Robust State Dependant Switching of Linear Systems with Dwell Time
Liron I. Allerhand, Tel Aviv University
A state dependant switching law that obeys dwell time constraints
and guarantees the stability of the system is designed. Sufficient
conditions are given for the stability of the linear switched systems
where the switching law is applied in presence of of polytopic type
parameter uncertainty. A Lyapunov function, in quadratic form, which
us non-increasing at the switching instants is assigned to each subsystem.
During the dwell time, this function varies piecewise linearly in time
after switching occurs. After the dwell time, the system switches only
if the switching will result in a decrease in Lyapunov function.
The method presented is extended to guarantee a prescribed L2-gain bound
on the switched system. It is then applied to stabilization via
state-feedback and to H-infinity control control design.
A Reduction of Planar Topological Infinite-Horizon Optimization to
Ido Bright, Weizmann Institute
The reduction of infinite-horizon optimization to periodic optimization is
desired, both for the simplicity of the solution obtained, and for the
richer tools available in the latter field.
We consider a general class of infinite-horizon optimization problems in the
plane, where the set of feasible curves satisfies general conditions similar
to, but weaker than, the solution set of a control system. We verify the
existence of an optimal periodic solution for this class of problems.
Applying this result we verify two results in this field: A known result for
infinite-horizon control systems, and a new result generalizing a previous
theorem for discounted infinite-horizon control problems.
A Maximum Principle for Positive Bilinear Control Systems
Lior Fainshil, Tel Aviv University
A Maximum Principle for Positive Bilinear Control Systems
We consider a positive bilinear control system, i.e. a control system
for which the first orthant is an invariant set. We derive a maximum
principle that provides a necessary condition for a control to be a
"most destabilizing" control. We use this to prove stability under
arbitrary switching laws for several classes of positive switched
Stochastic Calculus and an Associated Control
Problem for a Wide Class of Gaussian Processes
Alon Kipnis, Ben Gurion University
We develop an Ito-like stochastic calculus for a wide class of Gaussian
stationary increment processes
which include in particular the fractional Brownian motion. and colored
Gaussian noise. We use an approach
based on infinite dimensional analysis, and on Hida's white noise space
theory. Using the Bochner-Minlos theorem we
associate to each of these processes a probability space which is the
analog of the white noise space.
This allows us to study state space equations for linear systems
perturbed by these processes, and solve the classical linear quadratiic
optimal control problem in this setting.
A Pontryagin Maximum Principle for Boolean Control Networks
Dmitry Laschov, Tel Aviv University
A Boolean network consists of a set of Boolean variables whose state is
determined by other variables in the network. Cellular automata, with two
possible states per cell, are a particular case of Boolean networks.
Recently, Boolean networks gained considerable interest as models for
biological systems composed of elements that can be in one of two possible
states. Examples include genetic regulation networks, where the ON (OFF)
state corresponds to the transcribed (quiescent) state of a gene, and
cellular networks where the two possible logic states may represent the
open/closed state of an ion channel, basal/high activity of an enzyme, two
possible conformational states of a protein, etc.
Daizhan Cheng and his colleagues developed an algebraic state-space
representation for Boolean control networks using the semi-tensor product of
matrices. This representation proved quite useful for studying Boolean
control networks in a control-theoretic framework.
Using this representation, we consider a Mayer-type optimal control problem
for Boolean control networks. Our main result is a necessary condition for
optimality. This provides a analog of Pontrayagin's maximum principle to
Boolean control networks
A Minimum Acceleration Criterion for Object Manipulation
Raz Leib, Ben Gurion University
Many daily tasks, such as moving a cap of coffee or walking holding a suitcase, involve controlling and manipulating
objects. While simple reaching movements are easily controlled, the control of objects with complex dynamics and
additional degrees of freedom is much more challenging. In a simple unconstrained reaching movement it was demonstrated
that subjects implicitly
minimize the hand smoothness and a minimum Jerk model well account for the hand trajectory during such reaching movements.
A minimum acceleration with constraints model also fits typical reaching movement trajectories with interesting
prediction about the intermittent nature of the controller.
