynomials from the Askey-scheme to model various probability distributions. The basic idea of this approach is to approximate the stochastic system state in terms of finite-dimensional series expansion in the stochastic space. The completeness of the space allows for the accurate representation of any PDF using a suitable basis. Certain bases can be chosen to represent the random variable with a given PDF with the fewest number of terms. For example, Legendre polynomials can be used to represent a uniformly distributed random variable with only two terms. For dynamical systems described by mathematical equations, the

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unknown coefficients are determined by minimizing an appropriate norm of the residual. Let us consider a second order stochastic linear system given by Eq. (1):

˙ p) + Kx(t, p) = D(p)u(t) M¨ x(t, p) + C(p)x(t,

(9)

As mentioned earlier, p ∈ Rr is a vector of uncertain system parameters which is a function of random vector ξ with known probability distribution function (pdf) f (ξ), i.e., p = p(ξ). It is assumed that uncertain state vector x(t, p) and system parameters, Cij and Kij can be written as a linear combinations of basis functions, φi (ξ), which span the stochastic space of random variable ξ.

xi (t, p) =

N X

xil (t)φl (ξ) = xTi (t)Φ(ξ)

(10)

cij l φl (ξ) = cTij Φ(ξ)

(11)

kij l φl (ξ) = kTij Φ(ξ)

(12)

dij l φl (ξ) = dTij Φ(ξ)

(13)

l=0

Cij (p) =

N X l=0

Kij (p) =

N X l=0

Dij (p) =

N X l=0

where Φ(.) ∈ RN +1 is a vector of polynomials basis functionals orthogonal with respect to the pdf f (ξ) which can be constructed using the Gram-Schmidt Orthogonalization Process. The coefficients cij l , kij l and dij l are obtained by making use of following normal equations:

(14)

kij l

(15)

dij l where hu(ξ), v(ξ)i =

R

hCij (p(ξ)), φl (ξ)i hφl (ξ), φl (ξ)i hKij (p(ξ)), φl (ξ)i = hφl (ξ), φl (ξ)i hDij (p(ξ)), φl (ξ)i = hφl (ξ), φl (ξ)i

cij l =

(16)

u(ξ)v(ξ)f (ξ)dξ represents the inner product induced by the pdf f (ξ). Now, substi-

Rr

tution of Eq. (10), Eq. (11), Eq. (12) and Eq. (13) in Eq. (9) leads to

ei (ξ) =

n X

Mij

j=1

−

! x ¨jl (t)φl (ξ)

+

n N X X j=1

l=0

m N X X j=1

N X

! cij l φl (ξ)

l=0

N X

! x˙ jl (t)φl (ξ)

+

l=0

n N X X j=1

l=0

! kij l φl (ξ)

N X

! xjl (t)φl (ξ)

l=0

! dij l φl (ξ) uj ,

i = 1, 2, · · · , n

(17)

l=0

The n(N +1) time-varying unknown coefficients xik (t) can be obtained using the Galerkin projection method.

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Projecting the error onto the space of basis functions {φl (ξ)} and minimizing it leads to:

hei (ξ), φl (ξ)i = 0,

i = 1, 2, · · · , n,

l = 0, 1, · · · , N

(18)

This leads to following set of n(N + 1) deterministic differential equations:

M¨ xp (t) + C x˙ p (t) + Kxp (t) = Du(t)

(19)

 T where xp (t) = xT1 (t), xT2 (t), · · · , xTn (t) is a vector of n(N +1) unknown coefficients and M ∈ Rn(N +1)×n(N +1) , C ∈ Rn(N +1)×n(N +1) , K ∈ Rn(N +1)×n(N +1) and D ∈ Rn(N +1)×m . Let P and Tk , for k = 0, 1, · · · , N, denote the inner product matrices of the orthogonal polynomials defined as follows:

Pij = hφi (ξ), φj (ξ)i,

i, j = 0, 1, · · · , N

(20)

Tkij = hφi (ξ), φj (ξ), φk (ξ)i,

i, j = 0, 1, · · · , N

(21)

Then M, C and K can be written as n × n matrix of block matrices, each block being an (N + 1) × (N + 1) matrix. The matrix M consists of blocks Mij ∈ R(N +1)×(N +1) :

Mij = Mij P,

i, j = 1, 2, · · · , n

(22)

