a general class of sliding surface for sliding mode control in the conventional sliding mode control the relative degree of the chosen sliding surface is usually one this paper addresses a general class of sliding surface whose relative degree is no longer restricted to one we show that when the relative degree is more than two it is inevitable that there will exist a limit cycle for the nonlinear sign function and for the case of relative degree two the system possesses poor phase margin however these disadvantages can be remedied remarkably by smoothing out the control nonlinearity in a suitable boundary layer with a saturation function that ensures asymptotic stability of the overall system

design of variable structure controllers with continuous switching control a modified variable structure controller is proposed to deal with the conirol chattering problem which is one of the major difficulties for implementing the variable structure control vsc the new control strategy is effective for a class of uncertain nonlinear dynamical systems based on a suitably designed sliding sector a continuous switching controller whose control structure is switched whereas the control signal at switching remains continuous is constructed the complete attractiveness and the stability properties of the sliding sector in the sense of lyapunov stability ensure the retention of benefits achieved with the conventional vsc some typical examples are also investigated which show the improvement of system performance

brief paper output feedback variable structure control design with an h2 performance bound constraint in this paper we give an lmi based method for designing an output feedback variable structure control vsc system with an h 2 constraint the uncertain system model under consideration may have nonmatching disturbance input signals nonmatching disturbance input signal reduction is performed via h 2 norm minimization in terms of a constrained riccati inequality we give a sufficient condition for the existence of a linear sliding surface and a compensator guaranteeing an h 2 constraint and we show that the control parameters can be characterized by using the solution to the given inequality condition we also give a simple lmi based design algorithm of an output feedback vsc law guaranteeing an h 2 constraint finally by using a design example we show that our method can supplement the existing results

brief paper output feedback variable structure control for linear systems with uncertainties and disturbances this paper proposes a dynamic output feedback variable structure controller for linear mimo systems with mismatched and matched norm bounded uncertainties and matched nonlinear disturbances the proposed controller consists of nonlinear and linear parts similar to the unit vector type of controller the nonlinear part of the new controller takes care only of matched uncertainties and disturbances and on the other hand the linear part with full dynamics completely handles mismatched uncertainties designing such a linear part leads to use a lyapunov function associated with the full states which achieves the global stability against the mismatched uncertainties the resulting criteria are furthermore converted into solvable ones by using the so called cone complementary linearization algorithm for bi convex problems

stability radius and internal versus external stability in banach spaces an evolution semigroup approach in this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems these include autonomous and nonautonomous systems modeled with unbounded state space operators acting on banach spaces this approach allows one to apply the classical theory of strongly continuous semigroups to time varying systems in particular the complex stability radius may be expressed explicitly in terms of the generator of an evolution semigroup examples are given to show that classical formulas for the stability radius of an autonomous hilbert space system fail in more general settings upper and lower bounds on the stability radius are proven for banach space systems in addition it is shown that the theory of evolution semigroups allows for a straightforward operator theoretic analysis of internal stability as determined by classical frequency domain and input output operators even for nonautonomous banach space systems in particular for the nonautonomous setting internal stability is shown to be equivalent to input output stability for stabilizable and detectable systems for the autonomous setting an explicit formula for the norm of input output operator is given

analyzing displacement term s memory effect in a nonlinear boundary value problem to prove chaotic vibration of the wave equation abstract consider a nonlinear boundary value problem for the one dimensional wave equation with the left end boundary condition containing an extra linear displacement feedback term which induces a memory effect technical difficulty arises as to how to determine chaotic vibrations of the system in this paper we build new riemman invariants one of which contains time delay term and utilize method of characteristics to rigorously prove the onset of chaos in the sense of exponential growth of total variations of the bounded gradient

output feedback control based on a high order sliding manifold approach in this note we address the problem of output feedback control of multiple input multiple output plants the proposed approach is based on a high order sliding manifold strategy the resulting controller exhibits strong robustness properties similar to high gain control laws but avoids peaking phenomena thanks to the adoption of a time varying sliding surface moreover the dynamic control law is continuous and differentiable thus avoiding chattering problems

