Frequency domain based real-time performance ...

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Frequency domain based real-time performance optimization of Lur'e systems Agung Setiadi n, David Rijlaarsdam, Pieter Nuij, Maarten Steinbuch Eindhoven University of Technology, Department of Mechanical Engineering, Control Systems Technology, PO Box 513, GEM-Z -1.145, 5600 MB, Eindhoven, The Netherlands

a r t i c l e i n f o

abstract

Article history: Received 21 May 2012 Received in revised form 20 August 2013 Accepted 30 August 2013

Nonlinear effects can lead to performance degradation in (controlled) dynamical systems. This paper provides a practical method to optimally compensate performance degrading nonlinear effects in Lur'e-type systems in an automated way. Using novel frequency domain based techniques, a well defined performance measure is derived and real-time performance optimization is achieved by application of extremum seeking algorithm. This yields a new method for real-time compensation of performance degrading nonlinear effects, which is successfully demonstrated in both simulation and experiment. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Frequency domain methods Nonlinear systems Real-time performance optimization Extremum seeking

1. Introduction Frequency domain analysis plays an important role in the control of Linear Time Invariant (LTI) systems. For LTI systems, frequency domain methods are often used for modeling and performance analysis. Frequency domain methods provide a framework for optimal control design as well. In general frequency domain methods cannot be applied to nonlinear systems in straightforward manner and the notion of performance is difficult to define for nonlinear systems. However, when applied with care, frequency domain methods yield an insightful, useful and practically applicable way to assess and optimize the performance of certain nonlinear systems [1–6]. Several approaches for the analysis and modeling of nonlinear systems in frequency domain exist [2,7–9]. For example in [8], a nonlinear frequency response function is proposed for convergent nonlinear systems and used to assess the performance of a high precision motion system in [3]. Moreover, in [2,10,11] it is shown that an extension of the sinusoidal input describing function allows frequency domain methods to identify non-parametric models of nonlinear motion systems, subject to a sinusoidal input. Frequency domain methods have been used for performance optimization of controlled nonlinear systems as well. In [5], frequency domain methods are used to optimally design a feed forward friction compensator in a high precision motion stage of a transmission electron microscope. Recently, these results are extended in [6] yielding a frequency domain methodology that allows us to formulate a novel, practically applicable performance measure and corresponding performance optimization

n

Corresponding author. Tel.: þ31 402472796. E-mail addresses: [email protected] (A. Setiadi), [email protected] (D. Rijlaarsdam), [email protected] (P. Nuij), [email protected] (M. Steinbuch). 0888-3270/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2013.08.030

Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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problem for Lur'e-type systems. The work presented in this paper extends these results by presenting a novel method for realtime, automated performance optimization for Lur'e-type systems. The results in the following are intended for engineering practice and require minimal a priori knowledge of the nonlinearities while optimization is performed automatically. To motivate the practical use of the method consider a system as depicted in Fig. 1. The system consists of a mass connected with a linear spring and a linear damper. The mass interacts with a magnetic field which exerts a force to the mass. The magnetic force depends nonlinearly on the displacement of the mass and cannot be measured directly. Because of this, the nonlinearity profile is not known and therefore compensating the nonlinearity cannot be achieve by the common inverse approach of the nonlinearity. One of the benefits of the proposed method is that it can tackle this problem because it only requires the input–output measurement of the system instead that of the nonlinearity. This work is organized the following manners, first in Section 2 the required nomenclature and preliminaries are introduced. Next, Section 3 deals with performance optimization of Lur'e-type systems. This section consists of four main parts: analysis of Lur'e type systems, introduction of a frequency domain performance measure for nonlinear systems, design of static compensators to optimize this performance and finally, the application of extremum seeking algorithms to optimize the design of these compensators in real-time. The resulting framework allows for real-time performance optimization of Lur'e-type systems, based on a frequency domain representation of the systems output. This framework is applied in simulation in Section 4 and used in Section 5 to optimize the performance of a nonlinear amplifier in experiments. Finally, Section 6 presents conclusions and recommendations for future research.

