almost global stabilization of fully actuated rigid bodies this paper addresses the problem of stabilizing a fully actuated rigid body the problem is formulated by considering the natural configuration space for rigid bodies the special euclidean group se 3 the proposed solution consists of a landmark based controller for force and torque actuation that guarantees almost global asymptotic stability of the desired equilibrium point as such the equilibrium point is asymptotically stable and only a nowhere dense set of measure zero lies outside its region of attraction the controller uses velocity measurements and the position coordinates of a collection of landmarks fixed in the environment as an additional feature the control law is designed so as to verify prescribed bounds on the actuation

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

comment on discrete time optimal fuzzy controller design global concept approach in this correspondence we show that the relationship between the finite optimal trajectory and the infinite optimal trajectory in lemma 3 of the aforementioned paper cannot act as a theory basis for the further implementation of other theorems and the dynamic decomposition algorithm

vibrational control of nonlinear systems vibrational stabilizability in this work the theory of vibrational control for nonlinear systems is developed the present paper gives criteria for stabilizability of nonlinear systems by linear multiplicative vibrations by almost periodic forcing and by vector additive vibrations illustrative examples are also considered in a subsequent paper the problem of vibrational controllability and transient behavior will be addressed

brief almost global stabilization of the inverted pendulum via continuous state feedback continuous state feedback laws are presented for almost global stabilization of the planar inverted pendulum on a cart the synthesis is carried out exploiting energy shaping techniques and a suitable smooth switching between positive and negative feedback some of the employed ideas could be possibly generalized to the problem of stabilization of systems with multiple equilibrium points

representation of compositional relations in fuzzy reasoning abstract this paper deals with the representation of the compositional relations in fuzzy reasoning we show that the compositional relations of fuzzy reasoning under min zadeh operation can be represented by the largest solution of some fuzzy relational inequalities or relational equations

energy control of a pendulum with quantized feedback the problem of controlling a nonlinear system to an invariant manifold using quantized state feedback is considered by the example of controlling the pendulum s energy a feedback control law based on the speed gradient algorithm is chosen the main result consisting in precisely characterizing allowed quantization error bounds and resulting energy deviation bounds is presented

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

generalized receding horizon control of fuzzy systems based on numerical optimization algorithm the optimal control of fuzzy systems with constraints is still an open problem our focus concerns the optimal control problem of fuzzy systems derived from receding horizon control rhc schemes we consider methods to numerically compute the value function for general fuzzy systems the numerical method that is developed using the finite difference with sigmoidal transformation is a stable and convergent algorithm for the hamilton jacobi bellman hjb equation an optimization procedure is developed to increase the calculation accuracy with less computation time a parallel processing method is employed in the optimization procedure the optimization results are applied to the controller design of general fuzzy dynamic systems employing the principle of conventional rhc schemes rhc form controllers are designed for some classes of fuzzy dynamic systems the basic ideas are as follows first the value function is calculated by numerical methods then the value function is used as controller design parameters to redesign rhc controllers for fuzzy systems which is motivated by the inverse lyapunov function design method it is proven that the closed loop system is asymptotically stable an engineering implementation of the controller redesign scheme is discussed meanwhile the parallel processing framework that can improve the closed loop performance is also introduced

hypergraph partitioning for the parallel computing of fuzzy differential equations in our previous work fuzzy differential equations have been used to describe service composition and the solutions have been used for the performance analysis this paper presents a method to compute these fuzzy differential equations in parallel we first map the underlying petri net that generates fuzzy differential equations to a hypergraph and then partition the hypergraph to minimize inter processor communication while maintaining a good load balance based on the partition result we divide the fuzzy differential equations into several blocks finally we design parallel computing algorithm to compute these equations software hmetis is used to partition the hypergraph and software sundials is used to support the parallel computing of fuzzy differential equations online shop problem from ibm web site has been used to demonstrate the benefit of our method

some properties of infinite fuzzy relational equations on complete brouwerian lattices this paper deals with fuzzy relational equations email protected x b where both a a j j j and b are known b has an irredundant continuous join decomposition x x j j j t is unknown represents sup inf composition j is an infinite index set on complete brouwerian lattices at first a necessary and sufficient condition for existence of the attainable solutions the unattainable solutions respectively and some properties of the attainable solutions the unattainable solutions respectively are formulated then the solution set of the fuzzy relational equations is described