a new algorithm for the design of multifunctional observers this paper presents a general algorithm for low order multifunctional observer design with arbitrary eigenvalues the feature of this algorithm is that it can generate a functional observer with different orders which are no larger but usually much less than m v 1 where m is the number of functionals and v is the observability index of a c since the order needed for the observer varies with the functionals besides other system parameters this design approach should be practical the resulting observer system matrix is in its jordan form the key step of this algorithm is the generation of the basis for the transformation matrix which relates the system and observer states the computation of this algorithm is quite reliable it is based on the block observable lower hessenberg form of a c and all its initial and major computation involves only the orthogonal operations

sliding mode output feedback controller design using linear matrix inequalities considers a sliding mode controller which requires only output information and is static in nature the novelty of the approach is in the rationale and method used to synthesize the linear control component which involves a linear matrix inequality optimization the so called reachability condition used in sliding mode controller analysis is not required to be satisfied globally instead sliding is only expected to take place within a subset of the state space referred to as the sliding patch

analysis and design of variable structure systems using a geometric approach multivariable variable structure systems in the sliding mode are studies using a geometric approach the properties of system order reduction and disturbance rejection are proved using projector theory new design methods for the choice of switching hyperplanes are derived for the closed loop eigenvalue eigenvector assignment problem

reconstructing sliding surfaces for siso variable structure output feedback control systems using inverse method there exists a dynamic mapping relationship between the sliding variable and the output variable in a linear time invariant lti single input single output siso system this relationship is invariant of the matched disturbances which lies in the control space in this note a continuous time dynamic inverse compensator is proposed to reconstruct the sliding variable from the system output this inverse compensator is stable and causal provided that the original siso system is minimum phase and that the input output relative degree is less than or equal to one in case the system relative degree is greater than one a digital approximation for the inverse compensator is introduced to track the sliding variable in discrete time to the accuracy of o t where t is the sampling period thus the discrete time variable structure control vsc law can be implemented without full state accessibility

the operator equationax xb c admissibility and asymptotic behavior of differential equations it is shown that there exists a natural relationship between the regular admissibility of a translation invariant subspace m ofbuc r e the space of uniformly continuous bounded functions onrwith values in a banach spacee w r t the differential equationu t au t f t and the unique solvability of a special operator equation of lyapunov s typeax xdm 0 where dmis the restriction of the operator d d dtonto m and 0 f f 0 this leads to some spectral conditions for the regular admissibility applications to questions of exponential dichotomy exponential stability the existence of periodic and almost periodic solutions of equation as well as analogous questions for some nonlinear equations and functional differential equations are presented

numerical methods to design the reaching phase of output feedback variable structure control abstract two numerical methods are developed to design a variable structure control that satisfies the reaching condition using static output feedback the design is formulated as a nonsmooth convex optimization problem for which existing algorithms are available it is shown how the resulting control law can be modified to be robust in the presence of parameter uncertainty or a disturbance numerical examples successfully demonstrate the developed techniques

characterization of exponential stability of a semigroup of operators in terms of its action by convolution on vector valued function spaces overr abstract it is proved that a c 0 semigroup t t t t 0 of linear operators on a banach space x is uniformly exponentially stable if and only if it acts boundedly on one of the spaces l p r x or c 0 r x by convolution as an application it is shown that t is uniformly exponentially stable if and only if formula where ap r x is the space of x valued almost periodic functions on r

evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations abstract we investigate spectral criteria for the existence of almost periodic solutions to linear 1 periodic evolution equations of the formdx dt a t x f t with in general unbounded a t and almost periodicf using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set e isp f then the above equation has an almost periodic mild solutionxfwhich is unique if one requiressp xf 2 k k z sp f we emphasize that our method allows us to treat the equations without assumption on the existence of floquet representation this improves recent results on the subject in addition we discuss some particular cases in which the spectrum of monodromy operator does not intersect the unit circle and apply the obtained results to study the asymptotic behavior of solutions finally an application to parabolic equations is considered

