stabilization of nonlinear systems via forwarding mod l sub g v forwarding is a tool for constructing stabilizers for nonlinear systems a key step in this design technique is to find an explicit solution to a partial differential equation pde which may be hard to find actually the pde may even not be solvable at all in this paper we show that it is possible to provide an additional degree of freedom for the solution of the aforementioned pde hence effectively extending the realm of application of the forwarding methodology our contribution is illustrated with the example of an inverted pendulum with a disk inertia

speed observation and position feedback stabilization of partially linearizable mechanical systems the problems of speed observation and position feedback stabilization of mechanical systems are addressed in this paper our interest is centered on systems that can be rendered linear in the velocities via a partial change of coordinates it is shown that the class is fully characterized by the solvability of a set of partial differential equations pdes and strictly contains the class studied in the existing literature on linearization for speed observation or control a reduced order globally exponentially stable observer constructed using the immersion and invariance methodology is proposed the design requires the solution of another set of pdes which are shown to be solvable in several practical examples it is also proven that the full order observer with dynamic scaling recently proposed by karagiannis and astolfi obviates the need to solve the latter pdes finally it is shown that the observer can be used in conjunction with an asymptotically stabilizing full state feedback interconnection and damping assignment passivity based controller preserving asymptotic stability

analysis of the energy based control for swinging up two pendulums this note studies the energy based control for swinging up two pendulums on a cart and it presents an original analysis of the convergence of the energy and the motion of the two pendulums for the two pendulums with the different natural frequencies irrespective of initial state of the two pendulums it is shown that the energy of each pendulum converges to either of two values and the motion of the two pendulums can be described by four invariant sets moreover the stability analysis of the motion of the two pendulums is performed via the concept of stability of an invariant set the results obtained in this note not only show theoretically the effectiveness of the energy based control for swinging up the two pendulums but also pave one way for analyzing and designing the energy based control for more complicated underactuated mechanical systems

symmetry relations in exactly soluble models symmetry relations such as the star triangle or the inverse relation are very useful in determining the partition function of two dimensional exactly soluble models a common construction of the three dimensional equivalents of these symmetry relations is presented they are used to derive in a geometric way applying simultaneously to different kinds of spin models the consequent global properties i e the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function the usefulness of the inverse relation is illustrated by an application to the three dimensional ising model

interconnection and damping assignment passivity based control of port controlled hamiltonian systems passivity based control pbc is a well established technique that has shown to be very powerful to design robust controllers for physical systems described by euler lagrange el equations of motion for regulation problems of mechanical systems which can be stabilized shaping only the potential energy pbc preserves the el structure and furthermore assigns a closed loop energy function equal to the difference between the energy of the system and the energy supplied by the controller thus we say that stabilization is achieved via energy balancing unfortunately these nice properties of el pbc are lost when used in other applications which require shaping of the total energy for instance in electrical or electromechanical systems or even some underactuated mechanical devices our main objective in this paper is to develop a new pbc theory which extends to a broader class of systems the aforementioned energy balancing stabilization mechanism and the structure invariance towards this end we depart from the el description of the systems and consider instead port controlled hamiltonian models which result from the network modelling of energy conserving lumped parameter physical systems with independent storage elements and strictly contain the class of el models

stabilization of hamiltonian systems with nonholonomic constraints based on time varying generalized canonical transformations this paper is concerned with the stabilization of nonholonomic systems in port controlled hamiltonian formulae based on time varying generalized canonical transformations a special class of time varying generalized canonical transformations are introduced which modify the kinetic energy of the original system without changing the generalized hamiltonian structure with passivity utilizing these transformations time varying asymptotically stabilizing controllers for the nonholonomic hamiltonian systems are derived since the proposed method is a natural generalization of passivity based control for conventional holonomic systems it is expected that the tools developed for conventional systems will be applicable to nonholonomic systems based on the proposed method

the matching conditions of controlled lagrangians and ida passivity based control this paper discusses the matching conditions resulting from the controlled lagrangians method and the interconnection and damping assignment passivity based control ida pbc method both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an euler lagrange respectively hamiltonian system in the context of mechanical systems with symmetry the original controlled lagrangians method is reviewed and an interpretation of the matching assumptions in terms of the matching of kinetic and potential energy is given secondly both methods are applied to the general class of underactuated mechanical systems and it is shown that the controlled lagrangians method is contained in the ida pbc method the lambda method as described in recent papers for the controlled lagrangians method transforming the matching conditions a set of non linear pdes into a set of linear pdes is discussed the method is used to transform the matching conditions obtained in the ida pbc method into a set of quadratic and linear pdes finally the extra freedom obtained in the ida pbc method with respect to the controlled lagrangians method is used to discuss the integrability of the closed loop system explicit conditions are derived under which the closed loop hamiltonian system is integrable leading to the introduction of gyroscopic terms

