Roll control using discrete-time robust sliding hyperplanes and fast output sampling

Robustness to unmatched parametric uncertainty is prime requirement of roll control algorithm, especially when it is modelled in discrete time domain and implemented through on-board processor. Sliding mode control is a well established nonlinear control technique, which ensures a robust performance in presence of matched uncertainties and disturbances. In case of the discrete version of sliding mode control, due to finite operational sampling frequency, the system trajectories cannot be forced to slide on the switching manifold. The trajectories remain confined to certain domain around the sliding surface and this is known as Quasi Sliding Mode (QSM) motion. The bound of QSM decides the accuracy and performance of the discrete version of sliding mode. By design, the discrete-time sliding modes are robust to the matched bounded perturbations, however, unmatched perturbations directly affect the boundary layer width and hence the performance of the system. In the present paper, discrete time Lyapunov inequality based sliding hyperplane is designed, which enables robustness to unmatched perturbations arising due to uncertain system matrix A. Further, the requirement of full state-vector for the design of control and sliding surface is met through the multi-rate output feedback (MROF). This control strategy is then demonstrated with application to roll position control of missile with a bandwidth limited actuator.     

Orthogonal nonnegative matrix tri-factorization for co-clustering: Multiplicative updates on Stiefel manifolds

Matrix factorization-based methods become popular in dyadic data analysis, where a fundamental problem, for example, is to perform document clustering or co-clustering words and documents given a term-document matrix. Nonnegative matrix tri-factorization (NMTF) emerges as a promising tool for co-clustering, seeking a 3-factor decomposition X≈USV^?$\mathit{X}\approx {\mathit{USV}}^{?}$ with all factor matrices restricted to be nonnegative, i.e., U?0,S?0,V?0.$\mathit{U}?0\text{,}\mathit{S}?0\text{,}\mathit{V}?0\text{.}$ In this paper we develop multiplicative updates for orthogonal NMTF where X≈USV^?$\mathit{X}\approx {\mathit{USV}}^{?}$ is pursued with orthogonality constraints, U^?U=I,${\mathit{U}}^{?}\mathit{U}=\mathit{I}\text{,}$ and V^?V=I${\mathit{V}}^{?}\mathit{V}=\mathit{I}$, exploiting true gradients on Stiefel manifolds. Experiments on various document data sets demonstrate that our method works well for document clustering and is useful in revealing polysemous words via co-clustering words and documents.     

Separation of DNA by length in rotational flow: Lattice-Boltzmann-based simulations

