Robust Stabilization and ℒ2-Gain Sensitivity Specification for Nonlinear Systems with Stable Zero Dynamics and Matched Uncertainties

This paper deals with output feedback stabilisation of a class of nonlinear systems with nonlinearity of the unmeasured state variables. The nonlinearity of the unmeasured state complicates the zero dynamics of the nonlinear system and makes the control design more difficult. In this paper, the zero dynamics analysed for the class of nonlinear systems with respect to the original output and then with respect to a virtual output. It is demonstrated that nonlinear systems with unstable zero dynamics may have stable zero dynamics with the virtual output. Based on the analysis, conditions are then identified for nonlinear systems which can be stabilised by control design with estimated state, including the estimated virtual output, and this removes a restriction of a recent result that requires the transformed system has no zero dynamics. Examples are included to demonstrate that the proposed control design can be used to stabilise nonlinear systems with unstable nonlinear zero dynamics.

Dynamics of the aircraft configuration considered in this paper shows a unique characteristic that mere are no stable attractors in the entire high angle of attack flight envelope. As a result, once the aircraft has departed from die normal flight regime, no standard technique can be applied to recover me aircraft. In mis paper, using feedback linearisation technique, a nonlinear controller is designed at high angles of attack which is engaged after the aircraft departs from normal flight regime. This controller stabilises the aircraft into a stable spin. Then a set of synthetic pilot inputs are applied to cause an automatic transition from the spin equilibrium to low angles of attack where the second controller is connected. This controller is a normal gain scheduled controller designed to have a large domain of attraction at low angles of attack. It traps me aircraft into a low angle of attack level flight This entire concept of recovery has been verified using six degree of freedom nonlinear simulation. Feedback linearisation technique used to design a controller ensures internal stability only if the nonlinear plant has stable zero dynamics. Since zero dynamics depend upon the selection of outputs, a new method of choosing outputs is described to obtain a plant which has stable zero dynamics.

Global almost disturbance decoupling with stability for non minimum-phase single-input single-output nonlinear systems

Actuator failure compensation for nonlinear systems which can be transformed into parametric strict-feedback form with zero dynamics is investigated. Direct adaptive compensation control schemes are developed to ensure asymptotic output tracking and stabilize the system despite the uncertainties in actuator failures as well as in system parameters. Two main cases are studied for adaptive actuator failure compensation: systems with stable zero dynamics, and systems with extra controls for stabilization. Design conditions on systems suitable for actuator failure compensation are clarified.

We study the problem of output feedback stabilization for stochastic nonlinear systems. We consider a class of stochastic nonlinear systems in observer canonical form with stable zero-dynamics. We introduce a sequence of state transformations that transform the system into a lower triangular structure that is amenable for integrator backstepping design. Then we design the output feedback controller and prove that the closed-loop system is bounded in probability. Under an infinite-horizon risk-neutral cost criterion, the controller designed can guarantee an arbitrarily small long term average cost. Furthermore, when the disturbance vector field vanishes at the origin, the closed-loop system is asymptotically stable in the large, and the risk-neutral cost is guaranteed to be zero. With special care, the controller preserves the equilibrium of the nonlinear system.

This article addresses the robust stability for a class of nonlinear uncertain discrete-time systems with convex polytopic of uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic-type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discrete-time control systems is given. Numerical examples are included to illustrate the effectiveness of our results. MSC:15A09, 52A10, 74M05, 93D05.

This article presents a new approach to robust quadratic stabilization of nonlinear stochastic systems. The linear rate vector of a stochastic system is perturbed by a nonlinear function, and this nonlinear function satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of stochastic systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable. The new formulation provides a suitable setting for robust stabilization of nonlinear stochastic systems where the underlying deterministic systems satisfy the generalized matching conditions. Our sufficient conditions are written in matrix forms, which are determined by solving linear matrix inequalities (LMIs), which have significant computational advantage over any other existing techniques. Examples are given to demonstrate the results.