controlled lagrangians and the stabilization of mechanical systems ii potential shaping for pt i see ibid vol 45 p 2253 70 2000 we extend the method of controlled lagrangians cl to include potential shaping which achieves complete state space asymptotic stabilization of mechanical systems the cl method deals with mechanical systems with symmetry and provides symmetry preserving kinetic shaping and feedback controlled dissipation for state space stabilization in all but the symmetry variables potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables the approach also extends the method of controlled lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline

stabilization of a class of underactuated mechanical systems via interconnection and damping assignment we consider the application of a formulation of passivity based control pbc known as interconnection and damping assignment ida to the problem of stabilization of underactuated mechanical systems which requires the modification of both the potential and the kinetic energies our main contribution is the characterization of a class of systems for which ida pbc yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction the class is given in terms of solvability of certain partial differential equations one important feature of ida pbc stemming from its hamiltonian formulation is that it provides new degrees of freedom for the solution of these equations using this additional freedom we are able to show that the method of controlled lagrangians in its original formulation may be viewed as a special case of our approach as illustrations we design asymptotically stabilizing ida pbcs for the classical ball and beam system and a novel inertia wheel pendulum

swinging control of nonlinear oscillations the speed gradient method of control design used previously for problems of regulation and tracking is extended to oscillating systems with energy based objective functions the concept of swinging control is introduced meaning achievement of arbitrary large level of the objective function by arbitrary small control level the existence of swinging control for hamiltonian systems is established simulation results for pendulum swinging problem are demonstrated

the core variety of a multisequence in the truncated moment problem abstract let m i i z n i m 0 0 denote a real n dimensional multisequence of finite degree m the truncated moment problem concerns the existence of a positive borel measure supported in r n such that 0 1 i r n x i d i z n i m we associate to 2 d an algebraic variety in r n called the core variety v v the core variety contains the support of each representing measure we show that if v is nonempty then 2 d 1 has a representing measure moreover if v is a nonempty compact or determining set then 2 d has a representing measure we also use the core variety to exhibit a sequence with positive definite moment matrix and positive riesz functional which fails to have a representing measure

remarks on the star triangle relation in the baxter bazhanov model we show that the restricted star triangle relation introduced by bazhanov and baxter can be obtained either from the star triangle relation of the chiral potts model or from the star square relation proposed by kashaev mangazeev and stroganov and give a response to a guess of bazhanov and baxter

energy based balance control approach to the ball and beam system this article investigates the set point balance control problem of the ball and beam system acting on the ball which is an underactuated system the control law is designed based on the total mechanical energy and the passivity properties of the system according to the desired set point the zero equilibrium point and non zero equilibrium points are studied respectively for the zero case it is proved that a single pd feedback controller is sufficient to bring the state to zero from any initial condition on condition that the control parameters satisfy an inequality for the non zero case the control problem is much more complicated unlike the previous energy based control laws a new form of lyapunov function candidate is constructed a complete analysis of the convergence of the energy and the dynamics is given and the characteristics of the closed loop system with the proposed feedback control law are illustrated moreover it is proved that with the parameter choice rules proposed in this article the ball and beam system will eventually converge exactly to the desired non zero equilibrium point furthermore since the length of the beam is not unlimited the trajectory of the ball should be restricted within a limited range the balance control laws are modified to avoid the ball running beyond the joint range limitation for the zero case and the non zero case respectively simulation results show that the control laws proposed in this article are effective for the set point balance control problem of the ball and beam system

dynamic extension is unnecessary for stabilization via interconnection and damping assignment passivity based control interconnection and damping assignment passivity based control ida pbc is a technique that regulates the behavior of nonlinear systems assigning a desired port hamiltonian structure to the closed loop this basic idea introduced seven years ago has turned out to be very successful and has provided solutions to a wide variety of physical problems although ida pbc is originally formulated as a static state feedback technique a natural question that arises is whether it is possible to extend the realm of applicability of the method by considering dynamic controllers more precisely is the set of plants that is stabilizable with static state feedback ida pbc smaller than the one stabilizable with dynamic ida pbc the main contribution of this paper is to prove that the answer to this question is unfortunately negative

