Semi-global Practical Stability Research Articles

Uniform semiglobal practical asymptotic stability for non-autonomous cascaded systems and applications

It is due to the modularity they provide that results for cascaded systems have proved their utility in numerous control applications as well as in the development of general control techniques based on “adding integrators”. Nevertheless, the standing assumptions in most of the present literature on cascaded systems is that, when decoupled, the subsystems constituting the cascade are uniformly globally asymptotically stable (UGAS). Hence existing results fail in the more general case when the subsystems are uniformly semiglobally practically asymptotically stable (USPAS). This situation is often encountered in control practice, e.g. in control of physical systems with external perturbations, measurement noise, unmodelled dynamics, etc. After giving a rigorous framework for the analysis of such stability properties, this paper generalizes previous results for cascades by establishing that, under a uniform boundedness condition on its solutions, the cascade of two USPAS systems remains USPAS. An analogous result is derived for uniformly semiglobally asymptotically stable (USAS) systems in cascade. Furthermore, we show the utility of our results in the PID control of mechanical systems affected by unknown non-dissipative forces and considering the dynamics of the DC motors.

A Lyapunov Proof of an Improved Maximum Allowable Transfer Interval for Networked Control Systems

Simple Lyapunov proofs are given for an improved (relative to previous results that have appeared in the literature) bound on the maximum allowable transfer interval to guarantee global asymptotic or exponential stability in networked control systems and also for semiglobal practical asymptotic stability with respect to the length of the maximum allowable transfer interval

ON AUTOMATIC SEEKING OF OPTIMAL STEADY-STATES IN BIOCHEMICAL PROCESSES

It is discussed how the automatic seeking of optimal steady states in biochemical reactors can be achieved by using non-model based extremum-seeking control with semi-global practical stability and convergence properties. A special attention is paid to processes with multiple steady-states and multivalued cost functions.

Robustness of PID-controlled Manipulators vis-à-vis Actuator Dynamics and External Disturbances

We present a robustness analysis for PID-controlled robot manipulators. For robot manipulators under the influence of external disturbances and a wide class of neglected dynamics, we provide establish uniform semiglobal practical asymptotic stability and propose a tuning procedure of the PID gains in order to achieve any given precision from any given bounded set of initial conditions. In contrasts to other works on robust stability of PIDs, we do not use La Salle's principle but provide a strict Lyapunov function. The perturbations that we consider include discontinuous functions of the state, such as coulomb friction. As corollaries of our main results, one may conclude the same stability property for the case of motion control using linear PID.

ADAPTIVE CONTROL REDESIGN FOR NONLINEAR SAMPLED-DATA SYSTEMS

Semi-global practical stabilization of a general class of nonlinear sampled-data feedback control systems with unknown parameter is addressed. Due to the appearance of powers series of the unknown parameter in the analytic development of the exact discretized systems, an overparameterization is realized for the adaptive law. Then, sufficient conditions for the design of a high order controller are given, based on the Lyapunov analysis, that guarantees the desired stability properties.

Necessary and sufficient conditions for uniform semiglobal practical asymptotic stability: Application to cascaded systems

It is well established that, for a cascade of two uniformly globally asymptotically stable (UGAS) systems, the origin remains UGAS provided that the solutions of the cascade are uniformly globally bounded. While this result has met considerable popularity in specific applications it remains restrictive since, in practice, it is often the case that the decoupled subsystems are only uniformly semiglobally practically asymptotically stable (USPAS). Recently, we established that the cascade of USPAS systems is USPAS under a local boundedness assumption and the hypothesis that one knows a Lyapunov function for the driven subsystem. The contribution of this paper is twofold: (1) we present a converse theorem for USPAS and (2) we establish USPAS of cascaded systems without the requirement of a Lyapunov function. Compared to other converse theorems in the literature, ours has the advantage of guaranteeing a specific relationship between the upper and lower bounds on the generated Lyapunov function V and of providing a time-invariant bound on the gradient of V , which is fundamental to establish theorems for cascades.

