Invariant Set Analysis Research Articles

Optimal scheduling and control for constrained multi‐agent networked control systems

AbstractIn this paper, we study optimal control and communication schedule co‐design for multi‐agent networked control systems, with assuming shared parallel communication channels and uncertain constrained linear time‐invariant discrete‐time systems. To that end, we specify the communication demand for each system using an associated robust control invariant set and reachability analysis. We use these communication demands and invariant sets to formulate tube‐based model predictive control and offline/online communication schedule co‐design problems. Since the scheduling part includes an infinite dimension integer problem, we propose heuristics to find suboptimal solutions that guarantee robust constraints satisfaction and recursive feasibility. The effectiveness of our approach is illustrated through numerical simulations.

Invariant Sets in Quasiperiodically Forced Dynamical Systems

This paper addresses structures of state space in quasiperiodically forced dynamical systems. We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps. The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set. We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well-defined and reveals the invariant sets, namely, a finite number of basins of attraction. We provide a characterization of invariant sets in the quasiperiodically forced systems. A theoretical result on uniform boundedness of the invariant sets is presented. The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages. Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability. We show that our theoretical results can be used to understand stability regions in such complex systems.

Sustained oscillations in MIMO nonlinear systems through limit cycle shaping

SummaryThis paper studies on generating periodic behaviors through shaping stable limit cycles in multiple‐input‐multiple‐output nonlinear systems. For this purpose, first, limit cycles are shaped with respect to the desired sustained oscillations of the system's outputs. Then, the Lyapunov analyses, which are appropriate for stability analysis of invariant sets, are employed to design the control law and conclude the asymptotic convergence toward the predefined limit cycles. The problem is studied in two cases. In the first case, some assumptions in the mathematical model are assumed that leads to simplification in the design procedure. In the second case, the design procedure is discussed in more general cases. Finally, the validity and performance of the proposed method for shaping limit cycles with different geometric shapes are illustrated by computer simulations for practical and numerical examples.

Invariant set analysis for SISO discrete‐time polynomial systems with dynamic quantizers

SummaryThis paper deals with the stability analysis problem of discrete‐time polynomial systems subject to input quantization. The quantizer under consideration is supposed to be dynamic. Indeed, considering the quantization error as bounded disturbances, the regional input‐to‐state stability is studied. The approach developed consists in characterizing two nested invariant sets where the outer one is a region of attraction and the inner one is the robust invariant set with respect to the quantization error. By defining a performance index between the quantized system and the quantizer‐free system, a quasi‐convex problem is formulated to minimize an upper bound on such a performance index.

Generation of stable oscillations in uncertain nonlinear systems with matched and unmatched uncertainties

ABSTRACTIn this paper, a robust limit cycle control technique is proposed for generation of stable oscillations in a class of uncertain nonlinear systems with both matched and unmatched uncertainties. For this purpose, first, the modified Lyapunov function is introduced which is appropriate for stability analysis of invariant sets (instead of equilibrium points). The structure of the proposed Lyapunov function is related to the shape of the desirable limit cycle. Next, in order to design the robust limit cycle control input, the backstepping and Lyapunov redesign methods are employed, simultaneously. The The classical Lyapunov redesign controller is discontinuous and robust with respect to matched uncertainties. To overcome unmatched uncertainties, a modified version of the Lyapunov redesign controller is suggested in each step of backstepping which results in a continuous robust control law. Furthermore, the convergence of the phase trajectories of the uncertain closed-loop system to the target limit cycle is proved using the extended Lyapunov stability theorem. Finally, computer simulations are performed to show the applicability of the given approach. In this regard, two uncertain nonlinear practical systems are considered and robust stable oscillations are generated in these systems via the proposed controller. Simulation results confirm the effectiveness of the proposed technique.

Nonlinear analysis and control of a variable-geometry suspension system

The paper proposes methods for both the analysis and the synthesis of variable-geometry suspension systems. The nonlinear polynomial Sum-of-Squares (SOS) programming method is applied in the analysis and it gives the optimal utilization of the maximum control forces on the tires. Moreover, the construction of the system can be based on the nonlinear analysis. The variable-geometry suspension system affects the wheel camber angle and generates an additional steering angle, thus the coordination of steering and wheel tilting can be handled. An LPV (Linear Parameter-Varying) based control-oriented modeling and control design for lateral vehicle dynamics are also proposed. The novelty of the method is the combination of the LPV-based control design and the SOS-based invariant set analysis. The simulation example presents the efficiency of the variable-geometry suspension system and it shows that the system is suitable to be used as a driver assistance system. In the SIL (software-in-the-loop) simulation both the dSPACE-AutoBox hardware and the CarSim simulator are used as standard industrial tools.

