Proceedings of the 17th IFAC World Congress, 2008, 2008
A comprehensive approach to the input/output analysis of a class of secondorder sliding mode cont... more A comprehensive approach to the input/output analysis of a class of secondorder sliding mode control (2-SMC) systems is developed. Linear SISO plants with the "generalized suboptimal" 2-SMC algorithm are investigated. It is shown that in order to evaluate the properties of robustness against the presence of external perturbations the generalized suboptimal controller can be replaced by a proper constant gain, which is called "equivalent gain". The approximate DF method, and the theoretically exact LPRS method, are developed for computing the equivalent gain. It results from the presented analysis that the considered algorithm possesses an higher equivalent open-loop gain as compared to the standard relay. A thoroughly discussed worked example confirms the theoretical results and shows the feasibility of the presented procedures.
ABSTRACT The authors consider the equation αx '' (t)=-x ' (t)+F(x(t),t)-s... more ABSTRACT The authors consider the equation αx '' (t)=-x ' (t)+F(x(t),t)-signx(t-h), where F is a smooth function and α,h are positive constants. They study the dynamics of oscillations with emphasis on the existence, frequency and stability of periodic oscillations. The main conclusion of the paper is that, in the autonomous case F(x,t)≡F(x), for |F(x)|<1, there are periodic solutions with different frequencies of oscillations, though only slowly-oscillating solutions are (orbitally) stable. Also, the uniqueness of a periodic slowly-oscillating solution is shown under extra conditions, and a criterion for the existence of bounded oscillations in the case of unbounded function F(x,t) is given.
Proceedings of 35th IEEE Conference on Decision and Control, 2000
The relay control systems with time delay are considered. We show that the time delay does not al... more The relay control systems with time delay are considered. We show that the time delay does not allow to realize an ideal sliding mode, but implies oscillations, whose stability is determined by one discrete parameter-oscillation frequency. Any motion of scalar discontinuous delay system turns into a steady mode-a motion with a constant frequency. Such steady modes have all properties of
This paper highlights the interest of step-by-step higher order sliding mode observers for MIMO n... more This paper highlights the interest of step-by-step higher order sliding mode observers for MIMO nonlinear systems with unknown inputs. A structural matching condition, stating on the possibility to design such observers and to reconstruct the unknown inputs, is derived. A finite time sliding mode observer, based on the hierarchical use of the super twisting algorithm, is developed. Then, it is shown that this observer is of interest in the field of hybrid systems and systems with observability singularities. Lastly, it is shown through an example how to relax the usual matching condition by the means of this type of finite time sliding mode observer.
The singularly perturbed relay c o n trol systems (SPRCS) which h a ve stable periodic motion in ... more The singularly perturbed relay c o n trol systems (SPRCS) which h a ve stable periodic motion in reduced systems are studied. Slow motions integral manifold of such systems consists of parts which correspond to different v alues of control and the solutions contain the jumps from the one part of slow manifold to the other. Three classes of such systems are considered: systems without sliding modes, systems which c o n tain internal sliding modes and time delay systems. The theorems about existence and stability o f t h e s l o w periodic solutions are proved. The algorithm of asymptotic representation for this periodic solutions using boundary layer method is suggested.
This work is devoted to the design of interval observers for a class of Linear-Parameter-Varying ... more This work is devoted to the design of interval observers for a class of Linear-Parameter-Varying (LPV) systems. Applying High Order Sliding Mode (HOSM) techniques it is possible to decrease the initial level of uncertainty in the system, which leads to improvement of set-membership estimates generated by an interval observer and enlarges the class of the estimated systems. Next, this approach is applied to fault detection by verifying the consistency between the output trajectory and its estimated domain. The efficiency of the proposed approach is demonstrated through computer simulations of an electromechanical system.
ASME 2010 Dynamic Systems and Control Conference, Volume 1, 2010
ABSTRACT A planar linearized interception problem of a maneuverable target is considered. A novel... more ABSTRACT A planar linearized interception problem of a maneuverable target is considered. A novel continuous interception strategy, based on the super-twisting second-order sliding mode control, is constructed. The sufficient conditions guaranteeing that this strategy has the maximal capture zone are established. Simulation results demonstrate that the control expenditure of the new strategy is reduced without deteriorating the homing performance in comparison with a bang-bang strategy.
2014 13th International Workshop on Variable Structure Systems (VSS), 2014
This paper deals with the fault detection problem for nonlinear uncertain non-affine systems. The... more This paper deals with the fault detection problem for nonlinear uncertain non-affine systems. The variablestructure output-feedback stabilization based on observer is required. Depending on the type of uncertainties, conditions are given under which convergence to the unique solution of both the system and the observer, and fault detection are achieved. Simulation results illustrate the feasibility of the proposed approach.
