2016 American Control Conference (ACC) Boston Marriott Copley Place July 6-8, 2016. Boston, MA, USA
Multi-Delay Controller for Multi-Hazard Mitigation Liang Cao1 and Simon Laflamme1,2
Abstract— We present the concept of a variable input-space controller (VISC), for mitigation of multiple types of hazards. The VISC is a multi-delays feedback controller. The dimension and time delay of the feedback follow the construction of a delay vector based on the embedding theorem. These parameters are allowed to vary in time, which enables the controller to adapt to different types of excitations, or dynamic inputs. In this paper, we investigate the applicability of such controller by analyzing the selection procedures for the time delay, for a single-degreeof-freedom system subjected to a harmonic loading. This case leads to fixed parameters (i.e., dimension and time delay), and yields a specialized form of VISC: a multi-delay controller (MDC). The selection of the optimal time delay for a fixeddimension MDC is analyzed based on mutual information theory, and also based on the minimization of a transfer function. The comparison of both analyzes validates that a controller based on a delay vector is effective at mitigating vibrations, and conditions a given to ensure stability of such VISC. Results from this paper show that the controller has the potential to automatically adapt to different types of excitations, and can be based on limited measurements.
I. INTRODUCTION Civil infrastructures need to be designed and maintained to ensure daily operability and public safety under the effects of service and extreme loads. This can be done through a performance-based design (PBD) approach. In PBD, dynamic parameters (i.e, mass, damping, and stiffness) are designed to restrict motion to a given level of performance for a given load [1]. Supplemental damping is often used as an effective way to reduce structural response, amongst which passive technologies, such as viscous dampers and friction elements, have gained acceptance and popularity in the field of structural engineering [2]. However, these devices are known to be effective over a limited bandwidth [3], [4]. There is a push to develop damping strategies that are capable of covering a larger bandwidth in order to control for multiple hazards (e.g., strong winds, hurricanes, earthquakes) occurring either concurrently or sequentially in time. Devices capable of such high controllability belong to the group of semi-active, hybrid and active control devices [5]–[7]. They differ from passive solutions by necessitating power to operate, which power is used to adapt the device’s parameters (e.g., orifice size, normal force, liquid viscosity). However, these devices are not widely accepted due to applicability obstacles over the entire control loop (e.g., robust sensors, large-scale actuators, reliable power, etc.). In particular, 1 Department of Civil, Construction, and Engineering, Iowa State University, Ames, IA
Environmental 50011, USA
[email protected] 2 Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA
978-1-4673-8682-1/$31.00 ©2016 AACC
there are challenges in designing controllers for multi-hazard mitigation of civil structures, namely: 1) uncertainties on dynamic parameters are very large and excitations are unknown; 2) available measurements are limited and sensors have non-negligible probabilities of failure over time; 3) there is an immediate performance requirements; and 4) input-output data sets are unavailable [8], [9]. To address these challenges, several intelligent controllers have been proposed for structural control application. These include adaptive [10]–[12], fuzzy logic [13]–[15], and neuro[16]–[18] controllers. While these controllers are capable of a certain level of adaptability, they still heavily rely on tuning of their parameters, which is typically done to guarantee performance for a specific type of excitations. Here, we propose a new control strategy for multi-hazard mitigation, where the controller’s input states, for instance the type (e.g., acceleration, delayed force, size of delays) and number of inputs (e.g., number of delays, number of states), are allowed to vary in order to adapt to the different dynamics from different excitations. We term such controllers variable input space controllers (VISC), because their formulation will dynamically vary as a function of the measurements provided by the sensors. In related work, Pyragas proposed the time-delay autosynchronization (TDAS) method to stabilize the response of chaotic systems [19]. The method found limitations in the high period orbits. Instead, Socolar, Sukow and Gauthier [20] proposed a generalization of TDAS controllers, the extended TDAS (ETDAS), which applied to systems with large Lyapunov exponents and high period unstable period orbits. The ETDAS has difficulties controlling unstable steady states. Ahlborn and Parlitz [21] proposed a multiple delay feedback control (MDFC) to overcome TDAS and ETDAS limitations. The MDFC includes two or more delayed feedback signals with different delay times. The MDFC showed good improvement in performance, but introduced a significant numbers of control parameters [22]. Gjurchinovski and Urumov proposed a variable delay feedback control (VDFC) [23] to improve the performance of TDAS in controlling unstable steady state. The time delay is modulated during the control process. A limitation of the VDFC is the need to pre-defined time delay functions. The selection of the input space in the construction of representations (including control rules) is often overlooked despite its great importance [24], [25]. For instance, the choice of inputs influences computation time, adaptation speed, effects of the curse of dimensionality, understanding of the representation, and model complexity [24], [26], [27]. A proper selection of the input space may provide a more
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efficient representation. The proposed controller has input parameters that are selected based on the embedding theorem [28], [29]. With this strategy, the time series measurements of a single sensor are embedded in a delay vector, and this vector is used as the input space. The variable parameters are the embedding dimension d and time delay τ. The delay vector and its parameters have been used in several engineering applications, including structural health monitoring [30]–[32], and for model prediction, system identification, and control [8], [25], [33]–[35]. In this paper, we investigate the opportunity to use the embedding theorem to select the controller’s input parameters. In particular, we study the relationship between the optimal time delay selected using mutual information theory [36] for a harmonic signal and the time delay yielding to an optimal delayed controller for a single-degree-of-freedom (SDOF) system subjected to the same harmonic signal. Note that this specific case leads to a fixed control rule as a function of multiple delays of a single observation. It does not include a variable input space, but instead provides insights on the applicability and design of the control strategy. We refer to this specific controller as multi-delay controller (MDC). This paper is organized as follows. Section II describes the VISC strategy, and derive the optimal time delay using mutual information theory for a harmonic signal. Section III derives the optimal time delay in terms of minimization of the response function of an SDOF system subjected to a harmonic loading, and provides a stability analysis. Section IV validates the controller via a simple numerical simulation conducted on an SDOF system subjected to a harmonic excitation. Section V concludes the paper.
Consider the SDOF system of the form
u(t) = k1 y(t) + k2 y(t − τ) + ... + k(d−1) y(t − (d − 1)τ) d
A. Specialization of VISC: MDC To investigate the applicability and optimality of the control strategy proposed in (4), we study the specific application of the controller to a fix harmonic signal, yielding an MDC. Such application will lead to more trackable solutions. In this case, the external forcing f (t) in (1) is taken as f (t) = fˆ sin(Ωt)
(5)
where fˆ is the magnitude of the excitation and Ω the frequency of the excitation. The optimal embedding dimension of a harmonic signal is known to be d ∗ = 2 [39]. The optimal time delay τ ∗ can be computed based on information theory. As shown in Ref. [40], take two signals f1 (t) and f2 (t) that consist of two sinusoidal functions with a phase shift angle φ ∈ [0, 2π]. f1 = fˆ1 sin(θ ) f2 = fˆ2 sin(θ + φ )
(2)
τ ∗ = 0.25T
(6)
(7)
where T = 2π/Ω is period of the excitation and ν ∗ = [ y(t) y(t − τ ∗ ) ]
(8)
In the next section, we investigate the optimal value for τ that would provide the best control performance for an SDOF system. Results will be used to compare with τ ∗ found in this section, and assess the applicability of the control strategy.
i=1
III. OPTIMAL MDC
where y(t) is an observation, k are constants, d and τ can be time-varying. These parameters can be selected based on the embedding theorem. The embedding theorem states that there is an optimal d and τ that can be selected to construct a delay vector ν ν(d, τ) = [y(t) y(t − τ) y(t − 2τ) ... y(t − (d − 1)τ)]
(4)
where K is the vector of constants ki .