2. Nomenclatures and preliminaries Let Cp denote the space of piecewise continuous, bounded functions R↦R Z t0 ; t 0 A R. Then the analysis in this paper focusses on Lur'e-type systems which are defined as follows: Definition 1 (Lur'e-type system). Consider a Lur'e-type system depicted in Fig. 2, such that _ ¼ AxðtÞ þ Bu uðtÞ þ Bw wðtÞ; xðt 0 Þ ¼ x0 xðtÞ " # " # " # D1;w D1;u  w  y1 C1 ¼ xðtÞ þ D2;w D2;u y2 C2 u uðtÞ ¼ ϕðy2 ðtÞÞ

ð1Þ

with wA Cp an external input, y1 ðtÞ; y2 ðtÞ A R the outputs and xðtÞ A Rn the states of the system. Furthermore, A A Rnn ; Bu A Rn1 ; Bw A Rn1 ; C ℓ A R1n and Dℓ;u=w A R; ℓ ¼ 1; 2 constitute a state space representation (1) of the dynamics and uðtÞ A R is a nonlinear feedback generated by a static nonlinear mapping ϕ : R↦R.

⎯⎯

Fig. 1. Schematic depiction of a mass, subject to a nonlinear magnetic field.

Fig. 2. Lur'e-type system.

Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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A solution of (1) corresponding to an input w(t) is denoted by xw(t). Moreover, the corresponding limit solution (if it exists) is denoted by x w ðtÞ, such that limt-1 J x w ðtÞxw ðtÞ J ¼ 0 8 x0 A Rn . Finally, the output corresponding to the limit solution is denoted by y w ðtÞ ¼ gðx w ðtÞ; wðtÞÞ. When investigating the input–output mapping of a system prerequisites such as existence, stability and uniqueness of the solution should be considered. These properties are, for example, captured by the notion of convergence [12]. The formal definition of a uniformly convergent system follows from [12–14] and is provided below. Definition 2 (Uniformly convergent system). A time invariant system is said to be uniformly convergent for a class of input signals W if for every wA W: 1. all solutions xw(t) are well-defined for all t A ½t 0 1Þ and all initial conditions x0 A Rn ; 2. there exists a unique solution x w ðtÞ defined and bounded for all t A ð1 þ 1Þ; 3. the solution x w ðtÞ is uniformly globally asymptotically stable. To emphasize the dependence on the input w(t), the limit solution is denoted by x w ðtÞ. Sufficient conditions for uniform convergence of Lur'e-type systems follow from [3,12,13,15] and are summarized in the following lemma. Lemma 1 (Uniform convergence of Lur'e-type systems). Consider a Lur'e-type system according to Definition 1, with input w A Cp . Then, the system is uniformly convergent with respect to the class of piecewise continuous signals Cp if there exists a ϑ A R 4 0 such that: 1. 2. 3. 4.

A is Hurwitz; RfC 2 ð2πiξIAÞ1 Bu þ D2;u g 41=ϑ 8 ξ A R; 0 r ðϕðy2 Þϕðy1 ÞÞ=ðy2 y1 Þ rϑ 8 y1 ; y2 A R; u ¼ ϕðC 2 x þD2;u u þ D2;w wÞ has a unique solution for every x and w in the domain of interest.

Next, the following makes frequent use of sinusoidal signals, which are defined as follows: Definition 3 (Sinusoidal signal). Let S denote the set of sinusoidal signals. A signal z(t) is sinusoidal if zðtÞ ¼ γ cos ð2πξ0 t þφÞ; t A R, for some γ; ξ0 A R 4 0 ; φ A R. Finally, S ξ0  S denotes the subset of sinusoidal signals with frequency ξ0 . Moreover, the analysis presented in this paper frequency uses a spectral representation of scalar, real-valued signals. The single sided spectrum of such a signal is defined as follows: Definition 4 (Single sided spectrum). Consider the Fourier transform of a scalar real-valued signal y(t): Z ϝy ðξÞ ¼

1

yðtÞe2πiξt dt

ð2Þ

1

with ξ the frequency in ½Hz. Then the corresponding single sided spectrum equals 8 > < 2ϝy ðξÞ; YðξÞ ¼ ϝy ð0Þ; > : 0;

ξ 40 ξ¼0

ð3Þ

ξ o0

Finally, in the following, controller structures are parameterized using Chebyshev polynomials, which are defined as follows: Definition 5 (Chebyshev polynomials). Consider the set of Chebyshev polynomials of the first kind Tn, such that T 0 ðνÞ ¼ 1 T 1 ðνÞ ¼ ν T n þ 1 ðνÞ ¼ 2νT n ðνÞT n1 ðνÞ

Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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3. Theory 3.1. Frequency domain performance analysis of Lur'e systems The fact that a sinusoid is an eigenfunction of LTI systems is crucial in the application of frequency domain based analysis and controller design. Nonlinear systems, on the other hand, generally do not have a sinusoidal response to a sinusoidal input. However, for uniformly convergent nonlinear systems a sinusoidal input yields an output that, although not sinusoidal, is periodic with the same period time as the input [12]. For such systems, nonlinear effects can be observed as the deformation of the output signal compared to a pure sinusoid. Hence, in the frequency domain, this distortion due to nonlinear effects manifest itself as higher harmonic spectral components in the output spectra. Note that for uniformly convergent nonlinear systems, the nonlinearity generates solely higher harmonics in the output spectrum. Hence, no other spectral content is generated when the system is subject to a sinusoidal input. Minimizing the higher harmonics as such aims to achieve a sinusoidal response to a sinusoidal input. Whether this implies compensation of the nonlinear effects (linearization) is nontrivial. For uniformly convergent Lur'e-type systems, however, a direct connection exists between the presence of harmonics in the output spectrum and (non) linearity of the systems dynamics. This is investigated in [6] and is summarized in the following theorem: Theorem 1 (Linearity of Lur'e-type systems). Consider a Lur'e-type system according to Definition 1 with input w(t), outputs y1 ðtÞ; y2 ðtÞ and states x(t). Next, let w A S ξ0 be a sinusoidal input with frequency ξ0 , let υ A R 4 0 and assume that the following holds:

A1 : the system is uniformly convergent with respect to the class of sinusoidal input signals S; A2 : the steady state outputs y 1;w ðtÞ and y 2;w ðtÞ are nonzero for some sinusoidal input w A S ξ0 ; A3 : jC ℓ ð2πiωAÞ1 Bu þDℓ;u Þj a 0 8 ωA R; ℓ ¼ 1; 2. Then, the following statements are equivalent:

S1 : y 1;w A S ξ0 . S2 : y 2;w A S ξ0 with amplitude υ. ~ A R21 that constitute an LTI state space realization: S3 : there exist A~ A Rnn ; B~ A Rn1 ; C~ A R2n and D ~ ~ þ BwðtÞ; x~_ ðtÞ ¼ A~ xðtÞ ~ ~ ¼ C~ xðtÞ ~ þ DwðtÞ yðtÞ

~ 0 Þ ¼ x0 xðt ð4Þ

such that (4) is equivalent to (1), in the sense that for all ðx0 ; wÞ A fðx0 ; wÞ A Rn  Cp j J y2 ðtÞ J 1 r υg ~ ~ the solution and output of (1) and (4) satisfy xðtÞ ¼ xðtÞ and yðtÞ ¼ yðtÞ for all t A R Z t 0 . Proof. See [6].

□

Theorem 1 implies that for Lur'e-type systems, attaining a sinusoidal response to a sinusoidal input is necessary and sufficient to show the existence of an equivalent linear time invariant dynamical model that fully captures the system dynamics for a well defined set of input signals and initial conditions. To optimally compensate performance degrading nonlinear effects in Lur'e-type systems it therefore suffices to design a controller that suppresses the higher harmonics in the output spectrum. This yields a well defined frequency domain based performance measure as introduced in [6]: Definition 6 (Frequency domain based performance measure). Consider a Lur'e-type system such that assumptions A1 –A3 in Theorem 1 are satisfied and assume that the dynamics depend on a set of (controller) parameters κ A Rnκ . Now, let w A S ξ0 be a sinusoidal input with frequency ξ0 and consider the corresponding single sided spectrum of the steady state output y ℓ;w ðtÞ, denoted by Y ℓ;w ðξÞ A C; ℓ ¼ 1; 2. Then, the performance of the system is defined by the following frequency domain based cost function: VðκÞ ¼

1 K

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K jY ℓ ðkω0 ; κÞj2 ∑ ; 2 k ¼ 0;k Z 2 jY ℓ ðω0 ; κÞj