moving horizon control of linear systems with input saturation and plant uncertainty part 1 robustness we present a moving horizon feedback system based on constrained optimal control algorithms for linear plants with input saturation the system is a non conventional sampled data system its sampling periods vary from sampling instant to sampling instant and the control during the sampling time is not constant but determined by the solution of an open loop optimal control problem this is a two part paper in this part we show that the proposed moving horizon control system is robustly stable whether the state of the plant is measurable or not in the second part we show that the proposed moving horizon control system is capable of following a class of reference inputs and suppressing a class of disturbances experimental results show that the behaviour of the moving horizon control system is superior to that resulting from some alternative control laws

unattainable solutions of a fuzzy relation equation in a fuzzy relation equation the properties of the solution set have been investigated especially when the basic spaces are all finite sets however when the basic spaces are infinite a few properties are known in this paper we use the concepts of attainability and unattainability of a solution to clarify some properties of the solution set and show the condition for existence of a partially attainable solution and an unattainable one

latticized linear programming and fuzzy relation inequalities abstract a logical linear programming problem the latticized linear programming is proposed based on fuzzy lattice and fuzzy relation inequalities the proposed problem is essentially an optimization problem which should be useful under certain logical if then situations to illustrate the proposed approach numerical examples are solved

receding horizon control of nonlinear systems the receding horizon control strategy provides a relatively simple method for determining feedback control for linear or nonlinear systems the method is especially useful for the control of slow nonlinear systems such as chemical batch processes where it is possible to solve sequentially open loop fixed horizon optimal control problems online the method has been shown to yield a stable closed loop system when applied to time invariant or time varying linear systems it is shown that the method also yields a stable closed loop system when applied to nonlinear systems

reachable solution set of a fuzzy relation equation abstract the largest element of the solution set of a fuzzy relation equation has been found by e sanchez inform and control 30 1976 38 48 but the smallest element does not exist it is difficult to expose the solution of the fuzzy relation equation in the case of the determinate relation equations complete consequences have been found by wang peizhuang and yuan meng relation equation and relation inequalities selected papers on fuzzy subsets beijing normal university march 1980 in the case of the fuzzy relation equations wang and yuan have given a class of special solutions which probably possesses some minimality characterizations in this paper the reachable solution set of the fuzzy relation equation is given for the fuzzy relation equation on the finite sets a neat and efficient method for solving it is given

sets of solution set invariant coefficient matrices of simple fuzzy relation equations abstract a set of coefficient matrices which are invariant to a solution set of a simple fuzzy relation equation is considered the equipollency of the cardinal numbers of solution sets of simple fuzzy relation equations for two constant vectors is also shown

large scale complex eigenvalue problems abstract the nonsymmetric eigenvalue problem ax bx is discussed with special emphasis on linear algebra theory on algorithms implemented for solving large scale systems and on interpreting complex spectra obtained in applications from physics and engineering present day supercomputers make the required matrix computations feasible especially in view of the increase in memory

brief paper global stabilization of an inverted pendulum control strategy and experimental verification the problem of swinging up an inverted pendulum on a cart and controlling it around the upright position has traditionally been treated as two separate problems this paper proposes a control strategy that is globally asymptotically stable under actuator saturation and in addition locally exponentially stable the proposed methodology which performs swing up and control simultaneously uses elements from input output linearization energy control and singular perturbation theory experimental results on a laboratory scale setup are presented to illustrate the approach and its implementation

two new algorithms for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities this paper studies the optimization model of a linear objective function subject to a system of fuzzy relation inequalities fri with the max einstein composition operator if its feasible domain is non empty then we show that its feasible solution set is completely determined by a maximum solution and a finite number of minimal solutions also an efficient algorithm is proposed to solve the model based on the structure of fri path the concept of partial solution and the branch and bound approach the algorithm finds an optimal solution of the model without explicitly generating all the minimal solutions some sufficient conditions are given that under them some of the optimal components of the model are directly determined some procedures are presented to reduce the search domain of an optimal solution of the original problem based on the conditions then the reduced domain is decomposed if possible into several sub domains with smaller dimensions that finding the components of the optimal solution in each sub domain is very easy in order to obtain an optimal solution of the original problem we propose another more efficient algorithm which combines the first algorithm these procedures and the decomposition method furthermore sufficient conditions are suggested that under them the problem has a unique optimal solution also a comparison between the recently proposed algorithm and the known ones will be made

