A Brief Survey of Stochastic Resonance and Its Application to Control

Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India

A Brief Survey of Stochastic Resonance and Its Application to Control ? Jerome Moses M ∗ Ramakalyan Ayyagari ∗∗ ∗

(Research Scholar, e-mail: [email protected]) ∗∗ (Professor, e-mail: [email protected]) National Institute of Technology, Tiruchirappalli, TN 620015 IN. Abstract: Noise is generally considered as nuisance in engineering. Hence engineers routinely attempt to filter out noise. But quite remarkably a counter intuitive phenomenon was found in 1980s that changed the perception of noise, giving birth to a phenomenon called Stochastic resonance (SR). In this paper, we review various phenomena that are benefited by noise. Quite obviously, we extend those ideas to Control Systems design. We provide 3 examples: a randomized algorithm for a linear state feedback control law, (ii) minimizing limit cycles in nonlinear systems, and (iii) antilock braking system (ABS) design. Finally, we present some of our arguments in favour of stochastic resonance as a potential tool for system design in a broader perspective. Keywords: Stochastic resonance, Noise benefits, Control system synthesis 1. INTRODUCTION It has been nearly three decades, the idea of Stochastic Resonance (SR) has attracted attention. Hays et al. (1976) observed that the alternation between warm age and ice age was 105 years. Many attempts have been made by climatologists to find the cause for such recurrence. Benzi et al. (1981, 1982) addressed this paradox and coined the term SR as an explanation for the ice-age recurrence. They found that the earth’s orbit around the sun could not cause glaciation. Instead, they proposed that noise fluctuations might have amplified the effects of orbital eccentricity leading to periodic ice age. Their proposal was based on the idea that a feeble signal in a nonlinear system could be amplified with the help noise. Apparently, it is a counterintuitive phenomenon, since noise is generally considered as unwanted and we engineers routinely attempt to filter it out. An intuitive explanation of Benzi’s work is as follows. Consider a particle in a symmetric double-well potential given by a b ϑ(x) = − x2 + x4 2 4 The potential function curve p is described in fig 1. The minima are located at x = ± ab and the barrier height 2 is ∆ϑ = a4b . These minima represent the warm-age and the ice-age. Consider the motion of a particle in such a double-well potential in the presence of a weak periodic forcing signal Asin(ωt) and a Gaussian noise η(t). The weak periodic signal models the modulation of the earth’s orbital eccentricity. Climatic fluctuations because of solar radiation, atmospheric and oceanic circulations are modelled by gaussian white noise. The motion of the particle is described by ? The authors would like to thank National Institute of Technology, Tiruchirappalli for hosting the research.

978-3-902823-60-1 © 2014 IFAC

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Fig. 1. Potential Function x˙ = ϑ˙ + Asin(ωt) + η(t) where - A is the amplitude of the signal, ω is the modulation frequency, η(t) is Gaussian white noise of intensity D with zero mean and η(t)η(t + τ ) = 2Dδ(t − τ ). When A = 0 and η(t) = 0, the particle remains in one of the two states of the well; the weak periodic force alone cannot make the particle to transit between the two states. But when noise exists, there is a lowering of the barrier between two states, helping the particle to overcome the barrier and move from one minimum to other (cf. Fig 2). For an optimum value of noise intensity D, there is a synchronized hopping between the two minima of the well; in other words, the weak signal is assisted by noise causing transition between the two minima (warm and ice ages). Resonance occurs when the period of forcing signal equals the Kramer’s rate (Kramers, 1940) given by √ 2×a ∆ϑ rk = e(− ) π D where a, D, ∆ϑ are defined earlier, and rk is the transition rate. The noise intensity D can be tuned so that r1k matches the forcing signal frequency for SR to occur. 10.3182/20140313-3-IN-3024.00223

2014 ACODS March 13-15, 2014. Kanpur, India

assigned two possible states and SR was observed for an intermediate value of noise intensity. The notion of SR has widened over time and Collins et al. (1996) showed that SR could also occur when the input signal is aperiodic, and coined the term aperiodic stochastic resonance (ASR). ASR was quantified using power norm or normalized power norm. They demonstrated using ASR in bistable well system, an integrate-fire neuronal model and Hodgkin-Huxley neuronal model. Gammaitoni et al. (1998) wrote a comprehensive review of Stochastic resonance nearly after two decades of its invention. Harmer et al. (2002) presented a brief review SR and investigated SR in a number of bistable circuits.

