Input-to-state Stability Gain Research Articles

Robust impulsive observer design for infinite‐dimensional cell population balance models

AbstractThe observer design problem for a class of cell population balance models, describing the time evolution of the cell mass density distribution function and the substrate concentration in a continuous‐stirred tank bioreactor with irregular discrete‐time measurements of the cell mass distribution is considered. The model consists of a partial integro‐differential equation coupled with an ordinary differential equation. Using the theory of impulsive systems sufficient conditions for the input‐to‐state‐stability (ISS) of the observation error in the state‐space with respect to the measurement uncertainty are derived in terms of the maximum time between successive measurements and the ISS gain. In absence of measurement uncertainty the convergence conditions imply exponential stability of the observation error dynamics. Besides these rigorous conditions, application‐oriented tuning guidelines are established. The theoretical results are illustrated with numerical simulations including a comparison with a continuous–discrete extended Kalman filter based on the numerical approximation, showing that a similar accuracy is achieved when using a finite‐dimensional approximation of the proposed impulsive observer scheme with a considerably lower computational effort.

Robust nonlinear observer design based on impulsive dissipativity

The paper considers the design of a nonlinear dissipative impulsive observer based on non-periodic discrete-time measurements. Sufficient conditions are derived for (i) exponential convergence of the observer in absence of measurement uncertainty, and (ii) input-to-state stability (ISS) with respect to measurement uncertainty, by combining notions from impulsive and dissipative systems theory. The conditions mainly include constraints on the minimum and maximum time between measurements depending on system characteristics, the correction gain and the desired ISS gain. The results are illustrated with a numerical case example.

Input-to-state stability of switched systems with explicit gain functions

Input-to-state stability (ISS) for a switched system is a stronger property than stability in characterizing the external influence on system performance. In this paper, we study ISS for a linear switched system with marginally stable subsystems and hence with polynomially unstable subsystems. Moreover, we aim to calculate an explicit ISS gain which is critical in explicit controller design when the linear switched system is incorporated with other dynamic behavior in a complicated large-scale structure.

Event-Triggered Control of Nonlinear Systems with State Quantization

This paper studies event-based control of nonlinear systems with state quantization. Two configurations of the event-based quantized controller are considered: Quantization after sampling and quantization before sampling. The considered quantizer is assumed to have a finite quantization range. With input-to-state stability (ISS) tools, new event-triggering, and dynamic quantization mechanisms are designed to deal with the interaction of the quantizer and the sampler. For both of the configurations, infinitely fast sampling, and in particular the Zeno phenomenon, is avoided. And the system state asymptotically converges to the origin, if a growth condition is satisfied by the ISS gain of the controlled system.

Tradeoffs between quality-of-control and quality-of-service in large-scale nonlinear networked control systems

In this paper we study input-to-state stability (ISS) of large-scale networked control systems (NCSs) in which sensors, controllers and actuators are connected via multiple (local) communication networks which operate asynchronously and independently of each other. We model the large-scale NCS as an interconnection of hybrid subsystems, and establish rather natural conditions which guarantee that all subsystems are ISS, and have an associated ISS Lyapunov function. An ISS Lyapunov function for the overall system is constructed based on the ISS Lyapunov functions of the subsystems and the interconnection gains. The control performance, or “quality-of-control”, of the overall system is then viewed in terms of the convergence rate and ISS gain of the associated ISS Lyapunov function. Additionally, the “quality-of-service” of the communication networks is viewed in terms of the maximum allowable transmission interval (MATI) and the maximum allowable delay (MAD) of the network, and we show that the allowable quality-of-service of the communication networks is constrained by the required ISS gains and convergence rate of the hybrid subsystem corresponding to that network. Our results show that the quality-of-control of the overall system can be improved (or degraded) by improving (or relaxing) the quality-of-service of the communication networks. Alternatively, when relaxing the quality-of-service of one communication network, we can retain the quality-of-control of the overall system by improving the quality-of-service of one or more of the other communication networks. Our general framework will formally show these intuitive and insightful tradeoffs.