ℓ ¼ 1; 2

ð5Þ

The performance defined in (5) is said to be optimal if κ⋆ ¼ arg minVðκÞ κ

Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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The performance measure in Definition 6 allows us to quantify the performance degrading nonlinear effects based solely on output measurements. In the sequel, this performance measure will be applied to define and optimize the performance of Lur'e-type systems. 3.2. Performance optimization of Lur'e systems In the following, the problem of optimizing the performance of Lur'e-type systems is addressed. Examples of such systems include mechanical systems subject to friction or magnetic/electric fields and sensor/actuator nonlinearities. The analysis in the sequel is based on the premise that the nonlinearity is performance degrading and the remainder of this paper therefore focusses on optimal compensation of the nonlinear effects. The design of a compensator for a static nonlinearity usually focusses on approximating the nonlinear mapping or its inverse. As any sufficiently smooth function can be approximated by a combination of appropriate basis functions, a nonlinear mapping is often approximated by tuning the weights of the basis functions based on measurements of the input and output of the nonlinearity. Such approach, however, requires the output of the nonlinearity to be available, which is often not the case in practice. For example, the friction force and the force exerted by a magnetic/electric field is generally not available from measurements in mechanical applications. In contrast to conventional methods, the frequency domain performance measure in Definition 6 relies on the systems output rather than that of the nonlinearity. This allows us to assess and compensate nonlinear effects using Theorem 1 based solely on measurements of the output of a Lur'e-type system. Assuming that the signal y2 ðtÞ is available, two compensator configurations are proposed for compensation of the nonlinearity: cascade and parallel (Fig. 3). These configurations relate to the approximation of the inverse of ϕ or the nonlinearity itself. Moreover, the cascade configuration assumes that the input of the nonlinearity is accessible, e.g. nonlinear actuators, while the parallel compensator structure assumes that the control input and output of the nonlinearity are collocated. In the following, the compensator χ in Fig. 3 is parameterized using Chebyshev polynomials. These basis functions are selected as their orthogonality is expected to improve the conditioning of the optimization problem. Moreover, the polynomial structure allows us to retain control over the final solution of the optimization problem, which is defined as follows: Definition 7 (Chebyshev based performance optimization). Consider a uniform convergent Lur'e-type system subject to a cascade or parallel static compensator χ as in Fig. 3, where χ is given by N

χðνÞ ¼ ∑ βn T n ðνÞ ¼ λν þ n¼0

N

∑

n ¼ 0;n Z 2

bn νn

ð6Þ

with λ; βn ; bn A R and where χ is parameterized using Chebyshev polynomials of the first kind Tn (Definition 5). Moreover, λ is the linear feed-through of the compensator and is given by λðκÞ ¼

floorðN=2Þ

∑

m¼1

ð1Þm þ 1 ð2m1Þβ2m1

ð7Þ

with κ ¼ fβ0 ; β1 ; …; βN g. The optimization problem related to the performance VðκÞ as in Definition 6 is given by κ⋆ ¼ arg minVðκÞ κ

subject to: λðκÞ ¼ α

ð8Þ

where α is a user defined parameter which constraints the resulting linear feedback gain after compensation.

Fig. 3. Compensator configurations: (a) cascade configuration, (b) parallel configuration.

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Table 1 Gain gðαÞ after compensation if λ ¼ α and V ðκ⋆ Þ ¼ 0. COMPENSATOR Cascade Parallel

CONFIGURATION

GAIN gðαÞ ∂ϕðνÞ j α ∂ν ν ¼ 0 ∂ϕðνÞ j α ∂ν ν ¼ 0

The optimization problem (8) is well defined in the sense that the solution, i.e. the resulting linearized function after compensation, is unique. If compensation attains the optimal performance, i.e. Vðκ⋆ Þ ¼ 0, the nonlinear function is fully linearized and the linearized static nonlinear function equals ϕlin ¼ gðαÞy2 þ b;

y2 rυ

ð9Þ

with υ the level of excitation of the nonlinearity as in Theorem 1. Note that any gain g generates the same value of the performance measure. However, Definition 7 restricts the compensated nonlinear function to a unique value of g and the inclusion of a DC component in the performance measure (5) minimizes the value of jbj. The compensated function gain g depends on the selection of the user defined value α and compensator configuration. The feedback gain after compensation is given in Table 1 as derived in [6].