chaotic oscillations of a linear hyperbolic pde with a general nonlinear boundary condition abstract this article establishes a theorem that guarantees the occurrence of chaotic oscillations in a system governed by a linear hyperbolic partial differential equation pde with a nonlinear boundary condition nbc compared with the nbcs in all previous related literature such an nbc is more general both the left end and the right end of the system parameter interval for the occurrence of chaotic oscillations are precisely characterized the chaotic results obtained are further applied to two specific nbcs and the telegraph equation finally numerical examples verify the effectiveness of theoretical prediction

stable robust tracking by sliding mode control sliding mode control is examined from the perspective of obtaining stable and robust tracking of an arbitrary time varying reference by a multi input output linear time invariant system driven by a certain class of bounded errors nonlinearities and disturbances most existing schemes for such systems are subsumed by the one presented here the results are developed via the use of inverse models and make clear the constraints imposed by the finite and infinite zero structure of the system in particular stable and robust tracking is shown to be obtained by the scheme in this paper if and only if the system is minimum phase

discrete time output feedback sliding mode control design for uncertain systems using linear matrix inequalities an output feedback based sliding mode control design methodology for discrete time systems is considered in this article in previous work it has been shown that by identifying a minimal set of current and past outputs an augmented system can be obtained which permits the design of a sliding surface based upon output information only if the invariant zeros of this augmented system are stable in this work a procedure for realising discrete time controllers via a particular set of extended outputs is presented for non square systems with uncertainties this method is applicable when unstable invariant zeros are present in the original system the conditions for existence of a sliding manifold guaranteeing a stable sliding motion are given a procedure to obtain a lyapunov matrix which simultaneously satisfies both a riccati inequality and a structural constraint is used to formulate the corresponding control to solve the reachability problem a numerical method using linear matrix inequalities is suggested to obtain the lyapunov matrix finally the design approach given in this article is applied to an aircraft problem and the use of the method as a reconfigurable control strategy in the presence of sensor failure is demonstrated

output feedback stabilization of uncertain fuzzy systems using variable structure system approach based on the variable structure system theory an output feedback fuzzy control design method is developed for a class of uncertain nonlinear systems that can be represented by a t s takagi sugeno fuzzy model the uncertain fuzzy systems under consideration may have mismatched uncertainties in the state matrix as the local controller a compensator based sliding mode output feedback controller with a nonlinear discontinuous term is used in terms of linear matrix inequalities lmis a sufficient condition is derived for the existence of the fuzzy controller guaranteeing a stable sliding motion it is shown that the compensator gain and the switching surface parameter matrix can be characterized in terms of the solution of the lmi existence condition finally an lmi based design algorithm and numerical design examples are given

nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition abstract this paper studies the nonisotropic chaotic oscillations of the initial boundary value problem of one dimensional wave equation with a mixing transport term it separately considers that the boundary condition at the right end of the wave equation is a superlinear type and linear perturbation of such type each causing the total energy of the underlying system to rise and fall due to the interaction with a mixing transport term for each type of boundary condition the occurrence of nonisotropic chaotic oscillations is rigorously proved numerical examples verify the effectiveness of theoretical prediction

dynamic output feedback variable structure control for system stabilization simplex type dynamic output feedback variable structure control for stabilizing multivariable linear time invariant plants is analysed two types of controllers are considered a compensator type and an observer type it is shown that while the compensator type controller stabilizes the plant only if the plant is minimum phase the observer type controller does not face the same restriction the design of low order less than the order of the plant observer type controllers is also considered

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

output feedback sliding mode control for sampled data systems we address the output feedback sliding mode control problem for a sampled data linear system with external disturbances of the matching type by taking into account the disturbance compensation a deadbeat high gain output feedback control strategy with additional dynamics is able to attenuate the disturbances in the framework of singular perturbation analysis it is shown that the closed loop system exhibits good robustness against exogenous disturbances