We use a lattice-Boltzmann based Brownian dynamics simulation to investigate the separation of different lengths of DNA through the combination of a trapping force and the microflow created by counter-rotating vortices. We can separate most long DNA molecules from shorter chains that have lengths differing by as little as 30%. The sensitivity of this technique is determined by the flow rate, size of the trapping region, and the trapping strength. We expect that this technique can be used in microfluidic devices to separate long DNA fragments that result from techniques such as restriction enzyme digests of genomic DNA.The development of novel methods for manipulating biopolymers such as DNA is required for the continued advancement of microfluidic devices. Techniques such as restriction enzyme digests for genomic sequencing rely on the detection of DNA that differ in length by sometimes thousands of base pairs.¹ Methods that separate DNA strands with resolutions on the order of kilobase pairs are required to analyze the products of this technique. To gain an insight into possible techniques to separate polymers, it can be helpful to review the methods to separate particles in microfluidic devices. Experimental work has shown how hydrodynamic mechanisms can lead to separation of particles based on size and deformability.² Eddies, microvortices, and hydrodynamic tweezers have been used to trap and sort particles. The mechanism of the trapping and sorting arises from the differences between interactions of the particles with the fluid.^2–8 In particular, counter-rotating vortices have been used to sort particles and manipulate biopolymers. They have been used to deposit DNA precisely across electrodes⁹ and trap DNA.^10,11 Vortex flow may therefore be a good basis for a technique for sorting DNA by length.Streaming flow has been used in experiments to separate colloids of different size.^3,4 Particles are passed through a channel with a flow field driven by oscillating bubbles and pressure. The flow field becomes a combination of closed and open streamlines. The vortex flow is controlled by the accoustic driving of the bubbles while pressure controls the net flow of the fluid. Larger particles are trapped in the closed vortex flow created by the bubbles, while smaller particles can escape the neighborhood of a bubble in the open streamlines. This leads to efficient separation of particles with size differences as small as 1 μm.Previous work on DNA has shown that counter-rotating vortices can be used to trap DNA dynamically. Long strands of DNA have been observed to stretch between the centers of two counter-rotating vortices. The polymer stays trapped in this state for significant amounts of time.¹² In a different experiment, the vortices were used to thermally cycle the polymer and allow replication via the polymerase chain reaction (PCR). The DNA is also trapped against one wall by a thermophoretic force in these experiments.¹⁰ The strength of the trap is controlled by the gradient in temperature created by a focused infrared laser beam.Trapping DNA at one wall by counter-rotating vortices has also been explored in simulation and found to depend on the Peclet number, Pe = u_maxL/D_m, where u_max is the maximum speed of the vortex, L is the box size, and D_m is the diffusion coefficient of one bead in the polymer chain.¹¹ The trapping rate of the DNA was shown to depend on the competition between the flow compressing the DNA into the trap region and the diffusion of the DNA out of the trap. For the work presented here, Pe ≅ 2000, similar to the previous work done with the same simulation.We extend the previous work to investigate if counter-rotating vortices can be used to separate DNA of different lengths. We use the same type of simulation outlined in Refs. ¹¹ and ^13–17, based on the lattice-Boltzmann method. The simulation method has successfully modeled systems as diverse as thermophoresis of DNA,¹⁴ migration of DNA in a microchannel,¹⁶ and translocation of DNA through a micropore.^17,18 Using this method, the fluid is broken into a lattice with size, ΔL, chosen to be 0.5 μm, and is coupled to a worm-like chain model with Brownian dynamics for the polymer.^19,20 The fluid velocity distribution function, n_i(r, t), describes the fraction of fluid particles with a discretized velocity, c_i, at each lattice site.^21–24 A discrete velocity scheme with nineteen different velocities in three dimensions is used. The velocity distributions will evolve according to ni(r+ciΔτ,t+Δτ)=ni(r,t)+Lij[nj(r,t)−njeq(r,t)],(1)where L is a collision operator such that the fluid relaxes to the equilibrium distribution, nieq given by a second-order expansion of the Maxwell-Boltzmann distribution nieq=ρaci[1+(ci·u)/cs2+uu:(cici−cs2I)/(2cs4)],(2)where cs=1/3ΔLΔτ is the speed of sound. Δτ is the time step for the fluid in the simulation, Δτ = 8.8 × 10⁻⁵. The coefficients a^c_i are determined by satisfying a local isotropy condition ∑iaciciαciβciγciδ=cs4(δαβδγδ+δαγδbetaγ+δαδδβγ).(3)To simplify computation, the velocity distributions are transformed into moment space. The density ρ, momentum density j, and momentum flux density Π are some of the hydrodynamic moments of n_i(r, t). The equilibrium conditions for these three moments are given by ρ=∑nieq,(4) j=∑ci·nieq,(5) Π=∑nieq·cici.(6)L has eigenvalues τ0−1,τ1−1,…,τ18−1, which are the characteristic relaxation times of the moments. The Bhatanagar-Gross-Krook model is used to determine L:²⁵ the non-conserved moments have a single relaxation time, τ_s = 1.0. The conserved moments are density and momentum; for these, τ⁻¹= 0. Fluctuations are added to the fluid stress as in the method of Ladd.²⁴ We have also compared simulations with lattice sizes of 1 μm and 0.25 μm and found no significant differences in the results.The DNA used in the simulation is represented by a worm-like chain model parameterized to capture the dynamics of YOYO-stained λ DNA in bulk solution at room temperature.^15,16,26 Long, flexible DNA is modeled since techniques to separate long DNA molecules with kilobase pair resolution are necessary to complete techniques such as genomic level sequencing using restriction enzyme digests.¹ In addition, such DNA is often used in experiment. Its properties are similar to unstained DNA or DNA stained by other methods.