the notion of v detectability and stabilization of invariant sets of nonlinear systems the problem of global local stabilizability of an invariant set of passive nonlinear controlled system is considered new notion of detectability of the zero value set of a storage function is introduced it is shown that this property plays a fundamental role in solving of invariant set stabilization problems and in defining the global local stabilizing regulator sufficient conditions implying that passive nonlinear system has a detectable zero value set for a smooth storage function are proposed

star triangle relation for a three dimensional model the solvablesl n chiral potts model can be interpreted as a three dimensional lattice model with local interactions to within a minor modification of the boundary conditions it is an ising type model on the body centered cubic lattice with two and three spin interactions the corresponding local boltzmann weights obey a number of simple relations including a restricted star triangle relation which is a modified version of the well known star triangle relation appearing in two dimensional models we show that these relations lead to remarkable symmetry properties of the boltzmann weight function of an elementary cube of the lattice related to the spatial symmetry group of the cubic lattice these symmetry properties allow one to prove the commutativity of the row to row transfer matrices bypassing the tetrahedron relation the partition function per site for the infinite lattice is calculated exactly

linear optimization with cones of moments and nonnegative polynomials let mathcal a a be a finite subset of mathbb n n nn and mathbb r x mathcal a r x a be the space spanned by monomials x alpha x with alpha in mathcal a a let k k be a compact semialgebraic set of mathbb r n rn such that a polynomial in mathbb r x mathcal a r x a is positive on k k denote by fancyscript p mathcal a k pa k the cone of polynomials in mathbb r x mathcal a r x a that are nonnegative on k k the dual cone of fancyscript p mathcal a k pa k is fancyscript r mathcal a k ra k the set of all truncated moment sequences in mathbb r mathcal a ra that admit representing measures supported in k k first we study geometric properties of the cones fancyscript p mathcal a k pa k and fancyscript r mathcal a k ra k like interiors closeness duality memberships and construct a convergent hierarchy of semidefinite relaxations for each of them second we propose a semidefinite algorithm for solving linear optimization problems with the cones fancyscript p mathcal a k pa k and fancyscript r mathcal a k ra k and prove its asymptotic and finite convergence third we show how to check whether fancyscript p mathcal a k pa k and fancyscript r mathcal a k ra k intersect affine subspaces if they do we show how to get a point in the intersections if they do not we prove certificates for the empty intersection

positivity of riesz functionals and solutions of quadratic and quartic moment problems abstract we employ positivity of riesz functionals to establish representing measures or approximate representing measures for truncated multivariate moment sequences for a truncated moment sequence y we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set k r n if and only if the associated riesz functional l y is k positive for a determining set k we prove that if l y is strictly k positive then y admits a representing measure supported in k as a consequence we are able to solve the truncated k moment problem of degree k in the cases i n k 2 4 and k r 2 ii n 1 k 2 and k is defined by one quadratic equality or inequality in particular these results solve the truncated moment problem in the remaining open cases of hilbert s theorem on sums of squares

bounded stabilisation of stochastic port hamiltonian systems this paper proposes a stochastic bounded stabilisation method for a class of stochastic port hamiltonian systems both full actuated and underactuated mechanical systems in the presence of noise are considered in this class the proposed method gives conditions for the controller gain and design parameters under which the state remains bounded in probability the bounded region and achieving probability are both assignable and a stochastic lyapunov function is explicitly provided based on a hamiltonian structure although many conventional stabilisation methods assume that the noise vanishes at the origin the proposed method is applicable to systems under persistent disturbances

control of high order nonholonomic systems in power chained form using discontinuous feedback addresses the problems of almost asymptotic stabilization and global asymptotic regulation gar for a class of high order nonholonomic systems in power chained form this particular class of nonlinear systems is an extension of a nonholonomic system in chained form that has received considerable attention in the past few years the nonholonomic system considered in this paper is not necessarily affine in the control variables and therefore cannot be handled by existing methods sufficient conditions are presented under which a discontinuous state feedback control law or a switching controller can be recursively constructed using the so called adding a power integrator technique of c qian et al 2001 we also illustrate how the results can be extend to multi input systems in power chained form simulation examples are provided to demonstrate the effectiveness of the proposed controllers