Previous studies investigated the strategies used by humans to control one degree
of freedom objects which was simulated by mass attached to a spring. While considering
the new mechanical setup of the environment, that differ from unconstrained reaching
movements, new computational models were suggested to account for subjectís and
objectís trajectories. These strategies were based on solving an optimization
problem that minimizes the mean-squared force change of the hand trajectory or
the mean square crackle of object trajectory (crackle being the fifth time derivative
of the object position). Although providing logical solution for object manipulation,
both criteria could not account for subject performance in some cases. In addition
when trying to explain reaching movements with multiple objects, the minimum crackle
criteria is not applicable, while for simple reaching movement, the hand force
change criteria can be solved only numerically by iterative scheme.
Knowing the limitations of the two proposed criteria, we suggest another criterion.
Instead of defining the optimization problem while focusing entirely on the hand
or on the object we suggest optimizing the trajectory of the system center of
mass. More specifically we propose minimizing the acceleration of the center of
mass with the same constrains used for reaching movement optimization by Ben-Itzhak
and Karniel (2008).
We show that the minimum acceleration criteria with constrains can be expended to account for both reaching movements and object manipulation task without the need for two separate models. In the one degree of freedom object manipulation task we show that the extension of the model provide the same predictions for hand velocity profile as the other models for some values of mass and spring constant as reported in the literature. Moreover, it explains results from previous studies where other models fail to do so.
This model not only fits the observed behavior, it also suggests that the brain use intermittence control and a single cost function as it controls reaching as well as manipulating complex objects. The simplicity of explaining both simple movements and movements in complex environments without altering between models opens new possibilities in research on control of robotic systems.
L2 Optimal FIR Reconstructor
Yaron Levinson, Technion
In this talk I will present an L2-optimal FIR reconstructor, or
interpolator. The well known sinc interpolator was shown to be L2-optimal
also for non-bandlimited signals. However, the slow decay rate of the sinc
function prompted the use of suboptimal, fast decaying cardinal B-splines.
These splines are non-causal IIR filters, which sometimes are less adequate
for applications where the access to "future" information is limited.
Recently, causal IIR interpolators, with a prescribe amount of preview, were
derived by incorporating causality constraints into the optimization
problem. Some applications may benefit from a finite memory interpolator,
which, arguably, might handle better with abrupt changes in the signal and
I will present the closed-form expression for the optimal hold, having a
finite window length and arbitrary number of preview steps. The interpolator
can meet asymptotic constraints by the use of unstable weighing functions.
Quasi-analytical Describing Function for the Karnopp Stick-Slip
Arkady Lichtsinder, Technion
A quasi-analytical Describing Function (DF) for the Karnopp stick-slip
friction model is presented. The proposed DF is appropriate to describe
a quasi-linear stick-slip friction behavior in the frequency domain and
to predict limit cycling. It is due to the additional phase lag
(vs. Coulomb model) that is incorporated in this DF at low frequencies.
It is shown that this additional phase lag is dictated by the
static-to-dynamic friction ratio and it is generally limited by a
constant value of about 32.5 deg. This fact can facilitate robust
control strategy to avoid low-frequency limit cycling. The reliability
of the DF was verified by the common sine-scan technique which is based
on direct computation of the first Fourier coefficients. The capability
of the Karnopp DF to predict limit cycles was confirmed by time-domain
simulation. The proposed DF with rapid computation can be functional in
real-time control algorithms.
Near Optimal Control of a Grid-Connected Storage System
Doron Lifshitz, Technion
Two nearly optimal control algorithms for the charge and discharge of a grid connected photovoltaic system with
storage are proposed. These algorithms are based on Dynamic Programming (DP). The first uses a single state model
which describes a capacitor-like storage device. The second uses a two states storage model which approximates a
chemical battery. The control input is the storage device current. The cost function is based on the profit resulting
from operating the system, taking into account the cost of replacing batteries.
Two problems are discussed: 1) the continuous-time problem where the storage device current can change continuously
in order to realize the best charging policy.
2) The discrete-time problem where the storage device current must be constant over pre-defined time intervals
(the grid is managed via short period contracts).
