H∞ observer-based control for discrete-time one-sided Lipschitz systems with unknown inputs

This paper investigates the problem of robust observer-based stabilization for a class of one-sided nonlinear discrete-time systems subjected to unknown inputs. We propose a simple simultaneous state and input estimator. A nonlinear controller is then proposed to compensate for the effects of unknown inputs and to ensure asymptotic stability in a closed loop. Several mathematical artifacts are used to deduce stability conditions expressed in terms of linear matrix inequalities. To show high performances of the proposed technique, a relevant example is provided with comparisons to recent results.

H∞ observer-based control for discrete-time one-sided Lipschitz systems with unknown inputs

This paper investigates the problem of robust observer-based stabilization for a class of one-sided nonlinear discrete-time systems subjected to unknown inputs. We propose a simple simultaneous state and input estimator. A nonlinear controller is then proposed to compensate the effects of unknown inputs and to ensure asymptotic stability in closed loop. Several mathematical artifacts are used to deduce stability conditions expressed in terms of LMIs. To show high performances of the proposed technique, a relevant example is provided with comparisons to recent results.

Robust observer design for a class of one-sided Lipschitz systems subject to time delays and disturbances in both plants and outputs

In this paper, we consider the problem of designing state observers for a class of one-sided Lipschitz time-delay systems subject to disturbances. Not only the plants but also the outputs of the systems are subject to time delays and disturbances. The one-sided Lipschitz and quadraticallyinner-bounded conditions, which are more general than the traditional Lipschitz condition, are utilized to cover a broader class of nonlinearities. Based on the Lyapunov-Krasovskii approach, a robust observer design is proposed where synthesis condition is derived in terms of linear matrixinequality (LMI). A numerical example is given to show the effectiveness of the obtained results.

Unknown input observer design for one-sided Lipschitz discrete-time systems subject to time-delay

In this paper, we address the problem of unknown input observer design, which simultaneously estimates state and unknown input, of a class of nonlinear discrete-time systems with time-delay. A novel approach to the state estimation problem of nonlinear systems where the nonlinearities satisfy the one-sided Lipschitz and quadratically inner-bounded conditions is proposed. This approach also allows us to reconstruct the unknown inputs of the systems. The nonlinear system is first transformed to a new system which can be decomposed into unknown-input-free and unknown-input-dependent subsystems. The estimation problem is then reduced to designing observer for the unknown-input-free subsystem. Rather than full-order observer design, in this paper, we propose observer design of reduced-order which is more practical and cost effective. By utilizing several mathematical techniques, the time-delay issue as well as the bilinear terms, which often emerge when designing observers for nonlinear discrete-time systems, are handled and less conservative observer synthesis conditions are derived in the linear matrix inequalities form. Two numerical examples are given to show the efficiency and high performance of our results.

Non-linear observer design for a class of singular time-delay systems with Lipschitz non-linearities

In this paper, we consider a class of time-delay singular systems with Lipschitz non-linearities. A method of designing full-order observers for the systems is presented which can handle non-linearities with large-Lipschitz constants. The Lipschitz conditions are reformulated into linear parameter varying systems, then the Lyapunov–Krasovskii approach and the convexity principle are applied to study stability of the new systems. Furthermore, the observers design does not require the assumption of regularity for singular systems. In case the systems are non-singular, a reduced-order observers design is proposed instead. In both cases, synthesis conditions for the observers designs are derived in terms of linear matrix inequalities which can be solved efficiently by numerical methods. The efficiency of the obtained results is illustrated by two numerical examples.

Observer design for one-sided Lipschitz discrete-time systems subject to delays and unknown inputs

In this paper, we address the problem of observer design for a class of nonlinear discrete-time systems in the presence of delays and unknown inputs. The nonlinearities studied in this work satisfy the one-sided Lipschitz and quadratically inner-bounded conditions which are more general than the traditional Lipschitz conditions. Both H∞ observer design and asymptotic observer design with reduced-order are considered. The designs are novel compared to other relevant nonlinear observer designs subject to time delays and disturbances in the literature. In order to deal with the time-delay issue as well as the bilinear terms which usually appear in the problem of designing observers for discrete-time systems, several mathematical techniques are utilized to deduce observer synthesis conditions in the linear matrix inequalities form. A numerical example is given to demonstrate the effectiveness and high performance of our results.

Approximate Controllability Of Semilinear Systems Using Integral Contractors

In this article, a non-autonomous (time-varying) semilinear system is considered and its approximate controllability is investigated. The notion of 'bounded integral contractor', introduced by Altman, has been exploited to obtain sufficient conditions for approximate controllability. This condition is weaker than Lipschitz condition. The main theorems of Naito [11, 12] are obtained as corollaries of our main results. An example is also given to show how our results weaken the conditions assumed by Sukavanam[17].