Fig. 2. Transition between two states Generally SR is quantified using Signal Noise Ratio (SNR), or Spectral Amplification factor or Power Spectral Density. Evidently, noise is a bonus rather than nuisance. The rest of the paper is organized as follows. Section 2 briefly surveys Stochastic Resonance available in the literature. In section 3, we present three examples to speculate that the idea of stochastic resonance works well in control system design too. The paper is concluded in Section 4 with some of our arguments in favour of SR. 2. BRIEF SURVEY OF SR IN ENGINEERING In this section we present a survey of the results based on the idea of Stochastic Resonance since its conception by Benzi et al. (1981). Fauve and Heslot (1983) experimentally demonstrated the theory of SR using a Schmitt trigger. Roughly speaking its behaviour may be illustrated using the motion of a particle in the quartic potential. SR was observed in the power spectral density plot when twice the kramer’s rate is equal to the forcing frequency. The field remained inactive for sometime until McNamara et al. (1988) discovered the phenomenon in a ring laser experiment. In this experiment they were able to change the direction of light (clockwise to anti-clockwise and viceversa) in a closed loop. This is also similar to the mathematical model of a particle in a bistable potential. An acousto-optic modulator provides the sinusoidal modulation. It was observed that at an optimum value of noise intensity the SNR reaches maximum. Stochastic Resonance has also sparked interests among various research communities for further exploration. Douglass et al. (1993) investigated the phenomenon in the mechanoreceptors of crayfish and it was observed that at higher noise levels the signal to noise ratio decreased, making crayfish (which lives in a noisy environment) less vulnerable to predators. The hair cells on the tail were able to detect faint signals, which were amplified by the noisy environment and in turn activate the animal’s escape reflexes. Babinec (1997) studied the Weidlich model of public opinion formation. Only two kinds of opinions, either a “yes” or a “no”, can be made akin to the characteristics of a particle in a bistable system. The external influences on a particular person was characterized as noise. It was observed that noise could influence the collective opinion of the people. The idea of SR was further extended to opinion formation in small world networks by Kuperman and Zanette (2002). In this, each node in the network was 314

Coming to the field of electrical engineering, the idea of SR appears to have been explored by several researchers in signal processing. Chapeau-Blondeau (1997) studied a theoretical model for the transmission of a periodic signal and noise through a static nonlinearity. In this paper, he derived expression for SNR gain for a square pulse train and a sine wave, and demonstrated that for a stochastic resonator the gain is larger than unity. Rousseau and Chapeau-Blondeau (2005) demonstrated SR effects in various optimal detection strategies such as Bayesian, Neyman-Pearson, minimax, and minimum error probability detector. It was validated that the cost function of detectors achieved minimum for an intermediate value of noise. Saha and Anand (2003) studied SR effect in various quantizers and SNR gains were derived. It was noted that the performance of SR detector performed better than that of an optimal detector. Mitaim and Kosko (1998) introduced the notion of Adaptive Stochastic resonance and showed that adaptive systems can learn to add noise and increase the performance in nonlinear systems. This work provides a stochastic gradient ascent scheme to find the SR mode, and this learning scheme is depicted by few simulation studies on simple dynamical systems. They have also raised some interesting issues for further exploration – how SR learning algorithm will converge? and, which noisy dynamical systems show SR effects? Krawiecki and Holyst (2003) showed that bubbles and crashes in financial market can be modelled as phenomenon of stochastic resonance, much like the warm-age and ice-age cycles. It was shown that a piece of information carrying signal may be enhanced by an optimum value of noise via SR to induce a crash in a segment of the market. Fuentes et al. (2001) extended the idea of SR to Nongaussian noises. The key observation is that the SNR plot becomes broad and less sensitive to the actual value of the noise intensity. Priplata et al. (2002), Collins et al. (2003) found noise can enhance human balance control. Also, they showed that the postural sway of elderly persons (in their 70s) during quiet standing was reduced by applying subsensory mechanical noise to the feet. They demonstrated that the elderly people sway just as much as younger people (in the 20s) sway without noise. Badzey and Mohanty (2005) reported SR in bistable nanomechanical silicon oscillators. A radio frequency source controls the switching of beams between two states. It was observed that an addition of noise did amplify the modulation signal. This experiment perhaps enables us to