A Small-Gain Approach to Robust Event-Triggered Control of Nonlinear Systems

This paper presents a new approach to event-triggered control for nonlinear uncertain systems by using the notion of input-to-state stability (ISS) and the nonlinear small-gain theorem. The contribution of this paper is threefold. First, it is proved that infinitely fast sampling can be avoided if the system is input-to-state stabilizable with the sampling error as the external input and the corresponding ISS gain is locally Lipschitz. No assumption on the existence of known ISS-Lyapunov functions is made in the discussions. Moreover, the forward completeness problem with event-triggered control is studied systematically by using ISS small-gain arguments. Second, the proposed approach gives rise to a new self-triggered sampling strategy for a class of nonlinear systems subject to external disturbances. If an upper bound of the external disturbance is known, then the closed-loop system can be designed to be robust to the external disturbance, and moreover, the system state globally asymptotically converges to the origin if the external disturbance decays to zero. Third, a new design method is developed for event-triggered control of nonlinear uncertain systems in the strict-feedback form. It is particularly shown that the ISS gain with the sampling error as the input can be designed to satisfy the proposed condition for event-triggered control and self-triggered control.

On parameterized dissipation inequalities and receding horizon robust control

SUMMARYThis paper considers the standard input‐to‐state stability (ISS) inequality for discrete‐time nonlinear systems, which involves a candidate Lyapunov function and a supply function that dictates the ISS gain of the system. In order to reduce conservatism, a set of parameters is assigned to both the Lyapunov and the supply function, respectively. A set‐valued map, which generates admissible sets of parameters for each state and input, is defined such that the corresponding parameterized Lyapunov and supply functions enjoy the standard ISS inequality. It is demonstrated that the so‐obtained parameterized ISS inequality offers nonconservative analysis conditions, even when Lyapunov functions with a particular structure, such as quadratic forms, are considered. For bounded inputs, it is then shown how parameterized ISS inequalities can be used to synthesize a closed‐loop system with an optimized envelope of trajectories. An implementation method based on receding horizon optimization is proposed, along with a recursive feasibility and complexity analysis. The effectiveness of the proposed synthesis methodology is illustrated for a benchmark test case for model predictive control, that is, a continuous stirred tank reactor.Copyright © 2012 John Wiley & Sons, Ltd.

On Parameterized Dissipation Inequalities and Receding Horizon Robust Control

This paper considers the standard input-to-state stability (ISS) inequality for discrete-time nonlinear systems, which involves a candidate Lyapunov function (LF) and a supply function that dictates the ISS gain of the system. To reduce conservatism, a set of parameters is assigned to both the LF and the supply function. A set-valued map, which generates admissible sets of parameters for each state and input, is defined such that the corresponding parameterized LF and supply function enjoy the standard ISS inequality. It is demonstrated that the so-obtained parameterized ISS inequality offers non-conservative analysis conditions, even when LFs and supply functions with a particular structure, such as quadratic forms, are considered. For bounded inputs, it is then shown how parameterized ISS inequalities can be used to synthesize a closed-loop system with an optimized envelope of trajectories. An implementation method based on receding horizon optimization is proposed, along with a recursive feasibility and complexity analysis. The advances provided by the proposed synthesis methodology are illustrated for a continuous stirred tank reactor.

Input‐to‐state stabilizing sub‐optimal NMPC with an application to DC–DC converters

AbstractThis article focuses on the synthesis of computationally friendly sub‐optimal nonlinear model predictive control (NMPC) algorithms with guaranteed robust stability. To analyse the robustness of the MPC closed‐loop system, we employ the input‐to‐state stability (ISS) framework. To design ISS sub‐optimal NMPC schemes, a new Lyapunov‐based method is proposed. ISS is ensured via a set of constraints, which can be specified as a finite number of linear inequalities for input affine nonlinear systems. Furthermore, the method allows for online optimization over the ISS gain of the resulting closed‐loop system. The potential of the developed theory for the control of fast nonlinear systems, with sampling periods below 1 ms, is illustrated by applying it to control a Buck‐Boost DC–DC converter. Copyright © 2007 John Wiley & Sons, Ltd.

Input-to-state stability of networked control systems

A new class of Lyapunov uniformly globally asymptotically stable (UGAS) protocols in networked control systems (NCS) is considered. It is shown that if the controller is designed without taking into account the network so that it yields input-to-state stability (ISS) with respect to external disturbances (not necessarily with respect to the error that will come from the network implementation), then the same controller will achieve semi-global practical ISS for the NCS when implemented via the network with a Lyapunov UGAS protocol. Moreover, the ISS gain is preserved. The adjustable parameter with respect to which semi-global practical ISS is achieved is the maximal allowable transfer interval (MATI) between transmission times.