Proceedings of the 17th IFAC World Congress, 2008, 2008
A comprehensive approach to the input/output analysis of a class of secondorder sliding mode cont... more A comprehensive approach to the input/output analysis of a class of secondorder sliding mode control (2-SMC) systems is developed. Linear SISO plants with the "generalized suboptimal" 2-SMC algorithm are investigated. It is shown that in order to evaluate the properties of robustness against the presence of external perturbations the generalized suboptimal controller can be replaced by a proper constant gain, which is called "equivalent gain". The approximate DF method, and the theoretically exact LPRS method, are developed for computing the equivalent gain. It results from the presented analysis that the considered algorithm possesses an higher equivalent open-loop gain as compared to the standard relay. A thoroughly discussed worked example confirms the theoretical results and shows the feasibility of the presented procedures.
ABSTRACT The authors consider the equation αx '' (t)=-x ' (t)+F(x(t),t)-s... more ABSTRACT The authors consider the equation αx '' (t)=-x ' (t)+F(x(t),t)-signx(t-h), where F is a smooth function and α,h are positive constants. They study the dynamics of oscillations with emphasis on the existence, frequency and stability of periodic oscillations. The main conclusion of the paper is that, in the autonomous case F(x,t)≡F(x), for |F(x)|<1, there are periodic solutions with different frequencies of oscillations, though only slowly-oscillating solutions are (orbitally) stable. Also, the uniqueness of a periodic slowly-oscillating solution is shown under extra conditions, and a criterion for the existence of bounded oscillations in the case of unbounded function F(x,t) is given.
Proceedings of 35th IEEE Conference on Decision and Control, 2000
The relay control systems with time delay are considered. We show that the time delay does not al... more The relay control systems with time delay are considered. We show that the time delay does not allow to realize an ideal sliding mode, but implies oscillations, whose stability is determined by one discrete parameter-oscillation frequency. Any motion of scalar discontinuous delay system turns into a steady mode-a motion with a constant frequency. Such steady modes have all properties of
This paper highlights the interest of step-by-step higher order sliding mode observers for MIMO n... more This paper highlights the interest of step-by-step higher order sliding mode observers for MIMO nonlinear systems with unknown inputs. A structural matching condition, stating on the possibility to design such observers and to reconstruct the unknown inputs, is derived. A finite time sliding mode observer, based on the hierarchical use of the super twisting algorithm, is developed. Then, it is shown that this observer is of interest in the field of hybrid systems and systems with observability singularities. Lastly, it is shown through an example how to relax the usual matching condition by the means of this type of finite time sliding mode observer.
The singularly perturbed relay c o n trol systems (SPRCS) which h a ve stable periodic motion in ... more The singularly perturbed relay c o n trol systems (SPRCS) which h a ve stable periodic motion in reduced systems are studied. Slow motions integral manifold of such systems consists of parts which correspond to different v alues of control and the solutions contain the jumps from the one part of slow manifold to the other. Three classes of such systems are considered: systems without sliding modes, systems which c o n tain internal sliding modes and time delay systems. The theorems about existence and stability o f t h e s l o w periodic solutions are proved. The algorithm of asymptotic representation for this periodic solutions using boundary layer method is suggested.
This work is devoted to the design of interval observers for a class of Linear-Parameter-Varying ... more This work is devoted to the design of interval observers for a class of Linear-Parameter-Varying (LPV) systems. Applying High Order Sliding Mode (HOSM) techniques it is possible to decrease the initial level of uncertainty in the system, which leads to improvement of set-membership estimates generated by an interval observer and enlarges the class of the estimated systems. Next, this approach is applied to fault detection by verifying the consistency between the output trajectory and its estimated domain. The efficiency of the proposed approach is demonstrated through computer simulations of an electromechanical system.
ASME 2010 Dynamic Systems and Control Conference, Volume 1, 2010
ABSTRACT A planar linearized interception problem of a maneuverable target is considered. A novel... more ABSTRACT A planar linearized interception problem of a maneuverable target is considered. A novel continuous interception strategy, based on the super-twisting second-order sliding mode control, is constructed. The sufficient conditions guaranteeing that this strategy has the maximal capture zone are established. Simulation results demonstrate that the control expenditure of the new strategy is reduced without deteriorating the homing performance in comparison with a bang-bang strategy.
2014 13th International Workshop on Variable Structure Systems (VSS), 2014
This paper deals with the fault detection problem for nonlinear uncertain non-affine systems. The... more This paper deals with the fault detection problem for nonlinear uncertain non-affine systems. The variablestructure output-feedback stabilization based on observer is required. Depending on the type of uncertainties, conditions are given under which convergence to the unique solution of both the system and the observer, and fault detection are achieved. Simulation results illustrate the feasibility of the proposed approach.
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Papers by Leonid Fridman