2014 ACODS March 13-15, 2014. Kanpur, India

explore SR in quantum mechanical systems. Korneta et al. (2006) investigated SR in Chua’s circuit experimentally. In this work they observed that the dynamics switches between the two stable chaotic attractor when forced by a periodic signal and noise. The interplay of noise and nonlinearity was further explored in logical operation leading to logical stochastic resonance (Bulsara et al., 2010). Noise enhanced performance was observed as one increases noise intensity. SR techniques was used also in image processing (Jha et al., 2012) for enhancement of very low contrast images. Also the expression of noise intensity that maximizes SNR was derived. SR based techniques also find their application in spectrum sensing methods of Cognitive Radios. He et al. (2010) noticed SNR of a receiving signal was increased and the detection probability of SR based approach is higher than that of the traditional energy detector. Kawaguchi et al. (2011) showed SR also enhances the information transmission in Neural Networks. Simulations were run on different neuronal models and the observation was that the mutual information attained maximum for an optimum value of noise amplitude. In this section we attempted to showcase a very interesting concept and its multifarious applications. In the next section we shall attempt to demonstrate the benefits of SR in control system synthesis.

mechanical device can provide limited voltage, current, force or torque. So, searching for a suitably bounded K matrix takes an unpredictable amount of time, and hence the problem belongs to class NP-hard. It was proven by Blondel and Tsitsiklis (1997) that when m > 1 and when the elements of the matrix K are constrained such that k ij ≤ kij ≤ k ij , the problem belongs to the class NPhard. Problems belonging to this class may be solved using Randomized Algorithms which are most likely to solve the problem efficiently, i.e., in polynomial time. In our work (Moses and Ramakalyan, 2014), we employed randomization in a way different from what has been reported in literature (Vidyasagar and Blondel, 2001; Khargonekar and Tikku, 1996; Calafiore et al., 2007). We have a polynomial time algorithm to compute a K matrix given the matrices A and B (satisfying the rank condition mentioned earlier), and the roots of χ. We used this algorithm to generate a large pool of K matrices. Consequently, we showed that this pool has an interesting distribution w.r.t the size of K. This distribution allows us to pick up the right K matrix, i,e. K : k ij ≤ kij ≤ k ij in polynomial time. Algorithm In this section we present our algorithm. The algorithm consists of 3 steps and we use randomization in all the steps. Inputs : A ∈
3. BENEFITS OF NOISE IN CONTROL SYSTEMS In this section we present 3 examples. The first one is rather typical, but we interpret the results in terms of SR. The second one is a demonstration of the effect of noise on limit cycles. The third one is an example borrowed from (Ariyur and Krstic, 2003) which has connections with SR. We believe that SR is the potential tool for designing largescale heterogeneous control systems such as formation of agents, cyber physical systems and the like. 3.1 Linear Control Design Problem Randomized Algorithms Engineers aspire to design and control larger systems where complexity is enormous. To tame complexity randomization appears to be a promising key. And, the idea of randomization has been successfully applied in computer science, optimization, and in many diverse fields. During the past decade or so it has been shown that a number of problems in matrix theory do not have polynomial time algorithms (Blondel and Tsitsiklis, 1997). From a computational perspective these problem are proven to be NP-hard. One of the problems of interest to the control engineering community is defined as follows. Given a system matrix A ∈ 1 . If K is unbounded any one of the K matrices generated will work for us. But real world applications of feedback control involve control actuators having rate limitation and bandwidth problems. In particular, any electrical or 315

