generalized controller canonical form for linear and nonlinear dynamics a generalized controller canonical form for linear and nonlinear dynamics is proposed it is obtained using the theorem of the primitive element from differential algebra the derivation of the controller form does not apply to multivariable constant linear systems with the transformations all nonlinear dynamics can be exactly linearized by dynamic feedback the main departure from standard theory is that transformations may depend on input derivatives once differential algebraic tools are introduced the proofs of the results are easy

constrained control of siso bilinear systems relative degree and nonminimum phase difficulties limit the applicability of input output feedback linearization hence the need for approximations earlier work on predictive control of bilinear systems overcame these problems by means of interpolation between feedback linearization and state feedback the former providing optimality and the latter guaranteeing feasibility and stability through the use of invariant feasible polytopes the current work also makes use of polytopes in preference to ellipsoids but achieves distinctly different objectives first it is shown that feedback linearization can be used over particular polytopes without needing to resort to either approximation or interpolation then it is shown that invariant polytopes based on bilinear controllers can be much larger these two approaches are combined in an algorithm that guarantees stability over much larger initial condition sets and gives much improved closed loop performance

hunting for the fractal dimension of the kauffman model abstract the kauffman model is a random mixture of all possible cellular automata on the square lattice with nearest neighbor interactions simulations give a fractal dimension d above 1 8 compatible with the usual scaling law d d for damage spreading

input output feedback linearization for nonminimum phase nonlinear systems through periodic use of synthetic outputs abstract input output feedback linearization provides a convenient means of extending linear control strategies such as output zeroing or pole placement to the case of nonlinear affine in the input systems but such extensions cannot be applied in the presence of nonminimum phase characteristics this paper overcomes this difficulty through the periodic use of a finite number of synthetic outputs which are so constructed as to define embedded dynamics with stable zero dynamics the efficacy of the method is demonstrated by means of a numerical example

initiation of damage in the kauffman model an alternate method is presented for initiating damage to the kauffman model a numerical study suggests that this modification increases the efficiency of generating damage clusters without changing their fractal properties

nonlinear control via uniform system approximation abstract we introduce the notion of uniform system approximation and use it to solve the following problem given a single input nonlinear system f g x f x g x u that is linearly controllable on a manifold e of equilibrium points find vector fields f g such that f g and f g agree to first order on e and f g is input to state linearizable on a neighborhood of e the solution proceeds by constructing a linearizable uniform system approximation and yields as a by product a feedback control law that can be used to guarantee stable operation of the given system in a neighborhood of e this includes for example smooth transitions between operating points that are not necessarily close to each other a simple example is given to illustrate the approach

j inner outer factorization j spectral factorization and robust control for nonlinear systems the problem of expressing a given nonlinear state space system as the cascade connection of a lossless system and a stable minimum phase system inner outer factorization is solved for the case of a stable system having state space equations affine in the inputs the solution is given in terms of the stabilizing solution of a certain hamilton jacobi equation the stable minimum phase factor is obtained as the solution of an associated nonlinear spectral factorization problem as an application one can arrive at the solution of the nonlinear h sub spl infin control problem for the disturbance feedforward case

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

inner outer factorization for nonlinear noninvertible systems this paper considers inner outer factorization of asymptotically stable nonlinear state space systems in continuous time that are noninvertible our approach will be via a nonlinear analogue of spectral factorization which concentrates on first finding the outer factor instead of the inner factor an application of the main result to control of nonminimum phase nonlinear systems is indicated

constrained mpc using feedback linearization on siso bilinear systems with unstable inverse dynamics earlier work used the concept of partial invariance in order to identify regions in state space where feedback linearization can be used despite the presence of unstable internal dynamics the use of such regions as terminal regions in mpc has obvious advantages this paper recognizes that feedback linearization applied to siso bilinear systems steers the state to the kernel of the output map correspondingly by restricting attention to this kernel the paper develops results which allow for the significant enlargement of the relevant terminal regions expressions are also given for maximal partially invariant sets and this used recursively can lead to significant further enlargement