Arc exchange systems and renormalization

We exhibit the construction of stable arc exchange systems from the stable laminations of hyperbolic diffeomorphisms. We prove a one-to-one correspondence between (i) Lipshitz conjugacy classes of C(1+H) stable arc exchange systems that are C(1+H) fixed points of renormalization and (ii) Lipshitz conjugacy classes of C(1+H) diffeomorphisms f with hyperbolic basic sets Lambda that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Lambda. Let HD(s)(Lambda) and HD(u)(Lambda) be, respectively, the Hausdorff dimension of the stable and unstable leaves intersected with the hyperbolic basic set L. If HD(u)(Lambda) = 1, then the Lipschitz conjugacy is, in fact, a C(1+H) conjugacy in (i) and (ii). We prove that if the stable arc exchange system is a C(1+HDs+alpha) fixed point of renormalization with bounded geometry, then the stable arc exchange system is smooth conjugate to an affine stable arc exchange system.

State and input simultaneous estimation for a class of nonlinear systems

This paper addresses the problem of estimating simultaneously the state and input of a class of nonlinear systems. Here, the systems nonlinear part comprises a Lipschitz nonlinear function with respect to the state and input, and a state-dependent unknown function including additive disturbance as well as uncertain/nonlinear/time-varying terms. Upon satisfying some conditions, the observer design problem can be solved via a Riccati inequality or a LMI-based technique with asymptotic estimation guaranteed. A numerical example is included for illustration.

A new design of sliding mode control systems

A new sliding mode control technique for a class of SISO dynamic systems is presented in this chapter. It is seen that the stability status of the closed-loop system is first checked, based on the approximation of the most recent information of the first-order derivative of the Lyapunov function of the closed-loop system, an intelligent sliding mode controller can then be designed with the following intelligent features: (i) If the closed-loop system is stable, the correction term in the controller will continuously adjust control signal to drive the closed-loop trajectory to reach the sliding mode surface in a finite time and the desired closed-loop dynamics with the zero-error convergence can then be achieved on the sliding mode surface. (ii) If, however, the closed-loop system is unstable, the correction term is capable of modifying the control signal to continuously reduce the value of the derivative of the Lyapunov function from the positive to the negative and then drives the closed-loop trajectory to reach the sliding mode surface and ensures that the desired closed-loop dynamics can be obtained on the sliding mode surface. The main advantages of this new sliding mode control technique over the conventional one are that no chattering occurs in the sliding mode control system because of the recursive learning control structure; the system uncertainties are embedded in the Lipschitz-like condition and thus, no priori information on the upper and/or the lower bounds of the unknown system parameters and uncertain system dynamics is required for the controller design; the zero-error convergence can be achieved after the closed-loop dynamics reaches the sliding mode surface and remains on it. The performance for controlling a third-order linear system is evaluated in the simulation section to show the effectiveness and efficiency of the new sliding mode control technique.

Sliding mode-like learning control for SISO complex systems with T-S fuzzy models

In this paper, a sliding mode-like learning control scheme is developed for a class of single input single output (SISO) complex systems. First, the Takagi-Sugeno (T-S) fuzzy modelling technique is employed to model the uncertain complex dynamical systems. Second, a sliding mode-like learning control is designed to drive the sliding variable to converge to the sliding surface, and the system states can then asymptotically converge to zero on the sliding surface. The advantages of this scheme are that: 1) the information about the uncertain system dynamics and the system model structure is not required for the design of the learning controller; 2) the closed-loop system behaves with a strong robustness with respect to uncertainties; 3) the control input is chattering-free. The sufficient conditions for the sliding mode-like learning control to stabilise the global fuzzy model are discussed in detail. A simulation example for the control of an inverted pendulum cart is presented to demonstrate the effectiveness of the proposed control scheme.

A robust learning control for SISO Nonlinear systems with T-S Fuzzy Model : C02-robust control

In this paper, a robust learning control is developed for a class of single input single output (SISO) nonlinear systems with T-S fuzzy model. It is seen that the proposed sliding mode learning control with the powerful Lipshitz-like condition can guarantee the stability, convergence and robustness of the closed-loop system without involving any assumptions on uncertain system dynamics. In addition, theconcept that the local system with the maximum membership function dominates the system dynamic behaviours helps to greatly simplify the control system design. It will be further seen that the continuous learning control ensures the advantage of chattering-free that may occur in conventional sliding mode systems. Simulation examples are presented to demonstrate the effectiveness of the proposed learning control through the comparison with the H-infinity control.