On non-local stability properties of extremum seeking control

In this paper, we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted. This non-local stability result is proved by showing semi-global practical stability of the closed-loop system with respect to the design parameters. We show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction. Our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking. Simulation examples illustrate our results.

Output Feedback Stabilization of Uncertain Nonminimum-Phase Nonlinear Systems

An output feedback design technique is presented. The design technique achieves semiglobal practical stabilization for a class of nonminimum-phase nonlinear systems, subject to parameter uncertainties. This work is an extension of the result by Isidori in 2000. It provides a constructive controller design method for an auxiliary system, whose existence is crucial but is only assumed in Isidori's work. The control design technique is is used to regulate the benchmark van de Vusse reactor. Simulation results demonstrate satisfactory controller performance.

ON NON-LOCAL STABILITY PROPERTIES OF EXTREMUM SEEKING CONTROL

We revisit the extremum seeking scheme whose local stability properties were analyzed in (Krstić and Wang, 2000) and propose its simplified version that still achieves extremum seeking. We show under slightly stronger conditions that this simplified scheme achieves extremum seeking from arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted. This non-local convergence result is proved by showing semi-global practical stability of the closed-loop system with respect to the design parameters. Moreover, we show at the same time that reducing the parameters typically slows down the convergence of the extremum seeking controller. Hence, the control designer faces a tradeoff between the size of the domain of attraction and the speed of convergence when tuning the extremum seeking controller. We present a simulation example to illustrate our results.

Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback

It is shown, for general nonlinear systems, that asymptotic controllability and observability are sufficient for semiglobal practical asymptotic stabilization by output feedback. Indeed, as previously shown in the literature, asymptotic controllability implies the existence of a (discontinuous in general) state feedback that, when implemented by sample and hold, is semiglobally practically stabilizing and robust to measurement disturbance; moreover, a weak form of observability allows reconstruction of the state with arbitrary precision in an arbitrarily short amount of time. So, we can build an output feedback that operates periodically in two modes: an initial, small fraction of a sampling period is used to estimate the state, and the remainder of the sampling period is used to implement the state feedback control using the state estimate. Our stabilization results are presented not only for compact target sets (e.g., the origin) but also for noncompact target sets.

Output Feedback Stabilization with Nonlinear Predictive Control: Asymptotic properties

State space based nonlinear model predictive control (NMPC) needs the state for the prediction of the system behaviour. Unfortunately, for most applications, not all states are directly measurable. To recover the unmeasured states, typically a stable state observer is used. However, this implies that the stability of the closed-loop should be examined carefully, since no general nonlinear separation principle exists. Recently semi-global practical stability results for output feedback NMPC using a high-gain observer for state estimation have been established. One drawback of this result is that (in general) the observer gain must be increased, if the desired set the state should converge to is made smaller. We show that under slightly stronger assumptions, not only practical stability, but also convergence of the system states and observer error to the origin for a sufficiently large but bounded observer gain can be achieved.

A robust velocity field control

This paper is devoted to velocity field control (VFC) of uncertain robotic manipulators. We propose a proportional-integral (PI)-type controller derived from modeling error compensation ideas and singular perturbation theory, that requires a minimum knowledge of the plant (i.e., constant estimate of the inertia matrix). It is shown that semiglobal practical stabilization is achieved; that is, given any compact set of initial velocity field errors, there exist PI control gains which guarantee that the robot tracks a desired velocity field with arbitrary accuracy. The proposed controller was experimentally evaluated on a two degrees-of-freedom arm.

A Trajectory-Based Approach for the Stability Robustness of Nonlinear Systems with Inputs

We show, for two different definitions of semiglobal practical external stability, that the stability property holds on semi-infinite time intervals if and only if it holds on arbitrarily long but finite time intervals. These results have immediate applications in the analysis of the stability properties of highly oscillatory systems with inputs using averaging or for systems with inputs that are slowly varying. Results are stated for general flows and the stability is given with respect to arbitrary (not necessarily compact) sets.