on flat pseudo euclidean nilpotent lie algebras abstract a flat pseudo euclidean lie algebra is a real lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the lie bracket and such that the left multiplications are skew symmetric we show that the center of a flat pseudo euclidean nilpotent lie algebra of signature 2 n 2 must be degenerate and all flat pseudo euclidean nilpotent lie algebras of signature 2 n 2 can be obtained by using the double extension process from flat lorentzian nilpotent lie algebras we show also that the center of a flat pseudo euclidean 2 step nilpotent lie algebra is degenerate and all these lie algebras are obtained by using a sequence of double extension from an abelian lie algebra in particular we determine all flat pseudo euclidean 2 step nilpotent lie algebras of signature 2 n 2 the paper contains also some examples in low dimension

relaxation of output zeroing for bilinear non minimum phase systems linear control design techniques for output setpoint tracking of regulation problems can be extended through input output feedback linearization iofl to non linear input affine systems but are not applicable to the case of equilibrium points that exhibit non minimum phase characteristics rather than resort to minimum phase approximations this paper instead proposes a relaxation of iofl that does not require iofl to be implemented at every sampling instant the relaxation introduces degrees of freedom which in the case of bilinear systems can be used efficiently to achieve tracking regulation and to maintain this property to within perturbations caused by the relaxation until the desired state is reached a way for expanding the region of attraction is considered and the results of the paper are demonstrated through simulation examples

non linear minimal square spectral factorization recent work has identified a correspondence between minimal square spectral factors of a given spectral density invariant subspaces of an associated hamiltonian matrix solutions of an appropriate algebraic riccati equation and minimal unitary left divisors of a certain unitary function on the imaginary line the purpose of this report is to obtain a natural extension of these ideas to a non linear setting at least for the special case where only minimal square stable spectral factors are considered

shortest prediction horizon non linear model predictive control with guaranteed asymptotic stability this paper presents a continuous time shortest prediction horizon model predictive control method that provides optimal output regulation with guaranteed closed loop asymptotic stability within an assessable domain of attraction the closed loop stability is ensured by requiring plant state variables to satisfy a hard lyapunov inequality constraint whenever the output regulation alone cannot ensure asymptotic closed loop stability the closed loop system evolves while being at the hard constraint once the closed loop system enters a state space region in which the output regulation can ensure asymptotic stability the hard constraint becomes inactive consequently the non linear control method is applicable to stable and unstable plants whether non minimum or minimum phase a major shortcoming of unconstrained shortest prediction horizon model predictive control which is equivalent to input output linearization is its inapplicability to plants operating at a non minimum phase steady state this

boolean dynamics of networks with scale free topology the dynamics of boolean networks with scale free topology are studied the existence of a phase transition from ordered to chaotic dynamics governed by the value of the scale free exponent of the network is shown analytically by analyzing the overlap between two distinct trajectories the phase diagram shows that the phase transition occurs for values of the scale free exponent in the open interval 2 2 5 since the boolean networks under study are directed graphs the scale free topology of the input connections and that of the output connections are studied separately ultimately these two topologies are shown to be equivalent a numerical study of the attractor structure of the configuration space reveals that this structure is similar in both networks with scale free topologies and networks with homogeneous random topologies however an important result of this work is that the fine tuning usually required to achieve stability in the dynamics of networks with homogeneous random topologies is no longer necessary when the network topology is scale free finally based on the results presented in this work it is hypothesized that the scale free topology favors the evolution and adaptation of network functioning from a biological perspective

a normal form approach to approximate input output linearization for maximum phase nonlinear siso systems a control synthesis scheme is presented for nonlinear single input single output systems which have completely unstable antistable zero dynamics the approach is similar in spirit to linear approaches for nonminimum phase systems and involves the derivation of an input output linearizing controller for a suitably defined nonlinear minimum phase approximation to the original system the linearizing controller achieves an approximately linear input output response and internal stability