rate limited time optimal control of a planar pendulum abstract time optimal control problems often admit as solution bang bang controls but they present several drawbacks in real applications where additional degrees of regularity are desirable in this paper the rate limited time optimal swing up of an inverted pendulum is considered and necessary conditions that have to be satisfied by optimal trajectories in the presence of rate constraints are derived a comparison between this approach and the results from the standard time optimal problem is reported

robust receding horizon control of constrained nonlinear systems we present a method for the construction of a robust dual mode receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error the controller is dual mode in a neighborhood of the origin the control action is generated by a linear feedback controller designed for the linearized system outside this neighborhood receding horizon control is employed existing receding horizon controllers for nonlinear continuous time systems which are guaranteed to stabilize the nonlinear system to which they are applied require the exact solution at every instant of an optimal control problem with terminal equality constraints these requirements are considerably relaxed in the dual mode receding horizon controller presented in this paper stability is achieved by imposing a terminal inequality rather than an equality constraint only approximate minimization is required a variable time horizon is permitted robustness is achieved by employing conservative state and stability constraint sets thereby permitting a margin of error the resultant dual mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear constrained systems

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

a new iterative scheme for obtaining eigenvectors of large real symmetric matrices a rapidly convergent block iterative scheme for the computation of a few of the lowest eigenvalues and eigenvectors of a large dense real symmetric matrix is proposed the method is especially applicable to matrix eigenvalue problems that arise from the discretization of self adjoint partial differential equations one such application to certain symmetric matrices that arise in solid state band structure calculations is considered in detail the most timeconsuming parts of the present algorithm are a matrix multiplication and a gauss siedel relaxation step which are performed on each iteration these two parts can however be very efficiently implemented on a vector or parallel processing computer

some properties of sup min fuzzy relational equations on infinite domains abstract this paper deals with a sup min fuzzy relational equation on infinite domains some properties of its solution set will be obtained in particular a necessary and sufficient condition describing the attainable solution resp the unattainable solution is formulated in the end the solution set will be given under some conditions

non local stabilization of a spherical inverted pendulum we design a non linear stabilizing control law for a four degree of freedom spherical inverted pendulum the pendulum is a slim cylindrical beam attached to a horizontal plane via a universal joint the joint is free to move in the plane under the influence of a planar force the upright position is an unstable equilibrium of the uncontrolled system because of gravity the objective is to design a controller so that it stabilizes the upright position starting from any position in the upper hemisphere with arbitrary velocity we achieve this by first transforming the original system to an appropriate upper triangular form and then designing a controller which incorporates a high gain design with the method of non linear forwarding the control law is evaluated through computer simulations

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

brief paper a family of smooth controllers for swinging up a pendulum the paper presents a new family of controllers for swinging up a pendulum the swinging up of the pendulum is derived from physical arguments based on two ideas shaping the hamiltonian for a system without damping and providing damping or energy pumping in relevant regions of the state space a family of simple smooth controllers without switches with nice properties is obtained the main result is that all solutions that do not start at a zero lebesgue measure set converge to the upright position for a wide range of the parameters in the control law thus the swing up and the stabilization problems are simultaneously solved with a single smooth law the properties of the solution can be modified by the parameters in the control law control signal saturation can also be taken into account using the hamiltonian approach

sufficient conditions on general fuzzy systems as function approximators abstract in a constructive way we have found sufficient conditions under which general fuzzy systems can uniformly approximate any real continuous function on a compact domain to any degree of accuracy more importantly we have revealed the underlying mechanism of such function approximation the fuzzy systems can uniformly approximate any polynomials which according to the weierstrass approximation theorem can uniformly approximate any continuous function on a compact domain these findings lead to the following practical results 1 explicit fuzzy rules of fuzzy systems can now be easily obtained according to functions to be approximated and 2 formulas are derived which can calculate the number of input fuzzy sets output fuzzy sets and fuzzy rules needed in order to satisfy any given approximation accuracy the number increases as required approximation error decreases and as approximation error approaches zero the number approaches these formulas also reveal that the number is minimal when the maximum number of intersection between adjacent input fuzzy sets is one practical implications of these results will be discussed especially in the context of fuzzy control and fuzzy modeling two examples are provided to demonstrate how to design fuzzy systems to approximate given functions with required approximation accuracy