3.3. Real-time implementation The previous section yields a formalization of the performance optimization problem. Next, a numerical method to solve such optimization in a real-time setting is discussed. In the following, it is assumed that at least one optimal solution exists which represents a sufficient performance level and the system is assumed to remain convergent during optimization. Both these assumptions were observed to be satisfied in all performed simulations and experiments. In practice it is preferred to solve the optimization problem (8) with minimal a priori knowledge of the system. In this paper an Extremum Seeking Controller (ESC) is therefore proposed to solve the optimization problem. The ESC is an adaptive controller that drives a cost function, relating to the performance of a system, to its extreme value based on an online estimation of the derivative. It is model-free, and only requires the cost function to be available from measurements. Therefore, the ESC is considered a suitable candidate to solve the performance optimization problem (8). Recent advances in ESC yield a generalization of the controller structure as well as tuning guidelines [16–19,25–27,30]. Furthermore, in [20], ESC is extended to a class of nonlinear systems with periodic outputs. In general the ESC structure consists of three subsystems: the cost function, a gradient estimator and an optimizer block. Each subsystem can be designed separately, allowing the application of the ESC to a variety of engineering applications [22,23,28,29]. The ESC relies on time-scale separation of each of the three subsystems [19]. The cost function is a static function and is only computed after the dynamics have reached steady state. The gradient estimator requires the dynamics to have reached steady state as well and requires the cost function value to be available. Therefore, the gradient estimator has a slower time scale than both the systems dynamics and the cost function. Finally, the optimizer has the slowest time scale since it relies on gradient information and some optimizers are even required to be sufficiently slow to reach a precise optimal value. In this paper a multi-parameters ESC is proposed to solve the N-dimensional optimization problem in (8) and the design of each of the subsystems is discussed. The block diagram of the proposed ESC is shown in Fig. 4, where the corresponding list of parameters is shown in Table 2. Finally, A≔fa0 ; a1 ; …; aN g; W≔fωd0 ; ωd1 ; …; ωdN g and D≔fδ0 ; δ1 ; …; δN g denote sets containing the tunable parameters for the ESC. Next, implementation of the ESC to the optimization problem at hand is discussed by considering the implementation of the constraint, cost function selection, gradient estimation and the optimizer block separately.

3.3.1. Constraint Applying ESC to solve the optimization problem (8) is not straightforward because of the equality constraint. A proper strategy to handle the constraint is therefore required. Using (7), the equality constraint on λ can be used to remove one of variable βj from the optimization as the constraint λ ¼ α in (8), which yields   floorðN=2Þ α∑m ¼ 1;m a j ð1Þm þ 1 ð2m1Þβ2m1 ð10Þ βj ¼ ð1Þj þ 1 ð2j1Þ where βj is an arbitrary slack-variable, which is removed from the ESC update. For every time instance, the value of βj is given by (10) which forces the solution to stay on the feasible set. In general any odd Chebyshev coefficient can be selected, but for the remainder of this paper β1 is selected as the slack-variable. Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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Fig. 4. Multi-parameter extremum seeking control block scheme.

Table 2 Parameters in extremum seeking control in Fig. 4. PARAMETER

DESCRIPTION

κ V αn ωdn δn

Compensator parameters Cost function ((5) and (8)) Perturbation amplitude Perturbation frequency Step size of the gradient descent

3.3.2. Cost function selection The cost function is selected to equal the performance measure in Definition 6. This frequency domain cost function relies on spectral analysis of the output spectrum and therefore sufficient time is allowed for transients to vanish and the data block length is selected to attain leakage free measurements. 3.3.3. Gradient estimation The gradient estimator perturbs the cost function with periodic signals. A sinusoidal signal is commonly used as a perturbation signal in ESC. The period of this sinusoid characterizes the time-scale of the gradient estimator and should be larger than the time scale of the cost function computation. Furthermore, a moving average filter is used to improve the accuracy of the estimation and the approach in [20] is extended to the multi-parameter optimization case. The analysis of the gradient estimator is provided below. In general the analysis is valid for any number of parameters, but for simplicity the two parameters case is considered. Consider the two parameters ESC case. By means of Taylor series approximation the cost function can be written as κ ¼ ½β^ 0 þ a0 sin ðωd0 Þ; β^ 1 þa1 sin ðωd1 Þ VðκÞ ¼ Vðβ^ 0 þ a0 sin ðωd0 Þ; β^ 1 þa1 sin ðωd1 ÞÞ ∂  Vðβ^ 0 ; β^ 1 Þ þ a0 sin ðωd0 tÞ Vðβ^ 0 ; β^ 1 Þ ∂β^ 0 ∂ þ a1 sin ðωd1 tÞ Vðβ^ 0 ; β^ 1 Þ þOða0 ; a1 ; ωd0 ; ωd1 Þ ∂β^ 1