If the storage device has negligible resistance and is of the capacitor-like form, the solution of the first
problem is shown to be Bang-Off-Bang control. The sub-optimality of the solution is due to (a) Quantization of
the states for DP. The quantization can be chosen such that the difference between the solution and the actual
optimum is less than a pre-defined bound. (b) The difference between the actual storage device used and the
storage device model.
Sampled-Data Stabilization and Round Robin Scheduling:
A New Lyapunov-Based Approach
Kun Liu, Tel Aviv University
The input delay approach has now become one of the main methods for robust
sampled-data control. Modeling of continuous-time systems with digital control
in the form of continuous-time systems with delayed control input has
become also popular in Networked Control Systems (NCSs) (where the plant is controlled
via communication network). Until recently, only time-independent Lyapunov-Krasovkii
functionals (for systems with uncertain and bounded time-varying delays) were
applied. These functionals did not take advantage of the sawtooth evolution of
the delays induced by sampled-and-hold. The latter drawback was removed in Fridman
(2010), where time-dependent Lyapunov functionals for sampled-data systems were
introduced, significantly improving the existing results.
In the present paper novel discontinuous Lyapunov functionals are introduced for
sampled data systems with constant delay, which are based on the vector extension
of the Wirtinger's inequality. These functionals lead to simple and
efficient stability conditions in terms of Linear Matrix Inequalities (LMIs).
The new stability analysis is applied to a novel sampled-data static output-feedback
problem, where the delayed measurements are used for stabilization.
Next, we study stabilization of NCSs with communication constraints, variable
sampling interval and delay. We focus on static output feedback controllers
for linear systems. The system sensors nodes are supposed to be distributed
over a network. Data transmission over the network is subject to the Round-Robin
scheduling protocol. We present the closed-loop system as a switched system
with multiple delayed samples. By constructing an appropriate time-dependent
Lyapunov functional, which takes into account the switched system model and
the sawtooth delays induced by sampled-data control, we derive the exponential
stability conditions in terms of LMIs. We note that time-independent Lyapunov
functionals under Round-Robin scheduling protocol lead to the overlap of the
delay intervals and, thus, to overdesign. Polytopic uncertainties in the system
model can be easily included in the analysis. The efficiency of the method is
illustrated on the classical cart-pendulum benchmark problem.
Robot Navigation with Velocity Constraints
Gil Manor, Technion
The subject of on-line and off-line navigating an autonomous mobile robot through a maze of obstacles has been
well discussed in the literature. The most common approach to solving this problem is by modeling the robot's
configuration space of permitted positions and orientations, followed by a search for a free collision path in
this space. The method can be readily realized using a control scheme which is base on position and distance to
obstacles sensors. However, motion planning that takes into account the dynamic characteristics of the motion is
relatively an unexplored area in robotics control. This paper presents a novel method for modeling the robot's
dynamics, based on the position-velocity space geometry, in a way that facilitates
finding a free collision path in an augmented configuration space that takes into account the dynamic constraints.
Dynamic constraint such as braking distance is modeled in the augmented configuration space, called Velocity
Configuration Space. The Velocity Configuration Space is an extension to the classic configuration space by adding
the velocity magnitude and direction to the normal position and orientation. A method for modeling the dynamics is
presented, followed by a proposal of an algorithm to find the pseudo-optimal path using arc pricing. The path search
is not the main object of this paper. Rather explaining the modeling of the geometry of the dynamic constraints. Two
motion-with-velocity planning problems are discussed. A simple approach, which adds the magnitude of the velocity to
the robot's coordinates. And a more complex one, where both magnitude and direction are considered in a 4D space.
The Robot is assumed to be flat , like a disc robot, therefore its classic configuration space is a 2D space.
New Family of Topological Rings with Applications in Linear Systems
Guy Salomon, Ben Gurion University
The theory of distributions and theory of linear system has a long history.
In particular, it is possible to define a linear system as a continuous linear
operator, S->S', where S is the Schwartz space, or any other nuclear space of
test functions. A disadvantage of the Schwartz and other nuclear spaces used
in this theory is the fact that there is no natural continuous multiplication
action in this space, and thus, for example, there is no possible to consider
analytic calculus, or to have implementation theory, in some sense. Moreover,
in this manner, the stochastic and deterministic cases are examined separately,
as two different theories.