predictive control of switched nonlinear systems with scheduled mode transitions in this work a predictive control framework is proposed for the constrained stabilization of switched nonlinear systems that transit between their constituent modes at prescribed switching times the main idea is to design a lyapunov based predictive controller for each constituent mode in which the switched system operates and incorporate constraints in the predictive controller design which upon satisfaction ensure that the prescribed transitions between the modes occur in a way that guarantees stability of the switched closed loop system this is achieved as follows for each constituent mode a lyapunov based model predictive controller mpc is designed and an analytic bounded controller using the same lyapunov function is used to explicitly characterize a set of initial conditions for which the mpc irrespective of the controller parameters is guaranteed to be feasible and hence stabilizing then constraints are incorporated in the mpc design which upon satisfaction ensure that 1 the state of the closed loop system at the time of the transition resides in the stability region of the mode that the system is switched into and 2 the lyapunov function for each mode is nonincreasing wherever the mode is reactivated thereby guaranteeing stability the proposed control method is demonstrated through application to a chemical process example

optimal regulator for the inverted pendulum via euler lagrange backward integration we present a time invariant nonlinear controller for the inverted pendulum based on optimal control which is defined for all pendulum angles the paper provides a case study in the brute force method of calculating the extremal field by solving the euler lagrange equations backward in time thus exploring the stable manifold of the hamiltonian flow the controller is only given numerically on an interpolation grid we use this controller for closed loop simulations corroborating considerable gain robustness

brief stabilization of compact sets for passive affine nonlinear systems this paper is devoted to the problem of global local stabilization of a prescribed subset of the state space for a passive affine nonlinear system it is assumed that the desired attractive set can be described as an inverse image of zero value of some smooth nonnegative function and that this function does not increase along the solutions of the unforced system in terms of this function a class of state feedback regulators and new sufficient conditions guaranteeing global local goal set stabilization are obtained

the use of a refined error bound when updating eigenvalues of tridiagonals abstract the lanczos algorithm is used to compute some eigenvalues of a given symmetric matrix of large order at each step of the lanczos algorithm it is valuable to know which eigenvalues of the associated tridiagonal matrix have stabilized at eigenvalues of the given symmetric matrix we present a robust algorithm which is fast 20 j to 40 j operations at the j th lanczos step uses about 30 words of extra storage and has a fairly short program approxiamately 200 executable statements

lanczos versus subspace iteration for solution of eigenvalue problems in this paper we consider solution of the eigen problem in structural analysis using a recent version of the lanczos method and the subspace method the two methods are applied to examples and we conclude that the lanczos method has advantages which are too good to overlook

optimal fuzzy controller design local concept approach in this paper we present a global optimal and stable fuzzy controller design method for both continuous and discrete time fuzzy systems under both finite and infinite horizons first a sufficient condition is proposed which indicates that the global optimal effect can be achieved by the fuzzily combined local optimal controllers based on this sufficient condition we derive a local concept approach to designing the optimal fuzzy controller by applying traditional linear optimal control theory the stability of the entire closed loop continuous fuzzy system can be ensured by the designed optimal fuzzy controller the optimal feedback continuous fuzzy system can not only be guaranteed to be exponentially stable but also be stabilized to any desired degree also the total energy of system output is absolutely finite moreover the resultant feedback continuous fuzzy system possesses an infinite gain margin that is its stability is guaranteed no matter how large the feedback gain becomes two examples are given to illustrate the proposed optimal fuzzy controller design approach and to demonstrate the proved stability properties

optimal fuzzy controller design in continuous fuzzy system global concept approach we propose a design method for a global optimal fuzzy controller to control and stabilize a continuous fuzzy system with free or fixed end point under finite or infinite horizon time a linear like global system representation of continuous fuzzy system is first proposed by viewing a continuous fuzzy system in global concept and unifying the individual matrices into synthetical matrices based on this the optimal control law which can achieve global minimum effect is developed theoretically the nonlinear segmental two point boundary value problem is derived for the finite horizon problem and a forward riccati like differential equation for the infinite horizon problem the stability of the closed loop fuzzy system can be ensured by the designed optimal fuzzy controller the optimal closed loop fuzzy system cannot only be guaranteed to be exponentially stable but also be stabilized to any desired degree also the total energy of system output is absolutely finite moreover the resultant closed loop fuzzy system possesses an infinite gain margin