ð11Þ

If an is chosen to be relatively small, the higher order terms can be disregarded. Furthermore, in each channel of the gradient estimator, the cost function is modulated with a sinusoidal signal with a frequency corresponding to the respective channel. Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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For parameter β0 , using the approximation in (11), this yields e^ 0 ¼

2 ^ ^ ∂ Vðβ 0 ; β 1 Þ sin ðωd0 tÞ þ 2 sin 2 ðωd0 tÞ Vðβ^ 0 ; β^ 1 Þ a0 ∂β^ 0 þ

2 ∂ a1 sin ðωd1 tÞ sin ðωd0 tÞ Vðβ^ 0 ; β^ 1 Þ a0 ∂β^ 1

ð12Þ

As any multiplication of sinusoidal functions can be written as summation of sinusoids, this yields e^ 0 ¼

2 ^ ^ ∂ Vðβ 0 ; β 1 Þ sin ðωd0 tÞ þ 2 sin 2 ðωd0 tÞ Vðβ^ 0 ; β^ 1 Þ a0 ∂β^ 0 þ

2a1 ∂ Vðβ^ 0 ; β^ 1 Þ a0 ∂β^ 1

  1 ð cos ððωd0 ωd1 ÞtÞ cos ððωd0 þ ωd1 ÞtÞÞ 2

ð13Þ

At first glance, the estimated gradient does not seem to be a good estimate, however averaging over time and taking the limit: Z T ∂ e^ 0 dt ¼ lim e0 ¼ Vðβ0 ; β1 Þ lim T-1 0 T-1 ∂β0 yields that the estimated gradient will converge to the true value of the gradient. The analysis can be repeated for other parameters, which leads to the same conclusion. In practice, the length of the averaging block should be selected be sufficiently long to achieve an accurate estimate of the gradient. 3.3.4. Optimizer block The optimizer, which has the slowest time scale, is selected to be a gradient descent optimizer and is given by β^n ðt d Þ ¼ β^n ðt d 1Þδn eðt d Þ

ð14Þ

The gradient eðt d Þ provides the direction towards the minimum and δn determines the step size for each time instance. The convergence rate of the gradient descent therefore depends on the value of δn . The selection of δn is a trade-off between convergence rate and accuracy. Selecting a high value of δn can increase the rate of convergence but at the same time affects the accuracy of the optimization, which may result in oscillating behavior near the optimum. Summarizing, a multi-parameter ESC has been introduced to solve the optimization problem (8), based on the performance measure in Definition 6. The application of this procedure is demonstrated in both simulations and experiments in the remainder of this paper. 4. Simulation: linearization of a magnetic actuator This section presents the application of the method introduced in the preceding section in a simulation environment. A simplified model of a moving mass interacting with a magnetic field is used for the case study as in [6]. The system is depicted in Fig. 1 and consists of a mass m, with corresponding displacement yA ðηdηdÞ; η A ð01Þ, which is connected to a linear spring k and a damper b. Moreover, the system is subject to a linear Proportional – Differential (PD) feedback control force fc that aims to track a reference signal w(t). The magnetic field exerts a force Fm on the mass and varies nonlinearly with the position of the mass such that F m ðyÞ ¼

1 θ0 þθ1 yþ θ2 y2

ð15Þ

with θi A R such that Fm is monotonically decreasing. Alternatively the system can be represented in Lur'e form as depicted in Fig. 5. The LTI block consists of the mass–spring–damper system and the PD controller. Moreover, the system and the controller are given as follows: GðξÞ ¼