²⁷ Each molecule is represented by N_b beads and N_b − 1 springs. A chain composed of N_b − 1 springs will have a contour length of (N_b − 1) × 2.1 μm. The forces acting on each monomer include: an excluded volume force, a non-linear spring force, the viscous drag force, a random force that produces Brownian motion, a repulsive force from the container walls, and an attractive trapping force only at one wall as shown in Fig. ​Fig.11.¹³ The excluded volume interaction between beads i and j located at r_i and r_j is modeled using the following potential: Uijev=12kBTνNks2(34πSs2)exp(−3|ri−rj|24Ss2),(7)where ν=σk3 is the excluded volume parameter with σ_k = 0.105 μm, the length of one Kuhn segment, N_ks = 19.8 is the number of Kuhn segments per spring, and Ss2=Nks/6)σk2 is the characteristic size of the bead. This excluded volume potential reproduces self avoiding walk statistics. The non-linear spring force is based on force-extension curves from experiments and is given by fijS=kBT2σk[(1−|rj−ri|Nksσk)−2+4|rj−ri|nKσk−1]rj−ri|rj−ri|,(8)which applies when N_ks ≫ 1.Open in a separate windowFIG. 1.Simulation set-up. Arrows indicate direction of fluid flow. The region where the trapping force is active is shaded, and its width (X_stick) is shown. The region used to determine the trapping rate is indicated by the area labeled trap region. Figure is not to scale, the trap region and X_stick are smaller than shown.The beads are modeled as freely draining but subject to a drag force given by F_f = ?6πηa(u_p ? u_f).(9)The beads are also subjected to a random forcing term that is drawn from a Gaussian distribution with zero mean and a variance σ_v = 2k_BTζΔt.(10)The random force reproduces Brownian motion. To conserve total momentum, the momentum change imparted to the beads through their interactions with the fluid is balanced by a momentum change in the fluid. The momentum change is distributed to the three closest fluid lattice sites using a linear interpolation scheme based on the proximity of the lattice site to the polymer beads. Through this momentum transfer, hydrodynamic interactions between the beads occur.The beads are repelled from the walls with a force of magnitude Fwall=250kBTσk3(xbead−xwall)2, xbead>(xwall−1),(11)where the repulsion range is 1ΔL. Each monomer will also be attracted to the top wall by a force with magnitude Fstick=KstickkBTσk3(xbead−xwall+10)2, xbead>(xwall−Xstick)(12)and range X_stick (see Fig. ​Fig.1).1). The sticking force is turned off every one out of one hundred time steps of the polymer (1% of the simulation time steps). We vary both X_stick and K_stick to achieve separation of the polymers.In previous experiments, DNA has been trapped against one wall by using thermophoresis,¹⁰ dielectrophoresis,²⁸ and nanoplasmonic tweezers.²⁹ In the case of thermophoresis, the trap strength (K_stick) can be controlled by tuning the intensity of the temperature gradient and the trap extension (X_stick) can be controlled through the area over which the gradient extends. Both of these are set through focusing of the laser used to produce local heating. Similarly, the trap parameters can be controlled when using plasmonic tweezers by controlling the laser beam exciting the nanoplasmonic structures. In dielectrophoresis, the DNA is trapped by an AC electric field and can be controlled by tuning the frequency and amplitude of the field.In this work, the number of polymers, N_p, is 10 unless otherwise noted, and the container size is 25 ΔL × 50 ΔL × 2 ΔL. The time step for the fluid is Δτ = 8.8 × 10⁻⁵ s, and for the polymer is Δt = 3.7 × 10⁻⁶ s. The total simulation time is over 100 chain relaxation times, allowing sufficient independent samples to perform statistical analysis.Two counter-rotating vortices, shown in 1, are produced by introducing external forces to the fluid bound by walls in the x-direction and periodic in the y and z. Two forces of equal magnitude push on the fluid in the upper x region (12ΔL < x < 25ΔL): one in the +y-direction along y = 10ΔL, and one in the –y-direction along y = 40ΔL. Such counter-rotating vortices can be produced in microfluidic channels using acoustically driven bubbles,^3,4,30 local heating,¹⁰ or plasmonic nanostructures.⁵ The flow speed is controlled by very different external mechanisms in each case. We therefore choose a simple model to produce fluid flow that is not specific to one mechanism.The simulations are started using random initial conditions, and therefore, both lengths of polymer are dispersed throughout the channel. Within a few minutes, the steady state configurations pictured in Figs. ​Figs.22 and ​and33 are reached. We define the steady state as when the number of polymer chains in the trap changes by less than one chain (10 beads) per 1000 polymer time steps. Intermittently, some polymers may still escape and re-enter the trap even in the steady state. Three final configurations are possible: Both the lengths of DNA have become trapped, both lengths continue to rotate freely, or the shorter strand has become trapped while the longer rotates freely. Two of these states leave the polymers mixed; in the third, the strands have separated by size.Open in a separate windowFIG. 2.Snapshots at t = 0Δt (left) and t = 2500Δt (right) showing the separation of 15-bead strands (grey) from 10-bead strands (black) of DNA. For these simulations, K_stick = 55 and Y_stick = 0.7ΔL.Open in a separate windowFIG. 