control by interconnection and standard passivity based control of port hamiltonian systems the dynamics of many physical processes can be suitably described by port hamiltonian ph models where the importance of the energy function the interconnection pattern and the dissipation of the system is underscored to regulate the behavior of ph systems it is natural to adopt a passivity based control pbc perspective where the control objectives are achieved shaping the energy function and adding dissipation in this paper we consider the pbc techniques of control by interconnection cbi and standard pbc in cbi the controller is another ph system connected to the plant through a power preserving interconnection to add up their energy functions while in standard pbc energy shaping is achieved via static state feedback in spite of the conceptual appeal of formulating the control problem as the interaction of dynamical systems the current version of cbi imposes a severe restriction on the plant dissipation structure that stymies its practical application on the other hand standard pbc which is usually derived from a uninspiring and non intuitive ldquopassive output generationrdquo viewpoint is one of the most successful controller design techniques the main objectives of this paper are 1 to extend the cbi method to make it more widely applicable in particular to overcome the aforementioned dissipation obstacle 2 to show that various popular variants of standard pbc can be derived proceeding from a unified perspective 3 to establish the connections between cbi and standard pbc proving that the latter is obtained restricting the former to a suitable subset providing a nice geometric interpretation to standard pbc and comparing the size of the set of ph plants for which they are applicable

power based control canonical coordinate transformations integral and adaptive control power based modeling was originally developed in the early sixties to describe a large class of nonlinear electrical rlc networks in a special gradient form recently this idea has been extended for modeling and control of a larger class of physical systems in this paper first coordinate transformations are introduced for systems described in this framework such that the physical structure is preserved such a transformation can provide new insights for both analysis and control design second power based integral and adaptive control schemes are presented advantages of these schemes are shown by their application on standard mechanical systems

swing up control based on virtual composite links for n link underactuated robot with passive first joint this paper concerns the swing up control of an n link revolute robot moving in the vertical plane with the first joint being passive and the others being active the goal of this study is to design and analyze a swing up controller that can bring the robot into any arbitrarily small neighborhood of the upright equilibrium point with all links in the upright position to achieve this challenging objective while preventing the robot from becoming stuck at an undesired closed loop equilibrium point we first address the problem of how to iteratively devise a series of virtual composite links to be used for designing a coordinate transformation on the angles of all the active joints second we devise an energy based swing up controller that uses a new lyapunov function based on that transformation third we analyze the global motion of the robot under the controller and establish conditions on the control parameters that ensure attainment of the swing up control objective specifically we determine the relationship between the closed loop equilibrium points and a control parameter finally we verify the theoretical results by means of simulations on a 4 link model of a gymnast on the high bar this study not only unifies some previous results for acrobots and three link robots with a passive first joint but also provides insight into the energy and passivity based control of underactuated multiple degree of freedom systems

on the lambda equations for matching control laws we discuss matching control laws for underactuated systems we previously showed that this class of matching control laws is completely characterized by a linear system of first order partial differential equations for one set of variables lambda followed by a linear system of first order partial differential equations for the second set of variables widehat g widehat v here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the lambda equations we give four examples illustrating different features of matching control laws the last example is a system with two unactuated degrees of freedom that admits only basic solutions to the matching equations there are systems with many matching control laws where only basic solutions are potentially useful we introduce a rank condition indicating when this is likely to be the case

nonlinear control via approximate input output linearization the ball and beam example approximate input output linearization of nonlinear systems which fail to have a well defined relative degree is studied for such systems a method for constructing approximate systems that are input output linearizable is provided the analysis presented is motivated through its application to a common undergraduate control laboratory experiment the ball and beam where it is shown to be more effective for trajectory tracking than the standard jacobian linearization

the truncated moment problem via homogenization and flat extensions abstract this paper studies how to solve the truncated moment problem tmp via homogenization and flat extensions of moment matrices we first transform tmp to a homogeneous tmp htmp and then use semidefinite programming sdp techniques to solve htmp our main results are 1 a truncated moment sequence tms is the limit of a sequence of tms admitting measures on r n if and only if its homogenized tms htms admits a measure supported on the unit sphere in r n 1 2 an htms admits a measure if and only if the optimal values of a sequence of sdp problems are nonnegative 3 under some conditions that are almost necessary and sufficient by solving these sdp problems a representing measure for an htms can be explicitly constructed if one exists