on the stability and stabilization of quaternion equilibria of rigid bodies we study attitude control of rigid bodies on quaternion coordinates under three mathematically different perspectives depending on how the system dynamics are assumed to evolve in the first case we suppose that one equilibrium point is chosen a priori and a continuous controller is used under the assumption that the rigid body always spins in the same direction in the second case we relax the assumption that the sense of rotation is constant finally a third scenario is considered in which hybrid switching control is used to choose the direction in which to spin that is both equilibria are continuously considered with regard to less energy consumption it is showed that each of three scenarios must be treated in a different theoretical setting a comparative study in simulations is also provided

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

global set stabilisation of the spacecraft attitude using finite time control technique in this article we revisit the classical problem of attitude stabilisation for a rigid spacecraft with external disturbances a global set stabilisation method using finite time control technique is proposed in the absence of disturbances the states of the closed loop system will be stabilised in finite time to a set consisting of two equilibria in the presence of disturbances the states will be stabilised to a neighbourhood of this set by constructing a particular lyapunov function it is proved that the closed loop system satisfies global set stability the control method in this article is based on set control idea and thus is more natural and energy efficient numerical simulation results show the effectiveness of the method

stability and chaotic behavior of a pid controlled inverted pendulum subjected to harmonic base excitations by using the normal form theory abstract in this paper we investigate the stability and the onset of chaotic oscillations around the pointing up position for a simple inverted pendulum that is driven by a control torque and is harmonically excited in the vertical and horizontal directions the driven control torque is defined as a proportional plus integral plus derivative pid control of the deviation angle with respect to the pointing down equilibrium position the parameters of the pid controller are tuned by using the routh criterion to obtain a stable weak focus around the pointing up position whose stability is investigated by using the normal form theory the normal form theory is also used to deduce a simplified mathematical model that can be resolved analytically and compared with the numerical simulation of the complete mathematical model from the harmonic prescribed motions for the pendulum base necessary conditions for chaotic motion are deduced by means of the melnikov function when the pendulum is close to the unstable pointing up position the pid parameters are changed and the chaotic motion is destroyed which is achieved by employing very small control signals even in the presence of random noise the results of the analytical calculations are verified by full numerical simulations

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

a quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability we present in this paper a novel nonlinear model predictive control scheme that guarantees asymptotic closed loop stability the scheme can be applied to both stable and unstable systems with input constraints the objective functional to be minimized consists of an integral square error ise part over a finite time horizon plus a quadratic terminal cost the terminal state penalty matrix of the terminal cost term has to be chosen as the solution of an appropriate lyapunov equation furthermore the setup includes a terminal inequality constraint that forces the states at the end of the finite prediction horizon to lie within a prescribed terminal region if the jacobian linearization of the nonlinear system to be controlled is stabilizable we prove that feasibility of the open loop optimal control problem at time t 0 implies asymptotic stability of the closed loop system the size of the region of attraction is only restricted by the requirement for feasibility of the optimization problem due to the input and terminal inequality constraints and is thus maximal in some sense

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

stabilization of the inverted pendulum around its homoclinic orbit the inverted pendulum has been used as a benchmark for motivating the study of nonlinear control techniques we propose a simple controller for balancing the inverted pendulum and raise it to its upper equilibrium position while the cart displacement is brought to zero the control strategy is based on an energy approach of the cart and pendulum system

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

a nonlinear small gain theorem for the analysis of control systems with saturation a nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation the theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear feedforward form this result in turn is applied to the control of 1 linear systems stable and unstable with inputs subject to magnitude and rate saturation and time delays 2 the cascade of globally asymptotically stable nonlinear systems with certain linear systems those that are stabilizable right invertible and such that all of their invariant zeros have nonpositive real part 3 the inverted pendulum on a cart and 4 the planar vertical takeoff and landing aircraft