(1)
typically used to represent the dynamic of a civil structure of mass m, damping c, and stiffness k, subjected to an external forcing f (t) and control force u(t), and where x denotes displacement and the dot a time derivative. A VISC can be written in the form
= ∑ ki y(t − (i − 1)τ)
u(d, τ) = Kν
where θ is uniformly distributed over [−π, π] and can be taken as θ = Ωt. It can be shown that the optimal entropy is obtained when φ = ±π/2 [40]. This is equivalent to a quarter of the excitation period 2π, or
II. VARIABLE INPUT SPACE CONTROLLER
mx(t) ¨ + cx(t) ˙ + kx(t) = f (t) + u(t)
extended to a general class of nonautonomous systems with deterministic forcing [29], state-dependant forcing [37], and stochastic forcing [38]. It has also been shown that ν can be modified to include multivariate observations and unknown inputs [32]. Because ν preserves the essentially dynamics of the system of interest, it is argued here that ν can be used as inputs for a controller. Note that the control rule in (2) can be rewritten
(3)
which vector is considered as topologically equivalent to the original system from which y(t) is taken. The theorem has been developed for autonomous systems [28], and has been
Consider the observation y(t) = x(t) for the SDOF dynamics in (1), which makes u(t) a multi-delay displacement feedback controller: u(t) = −k1 x(t) − k2 x(t − τ)
(9)
Take ρk1 = k1 /k and ρk2 = k2 /k. Equation (1) is rewritten x(t) ¨ + 2ξ ωn x(t) ˙ + ωn2 x(t) + ρk1 ωn2 x(t) + ρk2 ωn2 x(t − τ) = f (t)/m (10)
p c where ωn = k/m and ξ = 2mω are the natural frequency n and fundamental damping ratio of system, respectively. An
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optimal delay τ ∗ can be found such that the response in (10) is minimized for the harmonic excitation given in (5). To do so, two dimensionless variables ρ = Ω/ωn and ρτ = τ/T are introduced, and the steady-state response is considered: x(t) = A sin Ωt + B cos Ωt
1 =p [1 − ρ 2 + ρk1 + ρk2 cos(2πρτ )]2 + [ρk2 sin(2πρτ ) − 2ξ ρ]2
τλ 3 +(2 + 2ξ ωn τ)λ 2 + (4ξ ωn + ωn2 τ + ρk1 ωn2 τ − ρk2 ωn2 τ)λ + 2ωn2 + 2ρk1 ωn2 + 2ρk2 ωn2
(11)
A transfer function H that represents the displacement amplification is obtained: x(Ω) |H(Ω)| = k fˆ
which yields to a third degree polynomial in λ :
(12)
The objective is therefore to select ρk1 , ρk2 , and ρτ to minimize H. Before doing so, we look at the bounds on these parameters to ensure stability. A. Stability
(20)
=0
Fig. 1 is a stability study using (20) under various feedback coefficients (ρk1 = 1 and ρk2 = {0, −0.1, −0.2, −0.3, −0.4, −0.5}) applied to an SDOF of period T = 2 with a the fundamental damping ratio ξ = 2% (typical for a civil structure). Values for ρk1 and ρk2 were selected to meet the stability conditions from (16) and (18). The SDOF system moves from stable to unstable with increasing time delay. When the path shifts from the left half-plane to the right half-plane, the corresponding value of τ defines the maximum time delay limit, which varies with ρk2 . This maximum time delay corresponds to λR = 0, or when λ = λI i. Substituting for λ in (13) leads to
For the stability analysis, take the general form of the homogeneous solution x(t) = xe ˆ λt to obtain the characteristic equation λ 2 + 2ξ ωn λ + ωn2 + ρk1 ωn2 + ρk2 ωn2 e−τλ = 0
(13)
The exponential term can be expressed by the power series 1 1 e−τλ = 1 − τλ + (τλ )2 − (τλ )3 + ... 2 6
(14)
Retaining the first two terms only and solving (13), λ has two roots λR ± λI i which can be estimated as 1 λR = −ξ ωn + ρk2 ωn2 τ q 2 1 λI = ωn 4 + 4ρk1 + 4ρk2 − (2ξ − ρk2 ωn τ)2 2
(15)