K 4ðπξÞ2 þ4iπζωn ξ þ ω2n

ð16Þ

CðξÞ ¼ K p þ2iπK d ξ ð17Þ pffiffiffiffiffiffiffiffiffi with ωn ¼ k=m is the natural frequency and ζ ¼ b=2km is the dimensionless damping of the mass–spring–damper system and K p ; K d A R the parameters of the PD controller. The nonlinearity block consists of the magnetic field ϕðyÞ ¼ F m ðyÞ. Prior to applying the method introduced in the preceding sections Assumptions 1–3 in Theorem 1 need to be verified. First of all, application of Lemma 1 yields that the system is uniformly convergent for the simulation parameters presented in Table 3. Moreover, as jGðξÞ=ð1 þ CðξÞGðξÞÞj a 0 8 ξ A R and simulations yield a nonzero response to a sinusoidal input, the second and third assumptions in Theorem 1 are satisfied as well. Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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Fig. 5. Lur'e form of the PD-controlled mass subject to magnetic field with the addition of cascade compensator.

Table 3 Parameters of the system depicted in Fig. 5. PARAMETER

VALUE

ωn ζ K θ0 θ1 θ2 Kp Kd δ η

100 Hz 5 100 0.151 0.102 0.001 300 0.1 1 0.75

Table 4 Parameters for extremum seeking control (simulation). PARAMETER

VALUE

A W

an ¼ 0:04 8 n A f0; 1; …; 10g   2π 2π 2π 2π 2π 2π 2π 2π 2π 2π 2π ; ; ; ; ; ; ; ; ; ; 5 6 7 8 9 10 11 12 13 14 15 δn ¼ 0:01 8 nA f0; 1; …; 10g

D

The system is excited with a sinusoidal signal w(t) with amplitude 1 and frequency 2 Hz. The amplitude value is selected to excite the full operating range of the nonlinearity. The frequency can be selected arbitrarily, provided that output spectra are measured leakage free. As the input to the nonlinearity cannot be manipulated, a parallel compensator configuration is applied as shown in Fig. 3b. Finally since the effects of the magnetic field are to be linearized but not to be removed, the constraint on the linear feed-through is selected to λ ¼ 0 (see Table 1). The ESC is implemented in Matlab Simulink using Chebyshev polynomials up to order 10. The parameters for the extremum seeking are listed in Table 4. These parameters take the principle of separation of time scales into account. The perturbation signal for the gradient estimator was set to be sufficiently slower than the cost function by tuning the period of the perturbation. The input to the compensator is scaled by 1=2d to be in ½1 1 to improve numerical conditioning. The initial condition is chosen to equal the uncompensated state, i.e. βn ¼ 0 and the performance is optimized by solving the optimization problem in Definition 7. The results are depicted in Figs. 6 and 7. Simulation results show that the ESC is able to solve the performance optimization problem of Definition 7 in a real-time setting. Fig. 6 shows that the ESC converges to a minimum while satisfying the equality constraint. The performance improvement is shown in Fig. 6a, while Fig. 7a shows significant suppression of the higher harmonics in the output spectrum. The frequency domain results depicted in Fig. 7a correspond to the compensated profile in Fig. 7b which yields significant improvement in the sense that the compensated profile shows significant linearization. Furthermore, the compensated profile approaches the prescribed linear gain. Hence, as λ ¼ 0 the resulting linear feedback gains equal the gain of the nonlinearity at y2 ¼ 0. Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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5. Experiment: nonlinear amplifier In this section the application of the proposed method in an experimental setup is presented. The setup consists of an operational amplifier connected to a low pass filter. The amplifier exhibits nonlinear behavior as shown in Fig. 9. The nonlinearity profile consists of two different linear slopes with a smooth transition in between and the low pass filter has a cut-off frequency ωc ¼ 120 Hz. Finally, the Lur'e representation of the system is depicted in Fig. 8. As the system does not possess undamped zeros and shows a nonzero response to a sinusoidal input Assumptions 2 and 3 in Theorem 1 are satisfied. Furthermore, the system is assumed to be convergent based on the physical elements present in the system. Hence, all assumptions in Theorem 1 are satisfied and the optimization in Definition 7 can be applied.

Fig. 6. Performance of ESC in simulation of nonlinear magnetic field. (a) Cost function evolution of performance optimization of nonlinear magnetic field. (b) Linear feedthrough constraint.

Fig. 7. Comparison between uncompensated and optimally compensated nonlinear magnetic field. (a) Output spectrum at harmonics k of the excitation frequency with and without compensation. (b) Nonlinear magnetic field, optimal compensator and result after linearization.