One of the approaches to sampled-data controller design for nonlinear continuous-time systems consists of obtaining an appropriate model and then proceeding to design a controller for the model. Then, it is important to derive a good approximate sampled-data model because the exact sampled- data model for nonlinear systems is often unavailable to the controller designers. Recently, Yuz and Goodwin have proposed an accurate sampled-data model which includes extra zero dynamics, so-called the sampling zero dynamics, corresponding to the relative degree of the continuous-time nonlinear system. This paper shows that a more accurate sampled-data model is required for a controlled Van der Pol system with the relative degree two. The reason is that the closed-loop system becomes unstable when a controller design method based on cancellation of the zero dynamics is applied, and the phenomenon seems related to the instability of the sampling zero dynamics of the more accurate sampled-data model. Further, this paper derives a more accurate model than that of Yuz and Goodwin for continuous-time nonlinear systems with the relative degree two, and presents a condition which assures the stability of the sampling zero dynamics of the obtained model.

Tracking of reference signals $y_{\mathrm{ref}}(\cdot)$ by the output $y(\cdot)$ of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error $e=y-y_{\mathrm{ref}}$ and its derivative $\dot{e}$ within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller $u(t)=-k_0(t)^2e(t)-k_1(t)\dot{e}(t)$, where the simple proportional error feedback has gain functions $k_0$ and $k_1$ designed in such a way to preclude contact of $e$ and $\dot{e}$ with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control.

Pulse-width modulation (PWM) is a widely used strategy for operating switch controlled systems. A model for PWM systems is reported which provides the discrete-time response of the one-cycle-average value of any signal in the system. The model typically has unstable zero dynamics, as well as a nonlinear dependence on the duty ratio input. Using scheduled output redefinition, an approximate design model is derived which has stable zero dynamics. A controller based on discrete-time input-output linearization is then developed using this approximate design model. Although this approach leads to approximate rather than perfect output tracking, it guarantees internal stability.

Interval Criterion of Robust Stability for Nonlinear System and Its Application to Flight Control

Due to the existence of dynamical uncertainties, it is important to pay attention to the robustness of nonlinear control systems, especially when designing adaptive critic control strategies. In this paper, based on the neural network learning component, the robust stabilization scheme of nonlinear systems with general uncertainties is developed. Through system transformation and employing adaptive critic technique, the approximate optimal controller of the nominal plant can be applied to accomplish robust stabilization for the original uncertain dynamics. The neural network weight vector is very convenient to initialize by virtue of the improved critic learning formulation. Under the action of the approximate optimal control law, the stability issues for the closed-loop form of nominal and uncertain plants are analyzed, respectively. Simulation illustrations via a typical nonlinear system and a practical power system are included to verify the control performance.

Stability of Zero Dynamics of Sampled-Data Nonlinear Systems

In this paper, a second order sliding mode control law for regulation of rotational speed in a non-minimum phase drill string system is proposed. In this regard, first, the nonlinear dynamic model of a drill string which is in contact with non-Newtonian mud is conducted. Next, a super twisting sliding mode control is designed that ensures the asymptotic regulation of output during sliding motion in a finite time and stable zero dynamics. It is shown that the proposed super twisting controller is capable of adjusting the output on a desired value by generating a proper sliding surface in a finite time. The convergence of the proposed controller for the dynamic model with a higher number of mode shapes is also investigated, and it is found that by increasing the number of obtained modes, the desired output needs less computation time with more accuracy.

In this paper, a class of nonlinear large-scale systems with similar subsystems is studied. Both matched and unmatched uncertainties are considered by utilizing their bounding functions, and the interconnections take a more general form than considered previously. Based on a constrained Lyapunov equation, a nonlinear dynamic output feedback decentralized controller is presented. Unlike existing results, matched uncertainties are considered in the control design; by using a decomposition of the interconnections, the known and uncertain interconnections are treated separately; thus, the robustness is improved and conservativeness is reduced significantly. The computation effort for solving the Lyapunov equation is greatly reduced by taking into account the similar subsystem structure. Finally, simulation is used to illustrate the effectiveness of our results.