Step 1 Following Ramakalyan (2003) & Brogan (1991) we generate K ∈
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Step 3 3.1 From the large pool of generated K in step 1, we filter the matrices with norm N such that Navg − t ≤ N ≤ Navg + t where t ∈ 1, 2, .., n . And these are stored in a cell array Ce. t is chosen so that the minimum size of the cell array is 200 3.2 From Ce, we randomly choose a matrix and check element-wise, if it satisfies the constraint; if yes, we return the matrix. Note that all the matrices Ki in our original pool yield the required characteristic polynomial χ. It is easy to see that this algorithm has polynomial time complexity since it depends only on the pool size of the generated matrices.

Fig. 3. Distribution of 1-Norm Vs No. of Matrices(Ex:1)

Numerical Examples For all the examples below, the number of samples generated was 10000 in step 1. We executed the algorithm for 1000 trials. Example 1: Let     

A1 =

10 19 8 14 −10 17 −18 −9 −7 16 −10 5 −9 1 7 3 19 13 −1 11 11 6 −11 2 −11 −6 10 18 −14 10 −15 18 14 −5 −15 −16 −10 −14 −6 3 3 3 0 0 −10 −12 2 −17 −1

    

 B1 =

   

1 0 1 0 0 1 1

1 0 0 1 1 0 0

1 0 1 1 1 0 0

0 1 0 1 0 1 0

1 1 1 1 0 1 0

0 1 1 0 1 1 1

1 1 1 0 0 1 0

    

The desired characteristic polynomial is χ1 = λ7 + 41.5λ6 + 688λ5 + 5841λ4 + 26886λ3 + 65721λ2 + 77724λ + 33696. We chose kij ∈ [−20, 20] . The Navg was found to be 92.90 & 52.25 for 1−norm & 2−norm respectively, and the mean of iterations in which we obtained K was around 50.07 & 32.28. One of the K matrices picked up at random (from 1−norm distribution) is K =   −3.664

2.274 14.283 8.432 6.862 −1.948 −12.903 7.907 3.701 3.025 12.183 −5.396 −16.589 −8.874 −2.256 −1.415 12.103 18.031 10.602 −0.577 −2.724 4.075 −19.361 −7.207 −5.271 −3.017 9.865 8.977 10.874 −0.405 −6.939 0.729 −10.613 −9.619 −8.892 −1.558 11.901 −11.216

 −11.206  −4.375  10.223  3.320

so that A + BK =     

−2.484 11.633 −0.811 −6.024 −9.125 −9.183 11.267

is χ2 = λ8 + 38λ7 + 604λ6 + 5222λ5 + 26719λ4 + 82292λ3 + 147636λ2 + 139248λ + 51840. We chose kij ∈ [−20, 20] Example 3 : The desired characteristic polynomial is χ3 = λ9 + 45λ8 + 870λ7 + 9450λ6 + 63273λ5 + 269325λ4 + 723680λ3 + 1172700λ2 + 1026576λ + 362880. We chose kij ∈ [−20, 20]. Clearly, adding randomization was effective for a successful search. The results are summarized in the table below for a quick reference. Table 1. Summary of Results

   