Dynamic Quantizer Design Under Communication Rate Constraints

Feedback type dynamic quantizers such as delta-sigma modulators are typically effective for encoding high-resolution data into lower resolution data. The dynamic quantizers include a filter and a static quantizer. When it is required to control under a communication rate constraint, the data rate of the quantizer output should be minimized appropriately by quantization. This technical note provides numerical methods for the complete design of a type of dynamic quantizers, including the selection of all the quantizer parameters in order to minimize a specific performance index and satisfy a communication constraint. The design method of the dynamic quantizer is proposed using a particle swarm optimization (PSO) method. A part of the initial quantizers in PSO are designed based on an invariant set analysis and an iteration algorithm. Effectiveness of the system with the proposed quantizer is assessed through numerical examples.

Shock Propagation in Cascade Systems: Analysis and Applications

In this paper, we study shock propagation in cascade systems that can be a model of outage of power systems and that of spreading a fad. While existing research works have dealt with a certain relationship between global cascade and system properties, such as degree and distribution of threshold values, and interconnection topology, this paper addresses new engineering problems of the cascade systems. Mainly, we would like to 1) identify subsystems contributing to a large cascade, 2) develop a strategy that prevents all initial failures within a designated region from cascading out of the region, and 3) develop a strategy that prevents all initial failures outside of the region from cascading into the region. In order to solve these challenging problems, we develop an invariant set analysis method for cascade models and derive some features inherent to the cascade systems. As the first feature, supposing that we know the resulting cascade region from the given initial failures in specific subsystems, we will show that the cascading of initial failures of any combination of the subsystems cannot exceed the region. The second feature is a graph theoretic distance condition, which guarantees that two cascading regions are never combined together. Based on the analytical results, we develop algorithms solving the aforementioned engineering problems, in which the complexity of the computation is reduced.

Model Following Output Feedback Control for Discrete-Valued Input Systems

This paper considers an output feedback control problem for discrete-valued input systems. The proposed controller achieves model following control for networked control systems and embedded devices with low-resolution AD/DA converters. The synthesis problem the authors address is the simultaneous synthesis of the nominal controller and the delta-sigma modulator (where the modulators are called the dynamic quantizers). The proposed approach is based on the invariant set analysis and the LMI technique. First, this paper considers a controller structure for model following control. Second, this paper proposes a controller analysis condition for quantized feedback systems. Third, the authors provide a simultaneous synthesis condition that is recast as a set of matrix inequalities. Finally, to verify the validity of the proposed method, numerical examples are presented and then the contributions related to the existing dynamic quantizer synthesis are clarified.

Stability analysis of bilinear systems under aperiodic sampled-data control

This note considers the problem of local stability of bilinear systems with aperiodic sampled-data linear state feedback control. The sampling intervals are time-varying and upper bounded. It is shown that the feasibility of some linear matrix inequalities (LMIs), implies the local asymptotic stability of the sampled-data system in an ellipsoidal region containing the equilibrium. The method is based on the analysis of contractive invariant sets, and it is inspired by the dissipativity theory. The results are illustrated by means of numerical examples.

Synthesis of continuous-time dynamic quantizers for LFT type quantized feedback systems

This paper focuses on analysis and synthesis methods of continuous-time dynamic quantizers for LFT type quantized control systems. Our aim is to propose a numerical optimization design method of multiple (decentralized) quantizers such that a given linear system is optimally approximated by the given linear system with the multiple quantizers. Our method is based on the invariant set analysis and the LMI technique. In addition, we clarify that our proposed method naturally extends to multiobjective control problems similar to linear control. For implementation, this paper presents an analysis condition of the applicable interval of switching process of quantizer. Finally, it is pointed out that the proposed method is helpful through a numerical example.

On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach

This paper is devoted to a problem of robust control design for a class of continuous-time dynamic systems with bounded uncertainties. We study a family of nonlinearly affine control systems and develop a computational extension of the conventional invariant ellipsoid techniques. The obtained method can be considered as a powerful numerical approach that makes it possible to design a concrete stabilizing control strategies for the resulting closed-loop systems. The design procedure for this feedback-type control is based on the classic Lyapunov-type stability analysis of invariant sets for the given dynamic system. We study the necessary theoretic basis and propose a computational algorithm that guarantee some minimality properties of the stability/attractivity regions for dynamic systems under consideration. The complete solution procedure contains an auxiliary LMI-constrained optimization problem. The effectiveness of the proposed robust control design is illustrated by a numerical example.