We present a new family of topological rings, which defined by a family of
positive numbers. While one specific such a family represents the Schwartz
space (and its dual), another one represents Kondratiev Space of stochastic
test functions (and its dual), which is used to define the White Noise space,
and another one represents the Kondratiev spaces of Poissonian test functions
(and its dual). Examining these families, we prove a theorem which gives
sufficient and necessary condition, to the existence of multiplicative action
(in an appropriate manner) on these spaces. We also give some examples and
Fold Change Detection and Scalar Symmetry of Sensory Input Fields
Oren Shoval, Weizmann Institute
Recent studies suggest that certain cellular sensory systems display
detection (FCD): a responsewhose entire shape, including amplitude and
depends only on fold-changes in input, and not on absolute changes.
Thus, a step
change in input from, say, level 1 to 2, gives precisely the same
as a step from level 2 to 4, since the steps have the same fold-change.
We ask what
is the benefit of FCD, and show that FCD is necessary and sufficient for
search to be independent of multiplying the inputfield by a scalar. Thus
search pattern depends only on the spatial profile of the input, and not
amplitude. Such scalar symmetry occurs in a wide range of sensory inputs,
as source strength multiplying diffusing/convecting chemical fields sensed
in chemotaxis, ambient light multiplying the contrast field in vision, and
protein concentrations multiplying the output in cellular signalingsystems.
we demonstrate that FCD entails two features found across sensory systems,
adaptation and Weber†s law, but that these two features are not sufficie
nt for FCD. Finally, we present a wide class of mechanisms that have FCD,
including certain nonlinear feedback and feedforward loops. We find that
bacterial chemotaxis displays feedback within the present class, and hence
is expected to show FCD. This can explain experiments in which chemotaxis
searches are insensitive to attractant source levels. This study thus
suggests a connection between properties of biological sensory systems
and scalar symmetry stemming from physical properties of their inputfields.
Linear Optimal State Estimation in Systems with Independent
Daniel Sigalov, Technion
State estimation in dynamical systems with randomly switching
coefficients is an important problem with a variety of applications.
Some natural examples are maneuvering target tracking, and fault detection
and isolation (FDI), featured in, e.g., navigation systems. The standard
modeling presumes a discretely evolving mode variable that is accompanied
by a continuously varying state vector. This is the well known formulation
of hybrid systems. Various problems have been formulated in the past using
state space modeling with a randomly evolving mode process. In problems
involving uncertain observations the mode comprises the matrices of the
measurement equation. In maneuvering target tracking applications the
mode is usually defined to consist of the matrix parameters of the
We consider a more general state space representation of dynamical systems
with switching coefficients. First, we assume that the process equation
depends also on the latest estimate of the state vector, representing the
behavior of a closed loop control system featuring a state estimator.
Second, the measurement equation is allowed to depend also on the current
state estimate, representing, e.g., a tracking system whose observations
are acquired in a small validation window that is set around some predicted
state of the target. Third, we broaden the class of problems that may be
treated under the hybrid systems paradigm by allowing the mode to assume
values in a continuous rather than in a discrete domain. Finally, the
random (matrix) coefficients may change not only in their values, but
also in their dimensions.
As is well known, the MMSE optimal estimate of the state of a hybrid system
cannot be practically obtained. Significant efforts have therefore been
dedicated to obtaining suboptimal solutions. In this work we consider
optimality within a narrower family of linear filters and derive a linear
MMSE algorithm that may be conveniently implemented in a recursive form,
circumventing the theoretical need for unbounded memory, as required by
the unrestricted optimal filter. We also show that some well known results
in estimation of hybrid systems are conveniently obtained as special cases
of our linear optimal filtering algorithm, via an appropriate adjustment
of the problem's parameters. To illustrate the applicability of the
proposed framework we show how one can model and solve to optimality,
in the linear MMSE sense, the problem of tracking a target in clutter.
The derived filter is shown to attain comparable performance when
compared to popular nonlinear approaches. "
Modeling and Control of Traveling Waves in
Lea Sirota, Technion
Flexible structures are governed by Partial Differential Equations (PDE), hence have infinite dimension.