Fig. 1: Stability condition of an SDOF system with a period T = 2 s and damping ratio ξ = 2%
The system is stable if λR < 0, which gives an expression for ρk2 ρk2 < 2ξ /(ωn τ)
Also, for 4 + 4ρk1 + 4ρk2 − (2ξ − ρk2 ωn τ)2 < 0, the imaginary part vanishes and λ becomes 1 λ = − ξ ωn + ρk2 ωn2 τ 2 q 1 ± ωn 4 + 4ρk1 + 4ρk2 − (2ξ − ρk2 ωn τ)2 2
−λI2 + ωn2 +ρk1 ωn2 + ρk2 ωn2 cos(τλI )
(16)
+ (2ξ ωn λI − ρk2 ωn2 sin(τλI ))i = 0
The equation is satisfied when the real and imaginary terms vanish: −λI2 + ωn2 + ρk1 ωn2 + ρk2 ωn2 cos(τλI ) = 0
(17)
2ξ ωn λI − ρk2 ωn2 sin(τλI ) = 0
1 − 12 τλ 1+
1 2 τλ
+ O[(τλ )3 ]
λI4 + (4ξ 2 ωn2 − 2ωn2 −2ρk1 ωn2 )λI2 2 4 + (ωn2 + ρk1 ωn2 )2 − ρk2 ωn = 0
(18)
Equation (16) and (18) show that the stability of system is related to all three variables ρk1 , ρk2 and τ. In what follows, we will study the stability of τ in terms of ρk1 and ρk2 . The exponential term in (13) is further expanded: e−τλ =
(22)
yielding to
The maximum root of λ has to be negative for λ < 0, yielding 1 + ρk1 + ρk2 > 0
(21)
(19)
(23)
Equation (23) can be used to give a condition for stability independent of time delay. Such stability is guaranteed if the solution for λ has no imaginary components. This occurs when 2 ρk2 − 4ξ 2 ρk1 < 4ξ 2 − 4ξ 4 (24) Lastly, the maximum allowable time delay is obtained by solving (22) and (23) for τ in terms of λI . The procedure
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for determining maximum time delay τ|max can be found in Reference [1], where it can be shown that: r λI = ωn2 + ρk1 ωn2 − 2ξ 2 ωn2 +
1 2
q
2 ω4 16ξ 4 ωn4 − 16ξ 2 ωn4 − 16ξ 2 ρk1 ωn4 + 4ρk2 n
(25)
In the delay dependent region, the maximum time delay can be obtained by solving the first part of equation (22) using (25), which leads to −ρk1 ωn2 − ωn2 + λI2 1 (26) τ|max = cos−1 λI 2ρk2 ωn2 The value for τ|max is plotted in Fig. 2 for T = 2 s and ξ = 2%. As ρk1 increases, τ|max decreases, and varying ρk2 will influence τ|max .
Fig. 4: Response function for different time delay feedback control Fig. 3 is a plot of the transfer function H as a function of ρ and ρτ . For this analysis, ρk1 and ρk2 are taken as 2 and -1, respectively, to meet stability criteria 1 and 2, and ρτ is limited to 0.25, as any additional delay would provide redundant information in terms of topology of the phasespace. Fig. 4 is the plot of H versus ρ for various values of ρτ (slices of Fig. 3). Results from Figs. 3 and 4 show that increasing ρτ reduces H, until a critical frequency ratio ρcr is reached. An expression for ρcr can be obtained by substituting appropriate values for ρτ in (12): ρcr = ξ +
Fig. 2: Maximum time delay τ|max as a function of ρk1 and ρk2 for T = 2 s and ξ = 2% B. Optimal time delay Here, we numerically investigate the effect of various time delays on H. We define τ ∗ as the value that minimizes H. Furthermore, we enforce the stability criteria found in the last subsection, namely: 1) ρk2 < ξ /(ρτ π) 2) 1 + ρk1 + ρk2 > 0 −ρk1 ωn2 −ωn2 +λI2 3) τ|max = λ1I cos−1 2 2ρ ω
q
ξ 2 + ρk1 + 1
(27)
For values of ρ < ρcr , the time delay ratio ρτ = 0.25 yields the optimal performance. However, for low values of ρ, stability criterion 3 will govern. Fig. 5 is a plot of the optimal time delay ratio for the MDC. The optimal time delay ratio is governed by the stability limit (black line; ρτ = τ|max /T ) until it reaches the value obtained from optimal mitigation performance (blue dashed line, ρτ = τ ∗ /T ). Once the excitation ratio ρ is higher than the critical frequency ratio ρcr , no time delay (red dashed-dotted line, ρτ = 0) yields an optimal performance.