snapback repellers as a cause of chaotic vibration of the wave equation with a van der pol boundary condition and energy injection at the middle of the span a wave equation on a one dimensional interval i has a van der pol type nonlinear boundary condition at the right end at the left end the boundary condition is fixed at exactly the midpoint of the interval i energy is injected into the system through a pair of transmission conditions in the feedback form of anti damping we wish to study chaotic wave propagation in the system a cause of chaos by snapback repellers has been identified these snapback repellers are repelling fixed points possessing homoclinic orbits of the non invertible map in 2d corresponding to wave reflections and transmissions at respectively the boundary and the middle of the span points existing literature f r marotto j math anal appl 63 199 223 1978 on snapback repellers contains an error we clarify the error and give a refined theorem that snapback repellers imply chaos numerical simulations of chaotic vibration are also illustrated

on variable structure output feedback controllers for uncertain dynamic systems a class of variable structure output feedback controllers for uncertain dynamic systems with bounded uncertainties is proposed no statistical information about the uncertain elements is assumed a variable structure systems approach and a geometric approach to the analysis and synthesis of system zeros are employed in the synthesis of the proposed controllers

evolutionary semigroups and lyapunov theorems in banach spaces we present a spectral mapping theorem for continuous semigroups of operators on any banach space e the condition for the hyperbolicity of a semigroup on e is given in terms of the generator of an evolutionary semigroup acting in the space of e valued functions the evolutionary semigroup generated by the propagator of a nonautonomous differential equation in e is also studied a discrete technique for investigating the evolutionary semigroup is developed and applied in describing the hyperbolicity exponential dichotomy of the nonautonomous equation

output feedback stabilization of uncertain plants a variable structure systems approach the problem of output stabilization for uncertain s1so systems is considered using structural transformations uncertain systems can change to the form convenient for output feedback design synthesis of observer based variable structure control for asymptotical stabilization or uniform ultimate boundedness of the closed loop system is provided examples are considered and simulation results are given

a new characterization of nonisotropic chaotic vibrations of the one dimensional linear wave equation with a van der pol boundary condition the one dimensional linear wave equation with a van der pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations trans amer math soc 350 1998 4265 4311 internat j bifur chaos 8 1998 423 445 internat j bifur chaos 8 1998 447 470 j math phys 39 1998 6459 6489 internat j bifur chaos 12 2002 535 559 in this paper we characterize nonisotropic chaotic vibration by means of the total variation theory we obtain the classification results on the growth of the total variation of the snapshots on the spatial interval in the long time horizon with respect to two parameters entering different regimes in r2

brief on the limitations of some variable structure output feedback controller designs numerical methods to design control laws based on output information only which ensure a sliding mode is attained have recently been proposed in the literature this paper shows that the proposed methods are only applicable to certain classes of linear systems it is further shown that even if the system has the required structure then the numerical methods will only terminate satisfactorily for a specific class of switching surface although the class of systems for which the schemes are applicable cannot be identified explicitly a straightforward test can be applied which parameterises all possible appropriate hyperplanes the theoretical results are illustrated by numerical design studies taken from the relevant literature

low order state estimators and compensators for dynamical systems with unknown inputs abstract we propose a new type of state estimator for a class of dynamical systems with unknown inputs we synthesize the combined low order estimator controller compensator and show the stability of the closed loop system the results are illustrated with a numerical example

an lmi approach for designing sliding mode observers this paper presents a method to design sliding mode observers for a class of uncertain systems using linear matrix inequalities the objective is to exploit the degrees of freedom available in the design which have hitherto been ignored because of the lack of a tractable solution framework the relationship between the linear component of the sliding mode observer and a particular sub optimal observer arising from classical linear quadratic gaussian lqg theory is demonstrated this helps motivate how the design weighting matrices inherent in the method may be chosen in practice it will also be shown how the weighting matrices affect the dynamics of the sliding motion furthermore using pole placement and pole clustering methods the poles of the sliding motion can be forced to lie in a specified region in the complex plane so that the performance of the sliding motion can be tuned this paper will also present a more general solution where the eigenvalues of the linear part of the observer are forced

weak lp stability of a linear semigroup on a hilbert space implies exponential stability george weiss department of theoretical mathematics the weizmann institute rehovot 76100 israel received september 24 1987 we prove that a strongly continuous semigroup of linear operators on a hilbert space is weakly p stable for some pe cl co if and only if the semigroup is exponentially stable as an application we prove that the cauchy problems associated with the semigroup are well posed on the infinite time interval co 0 if and only if the semigroup is exponentially stable