Similarly, for the matrices C and K, the k th row each of their block matrices Cij , Kij ∈ R(N +1)×(N +1) is given by,

Cij (k, :) = cTij Tk ,

i, j = 1, 2, · · · , n

(23)

Kij (k, :) = kTij Tk ,

i, j = 1, 2, · · · , n

(24)

The matrix D consists of blocks Dij ∈ R(N +1)×1 :

Dij = P dij

i = 1, 2, · · · , n,

j = 1, 2, · · · , m

(25)

Eq. (10) along with Eq. (19) define the uncertain state vector x(t, ξ) as a function of random vector ξ and can be used to compute any order moment or cumulant of a function of uncertain state variables. For

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example, the first moment of residual energy can be computed as: Z V (TZ , ξ)f (ξ)dξ =

E[V ](TZ ) =

1 T 1 1 x˙ (TZ )Mx˙ p (TZ ) + xTp (TZ )K(p)xp (TZ ) − E[xTf xf − 2xT xf ] 2 p 2 2

(26)

Rr

Now, the problem of the design of a robust Input-Shaper can be posed as:

min α12 E[V (TZ )] +

Ai ,Ti

P X

i

αi2 E[(V (TZ ) − E[V (TZ )]) ]

(27)

i=2

Notice that P = 2 corresponds to minimization of mean and variance of residual energy. In the next section, we illustrate the whole procedure by considering a particular example.

IV.

Example (Uniform Distribution)

To illustrate the proposed technique of using Polynomial Chaos for the determination of a robust Inputshaper, consider the second order system: x ¨ + kx = ku

(28)

where k is a uniformly distributed random variable which lies in the range

0.7 ≤ k ≤ |{z} 1.3 . |{z} a

(29)

b

The random variable k can be represented by Legendre polynomials (Pi ) as:

k = k0 P0 (ζ) + k1 P1 (ζ)

where ζ is a uniformly distributed random variable which lies in the range [−1, 1] and k0 = k1 =

b−a 2

(30)

a+b 2

= 1 and

= 0.3. The displacement x is represented as:

x=

N X

xi Pi (ζ)

(31)

i=0

where xi are the coefficients of the Polynomial Chaos expansion. The equation of motion can now be represented as: N X i=0

Pi (ζ)¨ xi + (k0 P0 (ζ) + k1 P1 (ζ))

N X

Pi (ζ)xi = (k0 P0 (ζ) + k1 P1 (ζ))u.

i=0

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(32)

Using a Galerkin projection method, the dynamics of xi can be determined. Multiplying Eq. (32) by Pj and calculating the inner product Z

1

hPj (ζ), Pk (ζ)i =

Pj (ζ)Pk (ζ) dζ

(33)

−1

we can derive the dynamics for the Polynomial Chaos coefficients which are given by the equation:

M

     x ¨0              x ¨1  ..      .              x ¨n

+K

     x0              x1   ..      .              xn

= Du

(34)

where the elements of the M matrix are 2 δij where i = 0, 1, 2...N 2i + 1

(35)

2 2(i + 1) 2i , Ki,i+1 = k1 , Ki,i−1 = k1 , 2i + 1 (2i + 1)(2i + 3) (2i + 1)(2i − 1)

(36)

Mij =

and the elements of the K matrix are

Kii = k0

and 



 2k0    2   3 k1       D= 0        0     .  ..

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(37)

which can be rewritten as:      x ¨0                 x ¨ 1          .    ..                x ¨i−1 

 k0   k1   0      +        x ¨i                  x ¨i+1               .   .     .               x  ¨n

1 3 k1

0

k0

2 5 k1

..

..

.

..

.

     x0                 x 1            .    ..       k0             xi−1    k1     =    u (38)        xi   0           ..      x .   i+1       .     . n    .    2n+1 k1            k0 xn 

.

k0

i 2i+1 k1

0

i 2i−1 k1

k0

(i+1) 2i+3 k1

0

i+1 2i+1 k1

k0 ..

0

..

.

.

n 2n−1 k1

0

Equation Eq. (38) can be easily solved for a parameterized u (Equation Eq. (5)). The residual energy at the final time TZ can be represented as:

1 V (TZ ) = 2

N X

!T x˙ i Pi (ζ)

i=0

M

N X

! x˙ i Pi (ζ)

i=0

N X

1 + 2

!T xi Pi (ζ) − xf

K(p)

i=0

N X

! xi Pi (ζ) − xf

(39)

i=0

The mean of the residual energy can be easily calculated using the equation Z

1

E[V (TZ )] = µ =

V (TZ ) dζ

(40)

−1

and the higher moments by the equation:

E[(V (TZ ) − µ)n ] =

Z

1

(V (TZ ) − µ)n dζ

(41)

−1

A.