−16.539 −2.569 −14.651 8.493 4.262 11.585 −7.520 3.796 8.537 16.652 −8.568 4.112 −25.230 −12.064 1.533 3.311 −0.684 3.386 3.297 −5.868 0.663 3.962 −13.121 −8.011 −12.777 −11.079 15.026 13.672 −19.716 0.676 −12.807 12.421 7.829 6.498 −13.361 −6.867 7.812 0.935 3.864 8.975 −15.178 −12.549 6.201 11.252 15.157 −3.973 1.923 −18.218 −12.667

Fig. 4. Distribution of 2-Norm Vs No. of Matrices(Ex:3)

A

    

B

poly. χ

k limits

kKk

k Kavg k 92.20 52.25

50.07 32.28

Mean itr

A1

B1

χ1

[−20, 20]

1 2

A2

B2

χ2

[−20, 20]

1 2

105.74 56.76

3.16 1.18

A3

B3

χ3

[−20, 20]

1 2

118.40 61.04

127.21 2.460

and χ1 = λ7 + 41.5λ6 + 688λ5 + 5841λ4 + 26886λ3 + 65721λ2 + 77724λ + 33696. We typically have several matrices that would give us the needed χA+BK but, not within the bounds of K. However, the distribution of these matrices w.r.t kKk is rather surprising, and we have a larger pool candidate matrices with low kKk. In this example (cf. Fig. 3), we could see that the no. of matrices with smaller norm is larger compared to matrices with larger norm.

There are many more examples (Moses and Ramakalyan (2014)) we have generated; for brevity we have shown a few significant ones.

Example 2 : The system matrix A, input matrix B and the gain matrix K are mentioned in the appendix A for the remaining examples. The desired characteristic polynomial

It is noteworthy that, for a constant interval [k ij , k ij ] the number of mean iterations is significantly smaller for higher order system. But, given practical problems

316

Discussion The distributions of the candidate K matrices with respect to the norm kKk for examples 1 and 3 are shown in figure 3 and figure 4 respectively.

2014 ACODS March 13-15, 2014. Kanpur, India

such as actuator constraints in process control, bandwidth problem (Callier and Desoer, 1991), the very motivation for imposing the limits on kij , it is rather surprising to note that obtaining a matrix with smaller kKk can be pretty easy using randomization. In fact, we are skeptical about the NP-hardness of the problem. We are currently investigating this phenomenon further with higher order systems. We are also investigating a similar problem called partial pole-placement, whose complexity is not known till now. 3.2 Effect of Noise on limit cycle Limit cycles are isolated closed orbits. They are inherent in many nonlinear systems. And stable limit cycles are scientifically important, that is even without periodic forcing, the system oscillates. Stable limit cycles are the ones which attract all its neighbouring trajectories. Examples include heart beats and chemical oscillations.

Fig. 5. Dynamics with noise variance 0.4 − 0.6

From a control engineers perspective, it is desired to reduce the size of limit cycles. Wang and Krstic (2000) proposed a ES scheme for limit cycle minimization, which employs measuring the limit cycle amplitude and minimization is provided by ES algorithm. In this paper, we provide stochastic perturbation to the dynamics and study how noise influences the limit cycles. And we see the minimization effect from a probabilistic sense.

(1) At lower level of variance 0.1 − 0.3, there was no change in the phase portrait. The portrait remained as a unstable limit cycle. (2) At intermediate values of variance 0.4 − 0.6, the fixed point shifts either to the left or right of the origin. And the size of the limit cycle is minimized. (cf. fig 5) (3) At higher values of variance 0.7 − 1.0, the limit cycle vanished. (cf. fig 6) Thus one can observe limit cycle minimization in certain noise variance range. Evidently, we can exploit the benefits in the interplay of noise and nonlinearity. Nevertheless, much work has to be done to obtain the relationship between limit cycle amplitude and the noise intensity. 3.3 ABS Design In adaptive control theory literature, a non-model based method is Extremum seeking (ES). It is further classified into Deterministic ES and Stochastic ES. Deterministic ES finds its application in bio reactors (Wang et al., 1999), Impedance Matching (Zhang and Ordˆ ao¨ anez, 2012), etc. 317

x

<

^ x

>

y

f(x)