Since the PDE is not the standard starting point for control oriented modeling, a common practice is to use a
finite dimension approximation, e.g. Finite Element or truncated modal model. However there are several problems
associated with that approach. First, for reasonable accuracy the approximate models need typically a very high
order, which affects the order of model based control methods. Moreover, some important properties, and most of
the physical insight are lost in the approximation process.
A different modeling, and consequently control approach uses the accurate, infinite dimension Laplace transfer
function. The building blocks of that transfer function are time delays, representing the wave motion, and low
order rational expressions, representing the boundary phenomena. The approach was used to derive the absolute
vibration suppression (AVS) controller, which is a dedicated, low order control law that completely suppresses
the residual vibrations in the structure. While the controller was derived using mathematical considerations,
its physical interpretation is eliminating reflections of the waves from the boundary.
The AVS control law was designed for a system with zero initial conditions (I.C.). This is a common assumption
in control applications, especially in tracking. The main goal of this presentation is to investigate the
performance of the AVS controller in the presence of nonzero I.C. We suggest a method of obtaining an explicit
form of the solution for a the general case of nonzero I.C.. In this solution, the response is given in terms
of propagating waves and their reflections, hence its physical character is emphasized. Having the explicit
solution, the effect of the wave based AVS controller on the response to I.C. can be rigorously investigated.
It is shown that the vibration suppression properties of this controller can be extended to non-zero initial
conditions. In some cases the motion stops completely in finite time, and in others it decays exponentially
Investigation of Techniques to Improve High Weights Lifting
by Robotic Manipulators
Sergy Stepura, Technion
According to manufacturer specification, the maximal allowable payload of
most open chained robotic manipulators ranges between 5 to 20 percent of
the manipulator's self weight. Human beings are able to lift weights greater
than their own; therefore, it is encouraged to investigate the possibility
of improving weightlifting ability of industrial robotic manipulators. A
good example for the human ability is the record set by Charles Rigoulot
(France) in 1930. The weightlifter had lifted 115 kg, while weighing only
103 Kg. The weightlifting was performed using the One Hand Snatch technique.
Utilizing this technique, the human body acts similar to the open chained
robotic manipulator. Present research has examined possibilities to improve
weightlifting ability of industrial robotic manipulators. Three approaches
to improve manipulators weightlifting ability were examined: mimicking the
Olympic weightlifter's strategy; weightlifting along the minimal energy
trajectory and manipulators motors overloading.
To obtain the minimal energy trajectory, three optimization approaches were
tested: analytical approach (Euler-Lagrange equation, Calculus of
Variations); adaptive algorithm (Genetic Algorithm) and gradient based
iterative approach (Line-Search). The study was performed on a simple
pendulum, i.e. a rod with electrical motor connected to its upper tip.
All three approaches lead to the same solution: oscillatory trajectory with
growing amplitude up to the weight lifting completion. The analytical
approach is the most accurate, but there is significant problems to apply
the operational envelop constraints. The adaptive algorithm enables
implementation of any constraints; although solution converges slowly
(computation duration is very long). Gradient based iterative approach
converges the fastest and enables implementation of operational envelop
constraints by adding a barrier function (Interior-Point method), although
being very sensitive to the initial guess. In order to use Interior-Point
method specifically for industrial manipulator, it was suggested that the
initial guess would be the reversal of falling trajectory.
In contrast to most of the researches dealing with manipulators
weightlifting ability, the RV-M2 Movemaster (selected for the demonstration
in this research) is subjected to operational envelops constraints and is
unable to perform oscillating movements using its most powerful joints.
Implementation of techniques for improving weightlifting capabilities for
a manipulator with such limitation indicates that the suggested techniques
are applicable for a wide range of industrial manipulators, especially for
those that have lesser limitation.
This research has shown that implementation of any of the suggested
techniques substantially improves the weightlifting capabilities of the
open chained robotic manipulator. Best result is achieved by combination
of minimal energy trajectory with electrical motors overload.