3.Snapshots at t = 0Δt and t = 2500Δt showing the separation of 13-bead strands (grey) from 10-bead strands (black) of DNA. For these simulations, K_stick = 55 and Y_stick = 0.7ΔL as in Fig. ​Fig.2.2. Note that one long polymer is trapped, as well as all of the shorter polymers.By tuning the attractive wall force parameters and fluid flow, the separated steady state can be realized. We first set the flow parameters that allow the larger chains to rotate freely at the center of the vortices while the shorter chains rotate closer to the wall. The trap strength, K_stick, and extension, X_stick, are changed until the shorter polymers do not leave the trap. The same parameters were used to separate 10-bead chains from 15-bead and 13-bead chains.As shown in Fig. ​Fig.2,2, we have been able to separate shorter 10-bead chains from longer 15-bead chains. In the steady state, 97% of the rotating polymers were long polymers averaged over twenty simulations initialized with different random starting conditions. For three simulations, one small polymer would intermittently leave the trap region. In two of these simulations, one long polymer became stably trapped in the steady state. In another simulation, one 15-bead chain was intermittently trapped. On average, the trapped polymers were 5% 15-bead chains and 95% 10-bead chains. Again, 97% of the rotating polymers were 15-bead chains.Simulations conducted with 10-bead and 13-bead chains also showed significant separation of the two sizes as can be seen in Fig. ​Fig.3.3. In the steady state, 30% of the trapped polymers are 13-bead chains and 70% are 10-bead chains, averaged over twenty different random initial starting conditions and 1000 polymer time steps. Only 14.8% of the shorter polymers were not trapped, leading to 85.2% of the freely rotating chains being 13-bead chains. This is therefore a viable test to detect the presence of these longer chains.We have also separated 20-bead chains from 10-bead chains with all of the shorter chains trapped and all of the longer chains freely rotating in the steady-state. These results do not change for twenty different random initial starting conditions and 1000 polymer time steps. None of the longer polymers intermittently enter the trap region nor do any of the shorter polymers intermittently escape.The separation is achieved by tuning the trapping force and flow rate. Strong flows will push all the DNA molecules into the trap. The final state is mixed, with both short and long strands trapped. For flows that are too weak, the short molecules are not sufficiently compressed by the flow and therefore do not enter the trap region. The end state is mixed, with all polymers freely rotating. Separation is achieved when the flow rate is tuned so that the short strands are compressed against the channel wall while the long polymers rotate near the center of the vortices. The trap strength must then be set sufficiently high enough to prevent the short strands from being pulled by the hydrodynamic drag force out of the trap.The mechanism of the separation depends on the differences in the steady state configurations of the polymers and chances of a polymer escaping the trap. As shown in Fig. ​Fig.4,4, both longer and shorter chains are pulled into the trap region by the flow. However, the longer chains have a larger chance of a bead escaping into a region of the flow where the fluid velocity is sufficient to pull the entire strand out of the trap. As shown in Ref. ¹¹, the trapping rate depends on diffusion in a polymer depleted region near the trap, in agreement with classical theory which neglects bead-wall interactions. In addition, the theory depends on the single bead diffusion rate and does not take into account the elastic force holding the beads together. Diffusion becomes as significant as convection in the polymer depleted region leading to dependence on the Peclet number. Since longer polymers have more beads; they have more chances of a single bead diffusing through this layer into the region where convection is again more important. Thus, they are pulled out of the trap at a faster rate than the shorter chains.Open in a separate windowFIG. 4.N, number of beads in the trap region, versus time for 15-bead DNA strands (solid line) and 10-bead DNA strands (dashed line). Here, ΔT = 10000Δt. The simulation parameters are the same as in Fig. ​Fig.22.In addition, longer chains have a second trap resulting from the microflow. As shown in Ref. ¹², DNA in counter-rotating vortices can tumble at the center of one vortex or be stretched between the centers of the two vortices. We have observed both these conformations for the longer polymer strand. They are a stable trajectory for the longer polymer that remains outside of the trapping region. As seen in Fig. ​Fig.4,4, few monomers of the longer chains enter the trap region once the steady state has been reached. However, the shorter polymer rotates at a larger radius than the longer polymer as seen in Fig. ​Fig.5.5. The shorter polymers therefore are pushed back into the trap while the longer strands rotate stably outside the trapping region.Open in a separate windowFIG. 5.Trajectories of 15-bead DNA (grey) and 10-bead DNA (black). The position of each monomer is plotted for 100 consecutive time steps. Note that the longer polymers rotate in the center of the channel while the shorter polymers rotate at the edges. Simulation parameters are the same as in Fig. ​Fig.22.This mechanism is similar to the one proposed for the separation of colloids by size in Refs. ³ and ⁴. In that experimental work, the smaller colloidal particles rotated at larger radii. This allowed the smaller beads to be pushed out of the vicinity of the vortices by the streaming flow, while the larger beads continued to circle. However, in our simulations, we have the additional mechanism of separation based on the increased chance of a longer polymer escaping the trap region. This mechanism is important for maintaining the separation. Long polymers initially in the trap region or which diffuse into the trap would not be able to escape without it.We expect that this technique could be used to detect the sizes of DNA fragments on the order of thousands of base pairs. It relies on the flexibility of the molecule and its interaction with the flow. Common lab procedures such as restriction enzyme digests for DNA fingerprinting can produce these long fragments. Current techniques such as gel electrophoresis require significant time to separate the long strands that move more slowly through the matrix. This effect could therefore be a good candidate for developing a microfluidic analysis that is significantly faster than traditional procedures. Our separation occurs in minutes rather than in hours as for gel electrophoresis.As pointed out in Ref. ², hydrodynamic effects have been shown to be important for microfluidic devices for separation. We have demonstrated, in simulation, a novel hydrodynamic mechanism for separating polymers by length. We hope that these promising calculations will inspire experiments to verify these results.     