k2 n
Fig. 5: Optimal time delay ratio ρτ under different frequency input with ρk1 = 2 and ρk2 = −1 Fig. 3: 3D plot of H function with ρk1 = 2 and ρk2 = −1 1854
IV. VISC: T IME D ELAY S ELECTION RULES A comparison of results for τ ∗ obtained using mutual information (Section II) and optimal mitigation performance (Section III) can be used to establish time delay selection rules for a VISC. Consider a harmonic signal with time-varying Ω. The mutual information theory can be used, in real-time, to select the appropriate value for τ. Note that the development of a method to achieve such selection is out-of-the-scope of this paper. Using findings from this paper, the following rules can be followed to establish τ ∗ : • If ρ ≥ ρcr , τ ∗ = 0. • If ρ < ρcr , τ ∗ = 0.25T and bounded by stability criterion 3.
(a)
A. Numerical example The proposed controller is simulated using an SDOF system subjected to a harmonic excitation f (t) = fˆ sin(Ωt) with one specific frequency Ω = 0.5ωn . Three control cases are considered: 1) VISC 2) No time delay controller (NDC) The NDC is a negative displacement feedback controller (u(t) = −(ρk1 + ρk2 )kx(t)). 3) Linear quadratic regulator (LQR) The LQR is constructed using regulatory weights Q = qI with positive definite diagonal elements and actuation weight r taken as a positive definite constant. The model and control parameters are listed in Table I where the LQR control parameters are pre-tuned to optimal mitigation performance. The displacement responses and control forces are plotted in Fig. 6. For each control case, the maximum absolute actuation force is bounded by umax . Results show that the VISC performs significantly better than the NDC, which reduces displacement by 17% compared with the uncontrolled response after stabilization (not shown). The VISC reduces displacement by 88% while the LQR reduces displacement by 94%. The mitigation performance of the VISC shows promise of the controller as a datadriven method. Future work will include the investigation of an adaptive time delay selection to improve performance. TABLE I: List of simulation parameters Object model
input
Parameter class mass stiffness damping ratio gain gain weight weight maximum control force amplitude of excitation
Parameter m k ξ ρk1 ρk2 q r umax fˆ
Value 0.05 kg 2 kN/m 2% 6 -1 100 1 2 kN 2 kN
V. CONCLUSIONS In this paper, we have introduced the notion of variable input space controllers (VISC) for mitigation of multiple
(b)
Fig. 6: Time series of (a) displacement response and (b) control force. hazards. The principle of VISC is to base the control rule on a delay vector, which embedding and delay sizes are allowed to vary in time. The delay vector is constructed based on the embedding theorem, yielding parameters d and τ. We have specialized the VISC concept to a system subjected to a single harmonic input, leading to a fixed delay vector of dimension d and optimal delay τ ∗ . This specialized application was termed a multi-delay controller (MDC). The analysis of the MDC was used to validate that basing the input space of a controller on a delay vector could be effective at vibration mitigation, and to provide rules for the selection of the time delay τ. The analysis begun by investigating the optimal time delay provided by the mutual information theory. After, the optimal time delay was studied in terms of minimization of a transfer function, and stability bounds were evaluated. Results showed a good agreement between both studies, showing that the delay vector could effectively be used. Additional requirements were added on the selection of τ, there is a critical excitation frequency beyond which τ ∗ = 0, and τ is bounded at low excitation frequencies by a stability criterion. Two other stability criteria have been established for the selection of control gains. Results from this paper demonstrated the concept of VISC. Such controller has great promise for structural control, because it has the potential to automatically adapt to different