OUTPUT FEEDBACK NONLINEAR PREDICTIVE CONTROL -A SEPARATION PRINCIPLE APPROACH

Over the recent years many advances in the area of nonlinear model predictive control have been made. However, with respect to the output feedback problem and predictive control only a limited number of results are available. Most of the existing approaches do only guarantee local stability. In this note we propose the use of a high gain observer in combination with a sampled nonlinear predictive controller for a special MIMO system class. The resulting output feedback leads to semiglobal practical stability and recovery of performance of the state feedback controller. The results are valid for a wide class of NMPC schemes. Thus they can be considered as a special separation principle for NMPC.

Input-to-State Stability for Nonlinear Time-Varying Systems via Averaging

We introduce two definitions of an averaged system for a time-varying ordinary differential equation with exogenous disturbances (“strong average” and “weak average”). The class of systems for which the strong average exists is shown to be strictly smaller than the class of systems for which the weak average exists. It is shown that input-to-state stability (ISS) of the strong average of a system implies uniform semi-global practical ISS of the actual system. This result generalizes the result of [TPA] which states that global asymptotic stability of the averaged system implies uniform semi-global practical stability of the actual system. On the other hand, we illustrate by an example that ISS of the weak average of a system does not necessarily imply uniform semi-global practical ISS of the actual system. However, ISS of the weak average of a system does imply a weaker semi-global practical “ISS-like” property for the actual system when the disturbances w are absolutely continuous and \(w, {\dot w}\in;{\scr L}_\infty\). ISS of the weak average of a system is shown to be useful in a stability analysis of time-varying cascaded systems.

Further Results on the Nonlinear Separation Principle: The General 'Asymptotically Controllable' Case

It is shown, for general nonlinear systems, that asymptotic controllability and observability are sufficient for semiglobal practical asymptotic stabilization by output feedback. Indeed, as previously shown in the literature, asymptotic controllability implies the existence of a (discontinuous in general) state feedback that, when implemented by sample and hold, is semiglobally practically stabilizing and robust to measurement disturbance; moreover, a weak form of observability allows reconstruction of the state with arbitrary precision in an arbitrarily short amount of time. So, we can build an output feedback that operates periodically in two modes: an initial, small fraction of a sampling period is used to estimate the state, and the remainder of the sampling period is used to implement the state feedback control using the state estimate. Our stabilization results are presented not only for a compact target set (e.g. the origin) but also for general closed and nonempty target sets.

Semi-Global Practical Stabilization and Disturbance Adaptation for an Underactuated Ship

We consider the problem of stabilizing the position and orientation of a ship to constant desired values, when the ship has only two independent controls and also the ship is subject to an environmental force of unknown magnitude. We propose a time-varying feedback control law and a disturbance adaptation law, and show that this provides semi-global practical asymptotic stability. The control and adaptation laws are derived using a combined integrator backstepping and averaging approach. Simulation results are presented.

A tool for semi-global stabilization of uncertain non-minimum-phase nonlinear systems via output feedback

This paper presents a simple design method by means of which it is possible to robustly stabilize, using output feedback, a significant class of uncertain nonlinear systems whose zero dynamics are unstable. The proposed procedure leads to the construction of a dynamic controller yielding robust, semi-global practical stability.

A note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback

It is known that if a system can be (robustly) globally asymptotically stabilized by means of a feedback that is driven by functions that are uniformly completely observable (UCO), then this system can be practically semiglobally stabilized by means of (possibly dynamic) output feedback. This papers discusses a significant structural hypothesis under which the existence of a dynamic feedback driven by UCO functions is guaranteed. The class of systems which satisfy this hypothesis includes any stabilizable and detectable linear system and any relative degree one nonlinear system which is stabilizable by dynamic output feedback. In particular, the hypothesis does not require the system to be minimum phase.

Semi-global practical asymptotic stability and averaging

We prove a generalized Liapunov theorem which guarantees practical asymptotic stability. Based on this theorem, we show that if the averaged system x ̇ =f av (x) corresponding to x ̇ =f(x,t) is globally asymptotically stable then, starting from an arbitrarily large set of initial conditions, the trajectories of x ̇ =f(x,t/ε) converge uniformly to an arbitrarily small residual set around the origin when ε>0 is taken sufficiently small. In other words, the origin is semi-globally practically asymptotically stable.