-k/s <

<

>

s/s+h <

>

x˙2 = −x1 + x2 (x12 + x22 − 1) This system has an unstable limit cycle around unit circle. To the above mentioned system, we added noise with gaussian distribution. We varied the variance between 0.1 to 1.0 with a step size of 0.1. For each variance we generated 10 phase portraits. Out of the 100 phase portraits generated we present 3 plots from Fig 6-8. And, we observed the following:

Fig. 6. Dynamics with noise variance 0.7 − 1.0

>

Consider the nonlinear system from Slotine and Li (1991) with dynamics x˙1 = x2 + x1 (x12 + x22 − 1)

a

Perturbation Signal

Fig. 7. Perturbation based ES scheme Manzie and Krstic (2009) developed Stochastic ES. It find its applications ranging from robotics (Liu and Krsti´c, 2010) to game-theory (Liu and Krsti´c, 2011).

PERTURBATION BASED ES The basic schematic for perturbation based ES is shown in Fig 7. y is the output to be minimized or maximized. The perturbation enters additively in this scheme. The perturbations can either be deterministic (like sin(ωt)) or stochastic (gaussian noise) Then the output y is passed through the filter s/s + h, which removes the dc component. The output from the filter also gets multiplied by the perturbation, generating the derivative y 0 . It is then passed through an integrator, which estimates x ˆ guiding it in the direction of zero gradient. If k < 0 then this scheme will converge to the nearest maximum and if k > 0 it drives to the nearest minimum.

2014 ACODS March 13-15, 2014. Kanpur, India

>

Torque Conrol

>

Wheel Model

-k/s <

<

Friction Force Coeff

>

s/s+h <

>

>

<

>

a

Gaussian Noise

Fig. 8. SES Architecture for ABS Design ABS DESIGN VIA STOCHASTIC EXTREMUM SEEKING

Fig. 9. Time Vs Friction force coeff

Anti-lock braking system is an essential automobile safety feature. There are many approaches to ABS design such ES (Ariyur and Krstic, 2003), sliding mode based ES (Yu and Ozguner, 2003). We consider the model given in (Ariyur and Krstic, 2003), with dynamics mu˙ = −N µ(λ) I Ω˙ = −BΩ + N Rµ(λ) − TB Where u is linear velocity, Ω is angular velocity , TB is the breaking torque, N is the weight of the wheel, µ(λ) is the coefficient of friction, λ is slip coefficient. We use stochastic perturbation based ES for ABS system design. Figure 8 shows the block diagram of the scheme proposed. The parameters of the wheels & the initial conditions are taken from (Zhang and Ordˆ ao¨ anez, 2012). This function has a maximum at λ = λ∗ , whose value is µ(λ∗ ). We run the simulation for λ∗ = 0.25 and µ∗ = 0.6. The parameters of the wheel are chosen as m = 400 kg, B = 0.01 and R = 0.3m. The initial conditions are linear velocity x1 (0) = 33.33 m/s, and angular velocity x2 (0) = 400/3.6, which makes λ(0) = 0. The simulation employs the stochastic perturbation based extremum seeking scheme with a = 0.40, h = 0.50 and k = 1.5. For λ0 = 0.1. We use stochastic perturbations that follows a gaussian distribution with mean(µ)= 0.38 and variance(σ) = 0.01. In the case of Deterministic ES the gaussian noise is replaced by sinusoidal perturbations. The simulation results are plotted from Fig 9-12. We compared the stochastic ES with deterministic ES for the friction force coefficient and slip in Fig 9 & 10. The Deterministic case uses a = 0.01 and perturbation signal as sin(3t). We observed for the same set of initial conditions the convergence toward the desired value was faster in Stochastic ES. It is seen that during braking, maximum friction force is reached and the vehicle is stopped within the shortest time and distance. The tradeoff is that more braking torque is required as in Fig 12. Also there are no oscillations.