Fig. 8. Schematic depiction of the experiment setup in Lur'e form.

Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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Fig. 9. Operational amplifier characteristic (experimental setup).

Table 5 Parameters for extremum seeking control (experiment). PARAMETER

VALUE

A W

an ¼ 0:001 8 n A f0; 1; …; 7g   2π 2π 2π 2π 2π 2π 2π 2π ; ; ; ; ; ; ; 5 6 7 8 9 10 11 12 δn ¼ 0:075 8 nA f0; 1; …; 7g

D

The experiment is conducted with a sinusoidal excitation signal of amplitude 0.8 V. The selection of the amplitude of the excitation signal is crucial in this case, since a small amplitude does not excite the nonlinear region of the system. The selected amplitude in this experiment is sufficient to excite the nonlinearity of the amplifier. The excitation signal frequency is set to be 15.625 Hz due to technical specifications and to prevent leakage. In this experiment, access to the input of the nonlinearity is available and therefore a cascade compensator configuration is selected. The optimization is performed with Chebyshev polynomials up to order 7. Similar to the simulation case, the initial condition is set to be the uncompensated case, i.e. β1 ¼ 1; βn ¼ 0 8 n a 1. Finally, the parameters for the extremum seeking are listed in Table 5. The experiment shows that the ESC is able to solve the performance optimization problem of Definition 7 in a real-time setting. The ESC shows convergence to a minimum and feasibility of solution is assured during optimization, as can be seen in Fig. 10. Fig. 11a shows that the proposed method is able to reduce the higher harmonics in the output spectra and Fig. 11b shows the initial amplifier profile, the compensator profile and the compensated amplifier profile. To illustrate the contribution of the constraint, α is set to be 0.4375. In general, α can be set to any arbitrary value, although one has to consider the physical limitations of the system. In practice, if information about the nonlinearity is not available, one can always select α ¼ 1 for the cascade configuration which results in a linear gain feedback equal to the local linearization of the original nonlinear function around 0. An example of the benefit of the freedom to tune the parameter α in practice is for example provided in [6]. Finally, the uncompensated and compensated nonlinearity are compared using a least square linear approximation to show how well the initial and compensated profile match a linear function. Fig. 11c shows a significant reduction of the difference (error) between the linear approximation and the actual profile after the compensation.

6. Conclusions A novel method for real-time compensation of performance degrading nonlinear effects in Lur'e-type systems is presented. The method addresses the nonlinearity compensation problem in three steps: (1) frequency domain based performance quantification, (2) compensator structure and configuration selection/parametrization and (3) real-time performance optimization using extremum seeking control. Recent developments with respect to frequency domain based performance analysis of Lur'e-type systems allow optimal compensation of performance degrading nonlinear effects while requiring little knowledge about the systems dynamics. The resulting optimization problem is formulated using Chebyshev basis functions while an extremum seeking controller is used to solve the optimization problem in real-time. The method Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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Fig. 10. Performance of ESC in experiment of nonlinear amplifier. (a) Cost function evolution of performance optimization of nonlinear amplifier. (b) Linear feedthrough constraint.

Fig. 11. Comparison between uncompensated and optimally compensated nonlinear amplifier. (a) Output spectrum at harmonics k of the excitation frequency with and without compensation. (b) Nonlinear amplifier, optimal compensator and result after linearization. (c) Error relative to linear approximation of the experiment before and after compensation.

requires no a priori knowledge of the nonlinearity, it is based solely on output measurements and is successfully demonstrated in both simulation and experiments. The methodology proposed in this paper may be extended to include non-convergent systems and a more general class of nonlinear systems including multi-dimensional and non-static nonlinearities. Secondly, the case where different outputs are available and the control action is not collocated with the nonlinearity may be considered in the future as well. Finally, further analysis of the performance measure, optimal basis function selection and tuning of the extremum seeking algorithm should be included in future research. References [1] X. Jing, Z. Lang, S. Billings, G. Tomlinson, Frequency domain analysis for suppression of output vibration from periodic disturbance using nonlinearities, Journal of Sound and Vibration 314 (3–5) (2008) 536–557.

Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i

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Please cite this article as: A. Setiadi, et al., Frequency domain based real-time performance optimization of Lur'e systems, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.08.030i