nonlinear control of mechanical systems with an unactuated cyclic variable numerous robotic tasks associated with underactuation have been studied in the literature for a large number of these in the plane the mechanical models have a cyclic variable the cyclic variable is unactuated and all shape variables are independently actuated this paper formulates and solves two control problems for this class of models if the generalized momentum conjugate to the cyclic variable is not conserved conditions are found for the existence of a set of outputs that yields a system with a one dimensional exponentially stable zero dynamics i e an exponentially minimum phase system along with a dynamic extension that renders the system locally input output decouplable if the generalized momentum conjugate to the cyclic variable is conserved a reduced system is constructed and conditions are found for the existence of a set of outputs that yields an empty zero dynamics along with a dynamic extension that renders the system feedback linearizable a common element in these two feedback problems is the construction of a scalar function of the configuration variables that has relative degree three with respect to one of the input components the function arises by partially integrating the conjugate momentum the results are illustrated on two balancing tasks and on a ballistic flip motion

tetrahedron equations and the relativistic s matrix of straight strings in 2 1 dimensions the quantums matrix theory of straight strings infinite one dimensional objects like straight domain walls in 2 1 dimensions is considered thes matrix is supposed to be purely elastic and factorized the tetrahedron equations which are the factorization conditions are investigated for the special two colour model the relativistic three strings matrix which apparently satisfies this tetrahedron equation is proposed

limits of positive flat bivariate moment matrices the bivariate moment problem for a sequence 6 of degree 6 remains unsolved but we prove that if the associated 10 10 moment matrix m3 satisfies m3 0 and rank m3 6 then admits a sequence of approximate representing measures and 5 has a representing measure more generally let fd denote the closure of the positive flat moment matrices of degree 2d in n variables each matrix in fd admits computable approximate representing measures and in 2013 jiawang nie and the author began to study concrete conditions for membership in this class let 2d i i zn i 2d 0 0 denote a real n dimensional sequence of degree 2d if the corresponding moment matrix md md is the limit of a sequence of positive flat moment matrices m k d i e m k d 0 and rank m k d rank m k d 1 then i md 0 ii rank md d 1 dim r x1 xn d 1 and iii 2d 1 admits a representing measure we extend our earlier results by proving conversely that for n 2 if md satisfies certain positivity and rank conditions related to i iii then md is the limit of positive flat moment matrices

controlled lagrangians and the stabilization of mechanical systems i the first matching theorem we develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled lagrangians the procedure involves making structured modifications to the lagrangian for the uncontrolled system thereby constructing the controlled lagrangian the euler lagrange equations derived from the controlled lagrangian describe the closed loop system where new terms in these equations are identified with control forces since the controlled system is lagrangian by construction energy methods can be used to find control gains that yield closed loop stability we use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group the procedure is demonstrated for several underactuated balance problems including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

enhanced nonlinear damping for a class of singularly perturbed interconnected nonlinear systems in this paper we propose a method of enhanced nonlinear damping control for a class of singularly perturbed interconnected nonlinear systems spinss instead of simply canceling out the interconnection between slow and fast subsystems the proposed method transforms spins into a feedback connection of two output strictly passive subsystems then enhanced nonlinear damping is implemented to improve the transient behavior of slow subsystem the proposed method provides weighting factors for each tracking error which enable the energy function to be designed to shape the tracking performance for each state we also design an augmented state observer to estimate unknown disturbances and unmeasurable states using a composite lyapunov function we prove that the origin of the tracking error dynamics is globally exponentially stable it is also analyzed that the tracking errors are globally uniformly ultimately bounded for a time varying bounded disturbance experiments were performed to evaluate the tracking performance of the proposed method and our results validate the effectiveness of the proposed method

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

vss version of energy based control for swinging up a pendulum abstract a new algorithm ensuring global attractivity of the upright unstable equilibrium of a pendulum based on the variable structure system version of the energy speed gradient method is proposed it is shown that global attractivity cannot be obtained with continuous static state feedback a detailed global analysis of the transient behavior of the closed loop system is presented in addition it is shown that the global attractivity of the upright equilibrium can be achieved by applying a control of arbitrarily small magnitude

partial stabilization of underactuated euler lagrange systems via a class of feedback transformations abstract a new class of non trivial feedback transformations which can be applied to any underactuated lagrangian system is proposed as shown these transformations play an important role in solving a stabilization problem with respect to a part of the actuated variables while the given additional constraints are imposed on the rest of the variables theoretical results are illustrated by solving the problem of swinging up the pendubot

periodic motion planning for virtually constrained euler lagrange systems the paper suggests an explicit form of a general integral of motion for some classes of dynamical systems including n degrees of freedom euler lagrange systems subject to n 1 virtual holonomic