Fig. 10. Time Vs Slip

Fig. 11. Time Vs Linear and Angular Velocity

Fig. 12. Time Vs Braking torque Effect of Mean on convergence

4. CONCLUSIONS AND DISCUSSIONS

We studied the sensitivity of the mean. Figure 13, shows the sensitivity plot of mean towards the convergence towards the extremum value. The desired slip value is 0.25. Thus, the choice of an appropriate value of mean is vital. It is observed that the mean has an effect on the convergence towards the extremum value. 318

In this paper we primarily attempted to showcase a very interesting concept, called Stochastic Resonance, and its multifarious applications. Then, we presented the benefits of SR in control system synthesis. The first example provides a polynomial time algorithm to compute the gain matrix K using random noise. In the second example

2014 ACODS March 13-15, 2014. Kanpur, India

Fig. 13. Effect of mean on slip we show limit cycle minimization by means of noise. In the third example we show that Stochastic based ES is better than Deterministic ES. These three examples are provided to highlight the potential benefits of noise. Also these examples provide some insights in designing system that can exploit the randomness in noise. Nevertheless, much work has to be done to obtain the basic relationships between the parameters of our choice, say the limits of the Kmatrix, or the limit cycle amplitude, and the noise intensity. We propose that noise benefits not only occurs in the certain natural phenomena, but also in Randomized algorithms (Vidyasagar, 1998), simulated annealing (Kirkpatrick et al., 1983) etc. The presumption of engineers is to eradicate noise in a system as it is generally felt that it is inherently present in all practical systems. Instead of eliminating noise, one can constructively exploit the randomness by noise. There are subtle difficulties one would face when trying to widen the definition of Stochastic Resonance. We are presently working towards a general scheme for extracting the maximum noise benefits in designing systems from a broader perspective, for example opinion formation ideas and learning algorithms in the design of networked control systems or cyber physical systems. REFERENCES Ariyur, K. and Krstic, M. (2003). Real-Time Optimization by Extremum-Seeking Control. Wiley-interscience publication. Wiley. Babinec, P. (1997). Stochastic resonance in the weidlich model of public opinion formation. Physics Letters A, 225(1), 179–181. Badzey, R.L. and Mohanty, P. (2005). Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance. Nature, 437(7061), 995–998. Barab´ asi, A.L. and Albert, R. (1999). Emergence of scaling in random networks. science, 286(5439), 509–512. Benzi, R., Parisi, G., Sutera, A., and Vulpiani, A. (1982). Stochastic resonance in climatic change. Tellus, 34(1), 10–16. Benzi, R., Sutera, A., and Vulpiani, A. (1981). The mechanism of stochastic resonance. Journal of Physics A: mathematical and general, 14(11), L453. Blondel, V. and Tsitsiklis, J.N. (1997). Np-hardness of some linear control design problems. SIAM Journal on Control and Optimization, 35(6), 2118–2127. Brogan, W. (1991). Modern Control Theory. Prentice hall, 3rd edition. 319

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2014 ACODS March 13-15, 2014. Kanpur, India

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Watts, D.J. and Strogatz, S.H. (1998). Collective dynamics of small-worldnetworks. nature, 393(6684), 440–442. Yu, H. and Ozguner, U. (2003). Smooth extremumseeking control via second order sliding mode. In American Control Conference, 2003. Proceedings of the 2003, volume 4, 3248–3253. IEEE. Zhang, C. and Ordˆao¨anez, R. (2012). Extremum-seeking Control and Applications: A Numerical Optimizationbased Approach. Advances in Industrial Control. Springer-Verlag London Limited. Appendix A. STATE AND GAIN MATRICES FOR SECTION 3.1 Example 2 : Let  0.0040 0.8222 A2 =

    