Chaotic behavior of discrete-time linear inclusion dynamical systems

Given any finite family of real d-by-d nonsingular matrices $\{S_1,\dots ,S_l\},$ by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:

$\begin{array}{c}\hfill x_n\in \{S_k{x}_{n?1};1\le k\le l\},x_0\in {\mathbb{R}}^d\mathrm{and}n\ge 1,\end{array}$

we study the chaotic dynamics of the state trajectory (x_n(x₀, σ))_{n ≥ 1} with initial state $x_0\in {\mathbb{R}}^d,$ governed by a switching law $\sigma :\mathbb{N}\to \{1,\dots ,l\}$. Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all $x_0,y_0\in {\mathbb{R}}^d,x_0\ne y_0,$

$\begin{array}{c}\hfill \underset{n\to +\infty }{lim\; inf}\parallel x_n(x_0,\sigma )?x_n(y_0,\sigma )\parallel =0\mathrm{and}\underset{n\to +\infty }{lim\; sup}\parallel x_n(x_0,\sigma )?x_n(y_0,\sigma )\parallel =\infty .\end{array}$

This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (x_n(x₀, σ))_{n ≥ 1} are extremely sensitive to the initial states $x_0\in {\mathbb{R}}^d$. We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.     

Null controllability of some nonlinear degenerate 1D parabolic equations

The main goal of the present paper is twofold: (i) to establish the well-posedness of a class of nonlinear degenerate parabolic equations and (ii) to investigate the related null controllability and decay rate properties. In a previous step, we consider an appropriate regularized system, where a small parameter α is involved. More precisely, the usual nonlinear term b(x)uu_x is replaced by b(x)zu_x, where $z={(\text{Id}.?{\alpha }^{2}A)}^{?1}u$ and A is a Poisson–Dirichlet operator. We investigate the behavior of the null controls and their associated states as α → 0.     