brief stabilization of invariant sets for nonlinear non affine systems the problem of the global and local stabilization of invariant sets for general nonlinear controlled systems is considered new state feedback stabilizing controllers and sufficient conditions of asymptotic stability of a goal set with the specified region of attraction are proposed the proofs of the obtained results are based on the detailed analysis of the w limit sets of the closed loop system with the speed gradient or jurdjevic quinn controllers for affinized controlled system two illustrating examples are given

asymptotic smooth stabilization of the inverted 3 d pendulum the 3 d pendulum consists of a rigid body supported at a fixed pivot with three rotational degrees of freedom it is acted on by gravity and it is fully actuated by control forces the 3 d pendulum has two disjoint equilibrium manifolds namely a hanging equilibrium manifold and an inverted equilibrium manifold the contribution of this paper is that two fundamental stabilization problems for the inverted 3 d pendulum are posed and solved the first problem asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and attitude of the 3 d pendulum the second problem asymptotic stabilization of the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and a reduced attitude vector of the 3 d pendulum these control problems for the 3 d pendulum exemplify attitude stabilization problems on the configuration manifold so 3 in the presence of potential forces lyapunov analysis and nonlinear geometric methods are used to assess global closed loop properties yielding a characterization of the almost global domain of attraction for each case

a new splitting to solve a large hermitian eigenproblem abstract a particular case of rohe s convergent splitting is presented which gives a rapidly converging method of estimating the lowest eigenvalues and corresponding vectors of a large hermitian matrix when a leading principal minor of that matrix provides a good approximation to the desired eigensolution this is particularly relevant for example to the total energy calculations of solids

case studies on the application of the stable manifold approach for nonlinear optimal control design this paper presents application results of a recently developed method for approximately solving the hamilton jacobi equation in nonlinear control theory the method is based on stable manifold theory and consists of a successive approximation algorithm which is suitable for computer calculations numerical approach for this algorithm is advantageous in that the computational complexity does not increase with respect to the accuracy of approximation and non analytic nonlinearities such as saturation can be handled first the stable manifold approach for approximately solving the hamilton jacobi equation is reviewed from the computational viewpoint and next the detailed applications are reported for the problems such as swing up and stabilization of a 2 dimensional inverted pendulum simulation stabilization of systems with input saturation simulation and a sub optimal servo system design for magnetic levitation system experiment

time optimal swing up of the planar pendulum this paper presents qualitative results on the global structure of the time optimal trajectories of the planar pendulum on a cart this mechanical system is a benchmark to test nonlinear control methods and various papers addressed the problem of computing time optimal open loop controls relying on the theory of optimal synthesis we provide a discontinuous feedback giving optimal solutions for any initial data the approach is that of geometric control theory

output feedback control for stabilization on se 3 this paper addresses the problem of stabilizing systems that evolve on se 3 the proposed solution consists of an output feedback controller that guarantees almost global asymptotic stability of the desired equilibrium point in the sense that the equilibrium point is stable and we have convergence for all initial conditions except for those in a nowhere dense set of measure zero the output vector is formed by the position coordinates of a collection of landmarks fixed in the environment the resulting closed loop system exhibits the following properties i the position and orientation subsystems are decoupled ii the position error is globally exponentially stable and iii the orientation error is almost globally exponentially stable results are also provided that allow one to select landmark configurations so as to control how the position and orientation of the rigid body converge to their final equilibrium values

how to implement the spectral transformation the general linear eigenvalue equations h x m z 0 where h and m are real symmetric matrices with m positive semidefinite must be transformed if the lanczos algorithm is to be used to compute eigenpairs x z when the matrices are large and sparse but not diagonal some factorization must be performed as part of the transformation step if we are interested in only a few eigenvalues x near a specified shift then the spectral transformation of ericsson and ruhe 1 proved itself much superior to traditional methods of reduction the purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it it arises in the use of inverse iteration a second goal is to introduce a systematic modification of the computed ritz vectors which improves the accuracy when m is ill conditioned or singular we confine our attention to the simple lanczos algorithm although the first two sections apply directly to the block algorithms as well 1 overview this contribution is an addendum to the paper by ericsson and ruhe 1 and also 7 the value of the spectral transformation is reiterated in a later section here we outline our implementation of this transformation the equation to be solved for an eigenvalue a and eigenvector z is 1 h xm z 0 h and m are real symmetric n x n matrices and m is positive semidefinite a practical instance of 1 occurs in dynamic analysis of structures where h and m are the stiffness and mass matrices respectively we assume that a linear combination of h and m is positive definite it then follows that all eigenvalues x are real in addition one has a real scalar a distinct from any eigenvalue and we seek a few eigenvalues a close to a together with their eigenvectors z ericsson and ruhe replace 1 by a standard eigenvalue equation 2 c h am ct v y 0 where c is the choleski factor of m m ctc and y cz if m is singular then so is c but fortunately the eigenvector z can be recovered from y via z h am icty of course there is no intention to invert h a m explicitly the received may 14 1984 revised december 20 1985 1980 mathematics subject classification primary 65f15 this research was supported in part by the afosr contract f49620 84 c 0090 the third author was also supported in part by the swedish natural science research council the paper was written while this author was visiting the center for pure and applied mathematics university of california berkeley california 94720 1987 american mathematical society 0025 5718 87 1 00 25 per page