Numerical Results

For the spring-mass system, a two time-delay filter is parameterized as:

u = A0 + A1 H(t − T1 ) + A2 H(t − T2 )

(42)

with the constraint that A0 + A1 + A2 =1, to mandate the final value of the output of the time-delay filter subject to a step input will be the same as the magnitude of the input. The Polynomial Chaos expansion is used to represent the residual energy random variable V (T2 ). A constrained minimization problem is solved to minimize a series of cost functions which are listed in Table 1.

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Cost

A0

A1

A2

T1

T2

E[V (T2 )] = µ p µ + E[(V (T2 ) − µ)2 ] p p µ + E[(V (T2 ) − µ)2 ] + 3 E[(V (T2 ) − µ)3 ]

0.2545 0.2554 0.2557

0.4909 0.4892 0.4886

0.2545 0.2554 0.2557

3.1416 3.1416 3.1415

6.2831 6.2830 6.2836

Table 1. Optimal Input Shaper (2 Delays)

Figure 1 illustrates the performance of the Polynomial Chaos based design and the standard minimax controller design which endeavors to minimize the maximum magnitude of the residual energy over the domain of spring stiffness uncertainty. The distribution of the uncertain spring stiffness is assumed to be uniform. It is clear from Figure 1 that as higher moments are included in the design process, the resulting solutions tends towards the minimax solution. This is further evidenced in Figure 2 where a three time-delay Input Shaper design based on Polynomial Chaos and the minimax approach are compared. 0.03 Minimize mean Minimax Input Shaper Minimize mean+deviation Minimize mean+deviation+skew

Residual Energy

0.025

0.02

0.015

0.01

0.005

0

0.7

0.8

0.9 1 1.1 1.2 Uncertain Spring Stiffness

1.3

1.4

Figure 1. PC Uniform distribution (2 delays filter)

Cost

A0

A1

A2

A3

T1

T2

T3

E[V (T2 )] = µ p µ + E[(V (T2 ) − µ)2 ] p p µ + E[(V (T2 ) − µ)2 ] + 3 E[(V (T2 ) − µ)3 ]

0.1810 0.1810 0.1357

0.4167 0.4160 0.3747

0.3223 0.3231 0.3658

0.0799 0.0800 0.1238

3.1365 3.1380 3.1500

6.2869 6.2858 6.2926

9.4237 9.4240 9.4190

Table 2. Optimal Input Shaper (3 Delays)

To illustrate the ability of the Polynomial Chaos expansion to capture the distribution of the residual energy, the random coefficient of stiffness is sampled with 10,000 Monte Carlo samples and the distribution of the residual energy is calculated. The mean and variance of the Monte Carlo based sampling is compared to those of the Polynomial Chaos based expansion. These results are illustrated in Figure 3 where the Monte

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0.012 Minimize mean Minimize mean+deviation Minimize mean+deviation+skew Minimax Input Shaper

Residual Energy

0.01

0.008

0.006

0.004

0.002

0

0.7

0.8

0.9 1 1.1 1.2 Uncertain Spring Stiffness

1.3

1.4

Figure 2. PC Uniform distribution (3 delays filter)

Carlo based result is illustrated by the dashed line. The solid line is the estimate of the mean and variance as a function of the number of terms in the Polynomial Chaos expansion. It is clear that with a fourth order polynomial, the mean and variance have converged to the true values. −4

Mean of pdf

1.5

x 10

1 0.5 0 1

2

3

4 5 Polynomial Order

6

7

8

2

3

4 5 Polynomial Order

6

7

8

−8

Variance of pdf

4

x 10

3 2 1 0 1

Figure 3. Polynomial Chaos based Estimate of Mean and Variance

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V.