Dissipative consensus problems for multi-agent networks via sliding mode control

In this paper, dissipative consensus problems are discussed for multi-agent networks. Firstly, sufficient conditions are proposed to ensure $(Q,S,R)?$dissipative consensus for multi-agent networks with external disturbances. Then, by designing an integral-type sliding surface function, a controller is obtained and the corresponding sufficient conditions are given to guarantee $(Q,S,R)?$dissipative consensus for multi-agent networks with external disturbances. Moreover, the sliding mode control law is formulated such that multi-agent networks drive onto the predefined surface in finite time. Finally, an example is given to illustrate the effectiveness of the obtained results.     

Mixed complementarity problems for robust optimization equilibrium under l1∩l∞-norm

We consider robust optimization equilibrium for the bimatrix game based on robust optimization method. In this paper, each player may neither exactly estimate his opponent's strategies nor evaluate his own cost matrix accurately while may estimate a bounded uncertain set. We obtain that the robust optimization equilibrium problem can be formulated as a mixed complementarity problem (MCP) under _l1∩_l∞-norm$l_1\cap l_{\infty }\mathrm{-}\mathrm{norm}$. Some numerical examples are presented to illustrate the behavior of robust optimization equilibrium.     

Rapid highly sensitive general protein quantification through on-chip chemiluminescence

Protein detection and quantification is a routinely performed procedure in research laboratories, predominantly executed either by spectroscopy-based measurements, such as NanoDrop, or by colorimetric assays. The detection limits of such assays, however, are limited to μM concentrations. To establish an approach that achieves general protein detection at an enhanced sensitivity and without necessitating the requirement for signal amplification steps or a multicomponent detection system, here, we established a chemiluminescence-based protein detection assay. Our assay specifically targeted primary amines in proteins, which permitted characterization of any protein sample and, moreover, its latent nature eliminated the requirement for washing steps providing a simple route to implementation. Additionally, the use of a chemiluminescence-based readout ensured that the assay could be operated in an excitation source-free manner, which did not only permit an enhanced sensitivity due to a reduced background signal but also allowed for the use of a very simple optical setup comprising only an objective and a detection element. Using this assay, we demonstrated quantitative protein detection over a concentration range of five orders of magnitude and down to a high sensitivity of 10pgmL−1, corresponding to pM concentrations. The capability of the platform presented here to achieve a high detection sensitivity without the requirement for a multistep operation or a multicomponent optical system sets the basis for a simple yet universal and sensitive protein detection strategy.     

Nested backward compensation of unmatched perturbations via HOSM observation

This paper tackles the compensation problem of linear time invariant systems affected by unmatched perturbations. The proposed methodology exploits a high order sliding mode observer, guaranteeing theoretically exact state and perturbation estimation. A compensation based strategy is proposed to cope with the unmatched perturbations. The compensation of the desired coordinate is carried through a nested backward sliding surface design, which compensates some of the non-actuated state components, while the remaining states are maintained to be bounded. The feasibility of the technique was tested in an active suspension vehicle system.¹     

Filter design for Markovian jump repeated scalar nonlinear systems

This paper is concerned with the _?2−_?∞${?}_2-{?}_{\infty }$ filtering problem for discrete-time Markovian jump repeated scalar nonlinear systems. Our attention is focused on the design of full- and reduced-order filters that guarantee the filtering error system to be stochastically stable with a prescribed weighted _?2−_?∞${?}_2-{?}_{\infty }$ performance. By employing both the mode dependent Lyapunov function approach and the positive definite diagonally dominant Lyapunov function technique, a sufficient condition is firstly established, which guarantees the Markovian jump repeated scalar nonlinear system to be stochastically stable with an induced _?2−_?∞${?}_2-{?}_{\infty }$ disturbance attenuation performance. The corresponding full-order filter design is cast into a convex optimization problem, which can be efficiently handled by using standard numerical algorithms. Finally, a numerical example is presented to show the effectiveness of the proposed methods.