Optimal Covariance Selection for Estimation Using Graphical Models
Sergey Vichik, Technion
We consider a problem encountered when trying to estimate a Gaussian random
field using a distributed estimation approach based on Gaussian graphical
models. Because of constraints imposed by estimation tools used in Gaussian
graphical models, the a priori covariance of the random field is restricted
to embed conditional independence constraints among a significant number of
variables. The problem is, then: given the unrestricted a priori covariance
of the random field, and the conditional independence constraints, how
should one select the restricted covariance, optimally representing the
(given) a priori covariance, but also satisfying the constraints? In 1972,
Dempster provided a solution, optimal in the maximum likelihood sense, to
the above problem. Since then, many works have used Dempster's optimal
covariance, but none has addressed the issue of suitability of this
covariance for estimation problems. We show that Dempster's covariance
is not optimal in most minimum mean squared error (MMSE) estimation problems.
We propose a method for finding the MMSE optimal covariance, and study its
properties. We then illustrate the analytical results via a numerical
simulation, which demonstrates the benefits of using the optimal covariance
instead of Dempster's covariance.
Extremum Seeking Control Using Small Oscillations
George Weiss, Tel Aviv University
In extremum seeking control we try to bring a
nonlinear function of dynamically varying parameters to
a maximum. Typical applications are controllers that
seek to optimize the voltage on a battery of photovoltaic
cells, or the speed of a wind turbine. In these
applications the objective is to extract the maximum
power from the device. When the aim is to extract maximum
power from a device, them extremum seeking control
is usually called maximum power point tracking (MPPT).
We shall focus on the simple particular case of an (almost)
static nonlinear plant.
Broadly, extremum seeking control can be done by sweeping
(searching the entire range of possible control inputs) or
by gradient methods, each with its advantages and
shortcomings. Here we concentrate on the gradient method
based on adding small sinusoidal oscillations to the control
variables and "listening" to the effect of these
oscillations. This was introduced and analyzed in Drapper
and Li (1951), Blackman (1962), Krasovskii (1963), Meerkov
(1967) and others. Careful mathematical analysis has been
carried out in Krstic (2000) and Krstic and Wang (2000).
The literature on practical electronic implementations is
huge, we mention Leyva et al (2006) and Branton et al
(2010). We explain a shortcoming of the method and a very
simple idea to overcome it.
Motion Control of a Servo-Pneumatic Actuated Quadruped Robot
Eddie Zisser, Ben Gurion University
This research explores techniques to control a servo-pneumatic actuator designed to drive a 300kg quadruped robot. The main advantages of pneumatic actuators are low cost, cleanliness, reliability, high power to weight ratio, a simple mechanism and natural damping as a result of compressibility of the working fluid (Air). In contrast, the high order non-linearity that characterize the pneumatic actuators' dynamics, significantly restrict industry implementations to open loop systems.
The main goal of this research is to develop a control law of a servo-pneumatic actuator, which will demonstrate high accuracy and quick response, while resisting a continuously varying external load, significantly greater than those mentioned in previous papers. The system was assembled from widespread components and has a simple configuration. It consists of a longitudinal reciprocating pneumatic actuator, a proportional servo valve and a standard micro-controller.
A non-linear model of the system dynamics was developed as a basic step toward the control law. The model deals with special phenomena such as compressibility of the air, chocked flow and valve dynamics, mass flow attenuation caused by tube friction and response delays due to relatively slow expansion of air pressure. The derivation of the model was based on the mass and energy conservation law, the ideal gas law and Newton's equation of motion.
The control law was designed according to Lyapunov criterion which relates the energy of the system with its stability. The Integrator backstepping methodology that was chosen, defines the system's energy function in a recursive manner. The control sign (the system input), aimed to reduce system energy to the minimum, and is computed according to the state feedback that comprise the actuator's chambers' pressures and the piston's position and velocity.
A comprehensive simulation has been written to display the system's behavior under the influence of external load. Closed-loop test showed excellent results stabilizing and tracking a reference input and reaching the desired position within one second. An important conclusion, obtained from simulation, is that the non-linear control law greatly reduces dependency of the system's response on external load magnitude and mechanical friction magnitude.
Further research will focus on implementing the control law on a real experimental system. The control law capability will be tested by exposing the system to a continuously changing external load varying from 0 to 150kg, which resembles the robot's legs' working range.