Example (Gaussian Distribution)

One can determine a robust Input Shaper for normally distributed coefficient of stiffness similarly. Consider a random stiffness coefficient given by the equation

k = N (1, 0.12 )

(43)

which corresponds to a Gaussian distribution with a mean of unity and a standard deviation of σ=0.1. The variable k can be represented using Hermite polynomials as:

k = k0 H0 (ζ) + k1 H1 (ζ)

(44)

where k0 =1, and k1 = 0.1. The displacement can be represented as:

x=

N X

xi Hi (ζ)

(45)

i=0

where xi are the coefficients of the Polynomial Chaos expansion. The resulting equation of motion is: N X

Hi (ζ)¨ xi + (k0 H0 (ζ) + k1 H1 (ζ))

N X

Hi (ζ)xi = (k0 H0 (ζ) + k1 H1 (ζ))u.

(46)

i=0

i=0

The Galerkin projection leads to the equation:            x0  x ¨     0                    x   x1  ¨1 = Du +K M ..  ..            . .                         xn x ¨n

(47)

where the elements of the M matrix are √ Mij = i! 2πδij where i = 0, 1, 2...N

(48)

and the elements of the K matrix are √ √ √ Kii = i! 2πk0 , Ki,i+1 = (i + 1)! 2πk1 , Ki,i−1 = i! 2πk1 ,

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(49)

and

A.

  √  2πk0   √    2πk1       D= 0        0     .  ..

(50)

Numerical Results

Assuming a Gaussian distributed random spring stiffness with a mean of unity and a deviation of 0.1, the residual energy represented by the Polynomial Chaos is determined and different combinations of the moments are minimized. The minimax problem is also solved to provide a benchmark to compare the solution of the Polynomial Chaos expansion. Figures 4 and 5 illustrate the results of the optimization problem which minimizes the mean, mean+deviation, mean+deviation+skew, and compares it to the minimax solution which is considered the desired solution. It is clear for both the two time-delay filter in Figure 4 and the three time-delay filter in Figure 5 that minimizes the mean results in a solution that is closest to the minimax solution. Cost

A0

A1

A2

T1

T2

E[V (T2 )] = µ p µ + E[(V (T2 ) − µ)2 ] p p µ + E[(V (T2 ) − µ)2 ] + 3 E[(V (T2 ) − µ)3 ]

0.2515 0.2527 0.2540

0.4970 0.4947 0.4923

0.2515 0.2526 0.2537

3.1416 3.1416 3.1417

6.2831 6.2830 6.2829

Table 3. Optimal Input Shaper (2 Delays)

0.035 Polynomial Chaos (mean) Polynomial Chaos (mean+dev) Polynomial Chaos (mean+dev+skew) Minimax Input Shaper

0.03

Residual Energy

0.025 0.02 0.015 0.01 0.005 0

0.7

0.8

0.9 1 1.1 1.2 Uncertain Spring Stiffness

1.3

Figure 4. PC Gaussian distribution (2 delays filter)

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1.4

Cost

A0

A1

A2

A3

T1

T2

T3

E[V (T2 )] = µ p µ + E[(V (T2 ) − µ)2 ] p p µ + E[(V (T2 ) − µ)2 ] + 3 E[(V (T2 ) − µ)3 ]

0.1276 0.1229 0.1179

0.3729 0.3694 0.3629

0.3725 0.3768 0.3817

0.1271 0.1309 0.1374

3.1433 3.1409 3.1421

6.2813 6.2792 6.2780

9.4145 9.4153 9.4061

Table 4. Optimal Input Shaper (3 Delays)

−3

8

x 10

Polynomaial Chaos (mean) Polynomial Chaos (mean+dev) Polynomial Chaos (mean+dev+skew) Minimax Input Shaper

7

Residual Energy

6 5 4 3 2 1 0

0.7

0.8

0.9 1 1.1 1.2 Uncertain Spring Stiffness

1.3

Figure 5. PC Gaussian distribution (3 delays filter)

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1.4

VI.

Conclusions

This paper exploits the Polynomial Chaos approximation to represent the residual energy random variable. Two classes of problems are considered: the first where the uncertain spring stiffness is uniformly distributed and the second where the spring stiffness distribution is Gaussian. Legendre and Hermite polynomials are used in the Polynomial Chaos approximation for the two cases respectively. Simulation results for a single spring-mass system illustrate that minimizing the mean and combinations of the higher moments can result in an Input Shaper which closely approximates the minimax solution. The benefit of using Polynomial Chaos approximation to represent the random cost function compared to the minimax approach is the computational benefit, since one does not need to sample the multi-dimensional uncertain space which is required for the minimax approach.

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