growth rates of total variations of snapshots of 1d linear wave equations with nonlinear right end boundary conditions abstract this paper is concerned with the one dimensional wave equation on a unit interval where the left end boundary condition bc is linear while the right end bc is general nonlinearity we give the dynamical behavior of the riemann invariants u v of the wave equation in terms of the total variations and show that chaotic vibration occurs when enters a certain parameter range

on variable structure output feedback controllers in a recent work by zak hui 1993 a sliding mode controller for mimo systems using static output feedback was proposed very nice geometric conditions for how to design sliding surfaces were given however there are two restrictive assumptions in it one is that the uncertainties in the system must be bounded by a known function of outputs which excludes some possible uncertainties in the a matrix if the system is described by the triple a b c the other one requires a matrix equality to be held which may also be very difficult to satisfy in many systems in this paper we propose a modification of the sliding mode controller for a class of single input single output siso systems which can eliminate the above mentioned limitations and under certain conditions guarantee global closed loop stability hence the range of applicability of zak hui s method can be greatly broadened

comments and further results regarding on variable structure output feedback controllers this paper points out some misleading statements of the above mentioned paper kwan 1996 and shows that the proposed controller can be easily derived under less restrictive conditions from a previously proposed output feedback variable structure control scheme

chaotic vibrations of the one dimensional wave equation due to a self excitation boundary condition part i controlled hysteresis the study of nonlinear vibrations oscillations in mechanical and electronic systems has always been an important research area while important progress in the development of mathematical chaos theory has been made for finite dimensional second order nonlinear odes arising from nonlinear springs and electronic circuits the state of understanding of chaotic vibrations for analogous infinite dimensional systems is still very incomplete the 1 dimensional vibrating string satisfying wtt wxx 0 on the unit interval x 0 1 is an infinite dimensional harmonic oscillator consider the boundary conditions at the left end x 0 the string is fixed while at the right end x 1 a nonlinear boundary condition wx wt w3 t 0 takes effect this nonlinear boundary condition behaves like a van der pol oscillator causing the total energy to rise and fall within certain bounds regularly or irregularly we formulate the problem into an equivalent first order hyperbolic system and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary condition since the solution of the first order hyperbolic system depends completely on this nonlinear relation and its iterates the problem is reduced to a discrete iteration problem of the type un 1 f un where f is the nonlinear reflection relation we say that the pde system is chaotic if the mapping f is chaotic as an interval map algebraic asymptotic and numerical techniques are developed to tackle the cubic nonlinearities we then define a rotation number following j p keener 11 and obtain denseness of orbits and periodic points by either directly constructing a shift sequence or by applying results of m i malkin 17 to determine the chaotic regime of for the nonlinear reflection relation f thereby rigorously proving chaos nonchaotic cases for other values of are also classified such cases correspond to limit cycles in nonlinear second order odes numerical simulations of chaotic and nonchaotic vibrations are illustrated by computer graphics received by the editors july 20 1995 and in revised form october 16 1996 1991 mathematics subject classification primary 35l05 35l70 58f39 70l05 the first and third authors work was supported in part by nsf grant dms 9404380 texas arp grant 010366 046 and texas a m university interdisciplinary research initiative iri 96 39 work completed while the first author was on sabbatical leave at the institute of applied mathematics national tsing hua university hsinchu 30043 taiwan r o c the second author s work was supported in part by grant nsc 83 0208 m 007 003 from the national council of science of the republic of china c 1998 american mathematical society

a transformed lure problem for sliding mode control and chattering reduction the sliding mode control problem is investigated with the frequency domain approach the authors show that the sliding mode control can be transformed into a lure problem with this formulation it is shown that the conventional sliding mode control with a nonlinear sign function can only guarantee to make the overall system stable but not asymptotically stable the authors also consider a practical situation where the sign nonlinearity may contain a hysteresis loop for this situation they show that even if the hysteresis loop is very small there still exists a limit cycle however chattering phenomenon caused by the limit cycle behavior can be eliminated by a certain class of nonlinearities