analysis and design for a class of complex control systems part i fuzzy modelling and identification this paper is the first of two dealing with the analysis and design of a class of complex control systems this part deals with fuzzy modelling and identification the second paper is concerned with fuzzy controller design we believe that these two papers show a way to analyze and design complex control systems using a combination of fuzzy logic and modern control theory the complex systems studied in these two papers can be represented by a fuzzy aggregation of a set of local linear models the identification can be divided into two procedures the first procedure is the identification of the membership functions including the determination of the number of fuzzy rules and the estimation of the parameters in the membership functions the second is the identification of the local linear models including the determination of the structure and the parameters of the local rule maps and the global rule interpolation two examples are used to demonstrate the application of the modelling method

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

nonlinear control of a swinging pendulum we propose a nonlinear controller to regulate the swinging energy of the pendulum for a cart and pendulum system roughly speaking the controller is designed to regulate an output the swing energy while providing internal stability regulating the cart position it is difficult to apply many of the standard nonlinear control design techniques since the output zeroing manifold does not contain any equilibrium points and the relative degree of the system is not constant in contrast to controllers that use a command generator and possibly a time varying feedback our control law is a simple autonomous nonlinear controller we analyze the stability of the closed loop system using an lv small gain approach on a transverse linearization of the system about the desired periodic orbit one can easily extend this approach to analyze the robustness of the control system with respect to disturbances and parameter variations experimental results demonstrate the effectiveness of the nonlinear controller

global stabilization of spherical orientation by synergistic hybrid feedback with application to reduced attitude tracking for rigid bodies in this paper we propose a hybrid feedback based on a synergistic potential function that achieves global asymptotic stabilization of a desired orientation on the n sphere with a nominal robustness to measurement disturbances a task that is not possible by classical feedback be it smooth nonsmooth periodic or any combination thereof due to the topological structure of the sphere we extend this basic result to a tracking controller for the reduced attitude or pointing direction of a rigid body and provide a method to remove jumps in the controlled torque by backstepping the proposed hybrid feedback is compared with a similar smooth feedback in simulation where it is illustrated that the hybrid feedback overcomes performance limitations inherent to the smooth feedback we provide two examples of a synergistic potential function one defined on a general sphere and the other on the unit circle

controller synthesis of fuzzy dynamic systems based on piecewise lyapunov functions this paper presents a kind of controller synthesis method for fuzzy dynamic systems based on a piecewise smooth lyapunov function the basic idea of the proposed approach is to construct controllers for the fuzzy dynamic systems in such a way that a piecewise continuous lyapunov function can be used to establish the global stability with h sub spl infin performance of the resulting closed loop fuzzy control systems it is shown that the control laws can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software an example is given to illustrate the application of the proposed methods

an explicit formula for lanczos polynomials abstract the lanczos algorithm for tridiagonalizing a given matrix a generates a sequence of approximating matrices a p that can naturally be obtained as restrictions to subspaces the eigenvalues of these approximating matrices are well known to be good approximations to the extreme eigenvalues of a in this paper we produce explicit formulas for the characteristics polynomials of the a p in terms of the eigenvalues of a these formulas can be used to explain heuristically why these approximations are often quite good at present we have no concrete analytic argument that explains the quality of the approximation the main result draws on the formal relationship between the lanczos algorithm and pade approximations to the moment generating function of a this result is one of the few analytic results available for the unsymmetric lanczos algorithm