kauffman boolean model in undirected scale free networks we investigate analytically and numerically the critical line in undirected random boolean networks with arbitrary degree distributions including the scale free topology of connections p k ensuremath sim k ensuremath ensuremath gamma we explain that the unattainability of the critical line in numerical simulations of classical random graphs is due to percolation phenomena we suggest that recent findings of discrepancy between simulations and theory in directed random boolean networks might have the same reason we also show that in infinite scale free networks the transition between frozen and chaotic phases occurs for 3l ensuremath gamma l3 5 since most critical phenomena in scale free networks reveal their nontrivial character for ensuremath gamma l3 the position of the critical line in the kauffman model seems to be an important exception to the rule

relaxation and limit cycles in a global version of the quenched kauffman model abstract a combinatorial method is used to investigate all possible cellular automata defined on a finite set of w global configurations relaxation time is defined as the number of steps before entering a fixed point or a cycle for given w we derive an analytical formula for the distribution of relaxation times t pw t w ww 1 k 0w t 1 wk k the same distribution is proved to be valid for the length of limit cycles the results have some relevance to the quenched kauffman model

approximate state feedback linearization using spline functions abstract this paper presents two numerical methods for synthesizing an approximate feedback linearizing control for single input nonlinear systems both techniques use part of the pseudo linearizing coordinate transformation as an output function t1 the first synthesis uses a well known recursive algorithm to compute the control given t1 splines represent functions that cannot be computed in closed form the second method yields a simplified controller that can be computed regardless of the plant s complexity experimental results from a rotating inverted pendulum show an improved performance of the approximate feedback linearization compared with pseudo linearization while highlighting the practical numerical difficulties that arise in the first method simulations of an acrobot illustrate the second method which is also improved over pseudo linearization

inner outer factorization of nonlinear operators abstract a fundamental result in the theory of hardy spaces of analytic matrix and operator valued functions on the unit disk is inner outer factorization for the case of rational matrix functions explicit formulas for the inner and outer factors have recently been obtained in these formulas the functions are presented in terms of a collection of finite matrices as the transfer function of a linear system the key ingredient in the construction is the solution of a linear matrix equation called a stein equation in this paper we obtain extensions of these results to the setting of input output maps of nonlinear discrete time systems a particaular computable class of examples is presented for illustration

multi input sliding mode control of nonlinear uncertain non affine systems with mono directional actuation in this paper the multi input sliding mode control of nonlinear uncertain non affine systems with mono directional actuators is considered the geometric properties of the regular simplices of vectors are exploited by the mono directional actuation system the proposed full state feedback control method introduces integrators in the input channel and suitably modifies the sliding output to be steered to zero the resulting high frequency gain matrix hfgm which is the matrix multiplying the control vector turns out to be affected by time and state dependent uncertainties an effective use of the integral sliding mode method in connection with a non conventional simplex based switching logic is introduced the proposed simplex control procedure is adequately designed to avoid any possible unbounded increment of the control actions despite the presence of integrators in the input channel and the exploitation of mono directional devices

time evolution of damage in the ising model the time evolution of the damage in the ising model with the heat bath and glauber dynamics has been investigated with the damage spreading method the damage density d t and the average distance r t of damaged sites from the origin have been measured numerically from the simulational data the scaling forms have been established as d t tx r t t1z and d t r t df and the corresponding exponents x z and df have been estimated

reduced order observer in the sliding mode control of nonlinear nonaffine systems the note considers the variable structure control of nonlinear known nonaffine systems when the state vector is not completely available and the use of observers is required the strategy of introducing integrators in the input channel is exploited to enlarge the class of tractable control systems a new reduced order observer is proposed and conditions are found under which it is proven the convergence to the unique ideal solution of both system and observer the control problem is solved by forcing a sliding regime for the observer while satisfying an exponential stability criterion for the observation error state equation