dynamic output feedback for observed variable stucture control systems abstract a global exponential observer is built explicitly for some classes of nonlinear variable structure observed control systems by using piecewise continuous output feedback control laws or the continuous equivalent control for the observer the unavailable state vector is asymptotically kept on the sliding manifold with exponential decaying error some explicit examples are presented

existence of chaotic oscillations in second order linear hyperbolic pdes with implicit boundary conditions abstract this paper establishes rigorously mathematical theorems that guarantee the existence of chaotic oscillations in the systems of second order linear hyperbolic pdes it separately considers the systems with nonlinear explicit boundary conditions ebcs and nonlinear implicit boundary conditions ibcs as well as those with such ibcs subjected to small perturbations where ibcs include ebcs as special cases but the latter cannot in general be expressed by the former numerical examples are demonstrated to illustrate the effectiveness of theoretical results

lmi based nonlinear fuzzy observer controller design for uncertain mimo nonlinear systems based on the variable structure system theory we develop a nonlinear fuzzy observer controller design method for a class of uncertain nonlinear systems that can be represented by a takagi sugeno fuzzy model we use a sliding mode observer and a sliding mode controller as the local state observer and the local controller in terms of linear matrix inequalities lmis we derive sufficient conditions for the existence of the fuzzy observer and the fuzzy controller we also derive lmi conditions for the existence of the fuzzy observer and the fuzzy controller guaranteeing the alpha stability or generalized h2 hinfin performance constraints we show that the observer and the controller can be independently designed i e the separation property holds finally we give simple lmi based design algorithms together with an example

output feedback stabilization using variable structure control we consider the stabilization of a class of multivariable nonlinear systems about an equilibrium point at the origin using variable structure output feedback control in particular the system can be transformed into a normal form with no zero dynamics a robust high grain observer is used to estimate the state variables while rejecting the effect of disturbance a globally bounded discontinuous variable structure controller is designed to compensate for modelling error we show that the controller can stabilize the closed loop system and does not suffer from the peaking phenomenon that exists in previous designs

further results on variable output feedback controllers zak and hui 1993 proposed a sliding mode controller for linear multiple input multiple output mimo systems using static output feedback the author s previous paper 1996 provides an improvement of the output feedback controller of zak hui for a class of linear single input single output siso systems that eliminated two important limitations 1 system uncertainties must be bounded by the system output and 2 a requirement of a matrix inequality the controller developed previously can guarantee global closed loop stability this paper extends the previous results to linear mimo systems it is emphasized that the proposed mimo controller yields global closed loop stability whereas the one in zak hui can only guarantee local stability an application of the proposed mimo controller to an aircraft model is included to show the effectiveness of the method

nonisotropic spatiotemporal chaotic vibrations of the one dimensional wave equation with a mixing transport term and general nonlinear boundary condition nonisotropic spatiotemporal chaotic vibrations can happen for a linear wave equation with a mixing transport term on the unit interval and with a nonlinear van der pol boundary condition at one end when the parameter s enters some regime chen et al int j bifurcation chaos appl sci eng 12 535 2002 in this paper on the one hand the equation with general nonlinear boundary conditions is considered on the other hand parameter range for the route to chaos is more precisely classified the results obtained here generalize and sharpen those both in the above paper and the earlier work of li l l and huang y j math anal appl 361 69 2010 two examples and their corresponding numerical simulations of chaotic space time profiles are also illustrated

an ilmi approach to robust static output feedback sliding mode control this paper proposes a systematic methodology for designing robust static output feedback sliding mode control sofsmc for a class of systems with mismatched uncertainties this methodology consists of two parts one is a new sufficient and necessary condition for the existence problem in terms of two matrix inequalities the other is a direct solving method using the iterative linear matrix inequality ilmi technique the main advantages of this method are that only original system parameters are involved without any extra coordinate changes and that appealing to solving the static output feedback stabilization sofs problem is no longer required in addition the ilmi algorithm includes an optimal part to avoid high control efforts a numerical example also demonstrates the efficacy of the proposed approach