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types of excitations, and can be based on limited measurements. Future work include the development of algorithms for real-time selection of d and τ, the introduction of robust rules for varying d and τ, and the introduction of timevarying control gains to adapt to different magnitudes of excitations. ACKNOWLEDGEMENTS This material is based upon work supported by the National Science Foundation under Grants no. 1300960, 1463497, and 1537626. Their support is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. R EFERENCES [1] J. J. Connor and S. Laflamme, Structural Motion Engineering. Springer, 2014. [2] M. Symans, F. Charney, A. Whittaker, M. Constantinou, C. Kircher, M. Johnson, and R. McNamara, “Energy dissipation systems for seismic applications: current practice and recent developments,” Journal of Structural Engineering, vol. 134, no. 1, pp. 3–21, 2008. [3] J. N. Yang and A. K. Agrawal, “Semi-active hybrid control systems for nonlinear buildings against near-field earthquakes,” Engineering Structures, vol. 24, no. 3, pp. 271–280, 2002. [4] W. He, A. Agrawal, and J. Yang, “Novel semiactive friction controller for linear structures against earthquakes,” Journal of Structural Engineering, vol. 129, no. 7, pp. 941–950, 2003. [5] F. Ubertini, “Active feedback control for cable vibrations,” Smart Structures and Systems, vol. 4, no. 4, pp. 407–428, 2008. [6] I. Venanzi, F. Ubertini, and A. L. Materazzi, “Optimal design of an array of active tuned mass dampers for wind-exposed high-rise buildings,” Structural Control and Health Monitoring, vol. 20, no. 6, pp. 903–917, 2013. [7] L. Cao, A. Downey, S. Laflamme, D. Taylor, and J. Ricles, “Variable friction device for structural control based on duo-servo vehicle brake: Modeling and experimental validation,” Journal of Sound and Vibration, vol. 348, pp. 41–56, 2015. [8] S. Laflamme, J. Slotine, and J. Connor, “Self-organizing input space for control of structures,” Smart Materials and Structures, vol. (in press), 2012. [9] Z. Zou, Y. Bao, H. Li, B. F. Spencer, and J. Ou, “Embedding compressive sensing-based data loss recovery algorithm into wireless smart sensors for structural health monitoring,” Sensors Journal, IEEE, vol. 15, no. 2, pp. 797–808, 2015. [10] H. Li, J. Yu, C. Hilton, and H. Liu, “Adaptive sliding-mode control for nonlinear active suspension vehicle systems using t–s fuzzy approach,” Industrial Electronics, IEEE Transactions on, vol. 60, no. 8, pp. 3328– 3338, 2013. [11] S. Suresh, S. Narasimhan, and N. Sundararajan, “Adaptive control of nonlinear smart base-isolated buildings using gaussian kernel functions,” Structural Control and Health Monitoring, vol. 15, no. 4, pp. 585–603, 2008. [12] T. Senjyu, T. Kashiwagi, and K. Uezato, “Position control of ultrasonic motors using mrac and dead-zone compensation with fuzzy inference,” Power Electronics, IEEE Transactions on, vol. 17, no. 2, pp. 265–272, 2002. [13] B. N. Alajmi, K. H. Ahmed, S. J. Finney, and B. W. Williams, “Fuzzy-logic-control approach of a modified hill-climbing method for maximum power point in microgrid standalone photovoltaic system,” Power Electronics, IEEE Transactions on, vol. 26, no. 4, pp. 1022– 1030, 2011. [14] M. M. Algazar, H. A. EL-halim, M. E. E. K. Salem, et al., “Maximum power point tracking using fuzzy logic control,” International Journal of Electrical Power & Energy Systems, vol. 39, no. 1, pp. 21–28, 2012. [15] H. Chaoui and P. Sicard, “Adaptive fuzzy logic control of permanent magnet synchronous machines with nonlinear friction,” Industrial Electronics, IEEE Transactions on, vol. 59, no. 2, pp. 1123–1133, 2012.
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