a novel approach to control of nonminimum phase nonlinear systems we present an approach to control of nonminimum phase multiple input multiple output mimo nonlinear systems based on a singular perturbation like technique first we show that a mimo nonlinear system can be converted into a singularly perturbed system through magnitude and time scaling transformation in particular the proposed transformation technique can make the hidden or internal dynamics behave like a fast subsystem however the fast subsystem takes a weakly controllable form which implies that a small positive low gain parameter multiplies the control input in the fast subsystem for this reason the well known integral manifold approach is then utilized to decompose the resulting singularly perturbed system into two subsystems in separate timescales each of which is of a lower dimension and more importantly controllable in the sense that the control input will explicitly appear in the boundary layer system the proposed method is particularly useful to make the input output dynamic characteristics of a decouplable nonminimum phase nonlinear system decoupled and linear

dynamic non minimum phase compensation for siso nonlinear affine in the input systems non minimum phase difficulties prevent the use of input output feedback linearization to avoid this problem earlier work proposed the use of synthetic outputs in order to place zeros in prescribed locations under a condition of static equivalence the objective here is also to introduce a synthetic output but this is defined through the use of perturbations aimed at yielding maximum relative degree resetting of the initial conditions allows design freedom for closing the gap between the behaviour of the synthetic and actual output

evolution by damage spreading in kauffman model in kauffman s boolean automata model on the square lattice the darwinian fitness of survival can be defined as the fraction of elements which do not change from one iteration to the next biological mutations are simulated by filpping one bit in the rule of one site selection of the fitter mutant then optimizes the whole lattice completely this optimization is particularly effective near the critical point of the transition to chaos but is in itself not a critical phenomenon also a two dimensional spin glass can be optimized in this way

a new approach to the zero dynamics assignment problem for nonlinear discrete time systems using functional equations the present research work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear discrete time real analytic system by constructing the requisite synthetic output maps the problem under consideration is motivated by the need to adequately address the control problem of nonminimum phase nonlinear discrete time systems since the latter represent a rather broad class of systems due to the well known effect of sampling on the stability of zero dynamics in the proposed approach the above control objective can be attained through i a systematic computation of synthetic output maps that induce minimum phase behavior while being statically equivalent to the original output maps both vanish on the equilibrium manifold and ii the subsequent integration into the methodological framework of currently available nonminimum phase compensation schemes for nonlinear discrete time systems that rely on output redefinition the mathematical formulation of the zero dynamics assignment problem is realized via a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived the solution to the above system of functional equations can be proven to be locally analytic and this enables the development of a solution method that is easily programmable with the aid of a symbolic software package the synthetic output maps that induce the prescribed zero dynamics for the original nonlinear discrete time system can be explicitly computed on the basis of the solution to the aforementioned system of functional equations

on pseudo euclidean novikov algebras abstract a flat pseudo euclidean lie algebra is a left symmetric algebra endowed with a non degenerate symmetric bilinear form such that left multiplications are skew symmetric we observe that in many classes of flat pseudo euclidean lie algebras the left symmetric product is actually of novikov this leads us to study pseudo euclidean novikov algebras a that is novikov algebra a endowed with a non degenerate symmetric bilinear form such that left multiplications are skew symmetric we show that a lorentzian novikov algebra a must be transitive this implies that the underlying lie algebra a l must be unimodular in geometrical terms the left invariant metric on a corresponding lie group is geodesically complete in this case we show that the restriction of the product to a l a l is trivial using the double extension process we prove that a is a pseudo euclidean novikov algebra such that a l a l is lorentzian if and only if a l splits as a l a l a l a l a l where a l a l and a l a l are abelian a l a l is lorentzian and ad x is skew symmetric for any x a l a l this also implies in this case that a is transitive a l is unimodular and two solvable this result solves the problem for lorentzian novikov algebras such that a l a l is non degenerate if a l a l is degenerate we show that such lorentzian novikov algebra is obtained by the double extension process from a flat euclidean algebra and we give some applications in low dimensions