dynamics and control of a class of underactuated mechanical systems this paper presents a theoretical framework for the dynamics and control of underactuated mechanical systems defined as systems with fewer inputs than degrees of freedom control system formulation of underactuated mechanical systems is addressed and a class of underactuated systems characterized by nonintegrable dynamics relations is identified controllability and stabilizability results are derived for this class of underactuated systems examples are included to illustrate the results these examples are of underactuated mechanical systems that are not linearly controllable or smoothly stabilizable

monotonically positive matrices abstract a matrix a is monotonically positive mp if there exists a matrix u such that a u u t and each column of u is monotonically nonincreasing or nondecreasing following the approach in j nie 2014 15 we propose a semidefinite algorithm for checking whether or not a matrix is mp if it is not mp a certificate for it can be obtained if it is mp an mp decomposition can be obtained some computational experiments are presented to show how to do this

three dimensional vertex model in statistical mechanics from baxter bazhanov model we find that the boltzmann weight of the three dimensional baxter bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the wu kadanoff wegner duality between the cube and vertex type tetrahedron equations is obtained explicitly for the baxter bazhanov model then a three dimensional vertex model is obtained by considering the symmetry property of the weight function which corresponds to the three dimensional baxter bazhanov model the vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube we write down the symmetry relations of the weight functions under the actions of the symmetry groupg of the cube the six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected with the six spaces in which the vertextype tetrahedron equation is defined

control of nonlinear underactuated systems in this paper we introduce a new method to design control laws for nonlinear underactuated systems our method produces an infinite dimensional family of control laws whereas most control techniques only produce a finite dimensional family these control laws each come with a natural lyapunov function the inverted pendulum cart is used as an example in addition we construct an abstract system that is open loop unstable and cannot be stabilized using any linear control law and demonstrate that our method produces a stabilizing control law 2000 john wiley sons inc

stabilization lyapunov functions and dissipation for linear time invariant systems any stabilizing controller for a given plant can be associated with a supply rate with respect to which the plant in open loop is half line dissipative we also prove the equivalence between the stability of the interconnection of two systems and their half line dissipativity with respect to a supply rate and its negative

stabilization of systems with one degree of underactuation with energy shaping a geometric approach a geometric formulation for stabilization of systems with one degree of underactuation which fully solves the energy shaping problem for these system is given the results show that any linearly controllable simple mechanical system with one degree of underactuation is stabilizable by energy shaping even in the absence of gyroscopic force via a closed loop metric which is not necessarily positive definite an example of a system with one degree of underactuation is provided for which the stabilization by energy shaping method is not achievable in the absence of gyroscopic forces using a positive definite closed loop metric

energy shaping of mechanical systems via pid control and extension to constant speed tracking in a recent contribution it was shown that a class of mechanical systems which contains many practical examples can be stabilized via energy shaping without solving partial differential equations the proposed controller consists of two terms a partial linearizing state feedback and a linear pid loop around two new passive outputs in this brief note we prove that the first admittedly non robust step can be obviated leaving only the linear pid a second contribution of the note is to propose a slight modification to the controller to go beyond regulation tasks being able to follow ramp references in the actuated coordinates

new solvable lattice models in three dimensions in this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models the first is the zamolodchikov model which is the only nontrivial model on a three dimen sional lattice so far solved the second is the chiral potts model on the square lattice and its generalization associated with theu q sl n algebra which is of current interest due to its connections with high genus algebraic curves and with representations of quantum groups at roots of unity we show that this last sl n generalized chiral potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square lattice layers with ann valued n 2 spin at each site further in then 2 case this three dimen sional model reduces after a modification of the boundary conditions to the zamolodchikov model we mentioned above

an analogue of the riesz haviland theorem for the truncated moment problem abstract let 2 n i i 2 n denote a d dimensional real multisequence let k denote a closed subset of r d and let p 2 n p r x 1 x d deg p 2 n corresponding to the riesz functional l l p 2 n r is defined by l a i x i a i i we say that l is k positive if whenever p p 2 n and p k 0 then l p 0 we prove that admits a k representing measure if and only if l admits a k positive linear extension l p 2 n 2 r this provides a generalization from the full moment problem to the truncated moment problem of the riesz haviland theorem we also show that a semialgebraic set solves the truncated moment problem in terms of natural degree bounded positivity conditions if and only if each polynomial strictly positive on that set admits a degree bounded weighted sum of squares representation