0.0517 0.2852 0.7900 0.5658 0.3161 0.4527 0.3500

0.6709 0.1381 0.4475 0.7043 0.6058 0.8775 0.9840

0.9408 0.7447 0.1816 0.1587 0.4762 0.3187 0.3344 0.8224

 16 B2 =

    

01 00 14 11 03 21 23

0.8792 0.5143 0.4148 0.6839 0.1589 0.3663 0.9192 0.9962

03 14 18 13 23 18 19 04

18 12 18 05 14 16 05 15

22 05 00 25 02 04 10 12

0.2357 0.9110 0.6515 0.6312 0.9743 0.0986 0.4970 0.3603

0.8175 0.7775 0.6806 0.8176 0.9230 0.8939 0.3821 0.1781

23 18 16 02 09 00 01 20

18 14 19 00 25 06 05 09

07 08 16 14 01 00 17 14

02 22 09 07 21 23 06 18

0.3713 0.8170 0.5738 0.0944 0.2637 0.0912 0.3469 0.5308

0.2255 0.5205 0.4125 0.1119 0.4318 0.2609 0.0440 0.6831

     

     

One of the K matrices picked up at random (using 2− norm) is  0.661 −1.179 −0.036 −0.186 1.717 −0.231 0.146 −1.237  K =

1.315 −6.162 −0.538 −0.679 8.910 −2.195 −0.366 −2.270 −4.220 −0.465 −0.576 6.118 −1.510 −0.266 −1.442 2.387 0.294 0.138 −3.602 0.793 0.215 0.848 −7.870 −0.718 −0.863 11.260 −2.475 −0.240 −3.227 7.264 0.586 0.886 −10.642 2.573 0.028 3.119 −1.698 6.667 0.559 0.791 −9.906 2.370 0.164 2.990 −0.591 2.379 0.238 0.219 −3.571 0.720 0.080 1.109

 0.664  −0.447   1.700  −1.691

Example 3 :Let  −15.09 −19.08 A3 =

     

9 −19 −18 −8 23 −10 −15 19 −23 −17.060 5 −8 −16 19 −11 15 −19 13 14 −24 5 16 −21 −22 −11 20 25 19.04 −11 −21 −15 −11.05 5 10 −11 −16.04 −15 −12 16 18 4 −7 21 −3 −14 −3.001 −24.003 −22 −8 −24 −23 21 −5 −11 −3 −17 −17 24 19 7 10 9.005 −16 −12 2 −25 9 −13 −24 −15 24 19 2 −3 11 3 −21 −2 −18

 B3 =

     

9 24 15 20 10 25 24 13 4

21 0 2 15 18 7 25 17 24

14 12 13 2 16 1 2 25 07

3 15 19 4 10 25 13 10 3

24 14 2 1 0 16 21 09 1

21 22 15 13 14 7 14 6 11

23 14 21 18 19 19 18 17 1

25 5 16 2 14 16 2 3 21

12 8 17 8 6 21 16 21 18

    

      

      

One of the K matrices picked up at random using 2−norm is (kKk = 61.04 )  K =

     

19.19 0.11 4.83 −3.66 −3.49 −0.13 −14.63 4.72 −8.47

13.60 −0.04 3.98 −2.63 −2.39 −1.55 −9.11 4.56 −6.44

5.41 0.38 1.71 −0.18 −1.77 −0.84 −3.79 2.26 −2.97

11.29 −12.75 −10.85 −14.20 −0.69 −15.82 0.10 −0.60 −1.32 −0.24 −0.66 −0.19 2.39 −2.98 −2.96 −3.61 0.34 −2.77 −2.40 1.50 0.55 3.62 0.41 3.07 −2.04 4.19 2.47 2.06 0.21 4.45 −0.31 −1.29 −0.96 −1.18 −0.60 1.61 −8.01 11.06 10.10 10.94 0.90 9.28 3.11 −4.05 −1.50 −2.96 0.48 −5.36 −4.36 6.22 4.75 6.54 0.58 7.08

      