on the effect of memory in one dimensional k 4 automata on networks the effect of implementing memory in cells of one dimensional ca and on nodes of various types of automata on networks with increasing degrees of random rewiring is studied in this article paying particular attention to the case of four inputs as a rule memory induces a moderation in the rate of changing nodes and in the damage spreading albeit in the latter case memory turns out to be ineffective in the control of the damage as the wiring network moves away from the ordered structure that features proper one dimensional ca this article complements the previous work done in the two dimensional context

singular pdes and the assignment of zero dynamics in nonlinear systems the present research work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear system by constructing the requisite synthetic output maps the minimum phase synthetic output maps constructed can be made statically equivalent to the original output maps and therefore they could be directly used for nonminimum phase compensation purposes specifically the mathematical formulation of the problem is realized via a system of first order nonlinear singular pdes and a rather general set of necessary and sufficient conditions for solvability is derived the solution to the above system of singular pdes can be proven to be locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package the minimum phase synthetic output maps that induce the prescribed zero dynamics for the original nonlinear system can be computed on the basis of the solution of the aforementioned system of singular pdes moreover static equivalence to the original output map can be readily established by a simple algebraic construction

on the nonexistence of closed timelike geodesics in flat lorentz 2 step nilmanifolds the main purpose of this paper is to prove that there are no closed timelike geodesics in a compact or noncompact flat lorentz 2 step nilmanifold n where n is a simply connected 2 step nilpotent lie group with a flat left invariant lorentz metric and a discrete subgroup of n acting on n by left translations for this purpose we shall first show that if n is a 2 step nilpotent lie group endowed with a flat left invariant lorentz metric g then the restriction of g to the center z of n is degenerate we shall then determine all 2 step nilpotent lie groups that can admit a flat left invariant lorentz metric we show that they are trivial central extensions of the three dimensional heisenberg lie group h 3 if n g is one such group we prove that no timelike geodesic in n g can be translated by an element of n by the way we rediscover that the heisenberg lie group h 2k 1 admits a flat left invariant lorentz metric if and only if k 1

damage spreading in non frustrated phase of a triangular antiferromagnet abstract we investigate a first order phase transition from a non magnetic phase to a non frustrated phase of a two dimensional triangular lattice with antiferromagnetic bonds a cellular automaton is defined which chooses a magnetic state of subsequent lattice sites where an interphase boundary arrives as to minimise the local energy the ground state magnetic state is found to be unstable with respect to the point defects of magnetic structure

nonsquare spectral factorization for nonlinear control systems this paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time more specifically we obtain a parametrization of nonsquare spectral factors in terms of invariant lagrangian submanifolds and associated solutions of hamilton jacobi inequalities this inequality is a nonlinear analogue of the bounded real lemma and the control algebraic riccati inequality by way of an application we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system

brief iterative learning control for linear discrete time nonminimum phase systems this paper investigates iterative learning control for linear discrete time nonminimum phase systems first iterative learning control with advanced output data is considered for maximum phase systems next the results are extended to nonminimum phase systems the stability of the inverse mapping from the desired output to the input is proven based on the results for maximum phase systems the input should be updated with the output which is more advanced than the input by the sum of the relative degree of the system and the number of nonminimum phase zeros an example is given to indicate the importance of proper advances of output in the input update law

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

survey approximate linearization via feedback an overview fostered by a growing interest in nonlinear control theory and catalyzed by the discovery in the early 1980s of the exact conditions under which a nonlinear plant can be linearized by static state feedback and coordinate transformation in the last decades there has been a rapid increase of interest in the search for approximate solutions to the problem of linearizing nonlinear systems by state or output feedback main reason for that is the limited applicability of the rigorous methods and the complexity sensitivity and design difficulties of the exact linearizing compensators if any in the present paper the literature on the subject is reviewed and organized in what is believed to be a new and consistent perspective recent works especially in the area of data based techniques are in fact described and related whenever possible to fundamental results previously obtained by model based differential geometric methods this is expected to bring modern system linearization methods closer to the needs of practicing control engineers and to stimulate further research eventually able to fill visible gaps in this direction