Local Stability of Bilinear Systems with Asynchronous Sampling

This note considers the problem of local stability of aperiodic sampled-data bilinear systems, controlled via linear state-feedback. The sampling intervals are time-varying and upper bounded. It is shown that by solving linear matrix inequalities (LMIs) the local asymptotic stability of the sampled system is guaranteed in an ellipsoidal region containing the origin of the state space. The method is based on the analysis of contractive invariant sets, and it is inspired by the dissipativity theory. The results are illustrated by means of a numerical example.

Synthesis of continuous-time dynamic quantizers for quantized feedback systems

This paper focuses on analysis and synthesis methods of continuous-time dynamic quantizers for quantized feedback systems. Our aim is to find a quantizer such that a given linear system is optimally approximated by the given linear system with the quantizer in terms of invariant set analysis. In the case of minimum phase systems, this paper clarifies that an optimal dynamic quantizer and its performance are parameterized by a design parameter. Otherwise, this paper derives the stable quantizer synthesis condition that can be recast as parameter dependent linear matrix inequalities.

Performance-oriented adaptive neural augmentation of an existing flight control system

Control systems for safety-critical applications, including the ones relying on adaptive elements, have to be certified against strict performance and safety requirements. This study presents a practical approach for the design of a neuro-adaptive element with the specific purpose of safely recovering the performance of a reference model in presence of bounded uncertainties. The boundedness of the tracking error vector within an a-priori specified compact domain is enforced by applying robust invariant set analysis to the uncertain linear plant where the adaptive neural contribution is considered as an amplitude-bounded persistent disturbance. In this framework, tracking error requirements are specified via a set of LMI constraints and maximal allowed amplitudes for the adaptive control are computed using a numerical LMI solver. A specific neural network on-line learning and output confinement algorithm is also proposed to keep the adaptive control within selected amplitudes; as a result, the overall closed loop system has a guaranteed worst-case transient response. The proposed approach has been successfully applied to the design of a multi input multi output (MIMO) augmentation adaptive element that improves the performance of a pre-existing tracking controller for a research aircraft model.

Persistent bounded disturbance rejection for discrete-time delay systems

In this article, we provide a novel solution to the problem of persistent bounded disturbance rejection in linear discrete-time systems with time-varying delays. The solution is developed based on the tools of invariant set analysis and Lyapunov-function method. As an integral part of the solution, we derive less conservative sufficient conditions on robust attractor for discrete-time systems with delays in terms of strict linear matrix inequalities to guarantee the desired ℓ1-performance. A robust state-feedback controller is designed and the associated gain is determined using strict LMIs. The developed results are tested on a representative example.

Synthesis of decentralized dynamic quantizer within invariant set analysis framework

AbstractThis paper focuses on the synthesis of decentralized dynamic quantizers for linear systems with discrete-valued signal in terms of invariant set analysis. We derive the synthesis condition that can be recast as parameter dependent linear matrix inequalities and has a closed-form solution under some circumstances. In the case of minimum phase systems, our method can provide an optimal decentralized dynamic quantizer in the sense that the quantizer gives an optimal output approximation property. Also, in the case of non-minimum phase systems, our method can provide a suboptimal stable decentralized dynamic quantizer. As a result, this paper gives a unified design method of the dynamic quantizer based on the invariant set analysis.

Rejection of persistent‐bounded disturbances in linear time‐delay systems by output‐feedback

AbstractIn this paper, the problem of persistent‐bounded disturbance rejection of linear continuous‐time systems with time‐varying delays is investigated using the tools of invariant set analysis and Lyapunov‐function methodology. We derive less conservative sufficient conditions on robust attractor for time‐delay systems in terms of strict linear matrix inequalities (LMIs) to guarantee the desired ℒ︁1‐performance. A robust output‐feedback controller is designed and the associated gain is determined using strict LMIs. The developed results are tested on two representative time‐delay examples. Copyright © 2009 John Wiley & Sons, Ltd.

離散値入力型ＳＩＳＯシステムに対する不変集合解析に基づく動的量子化器設計

This paper proposes analysis and synthesis methods of dynamic quantizers for linear feedback single input single output (SISO) systems with discrete-valued input in terms of invariant set analysis. First, this paper derives the quantizer analysis and synthesis conditions that clarify an optimal quantizer within the ellipsoidal invariant set analysis framework. In the case of minimum phase feedback systems, next, this paper presents that the structure of the proposed quantizer is also optimal in the sense that the quantizer gives an optimal output approximation property. Finally, this paper points out that the proposed design method can design a stable quantizer for non-minimum phase feedback systems through a numerical example.

LMI-based design of a Neuro-Adaptive augmentation controller for an Unmanned Aerial Vehicle

This paper presents a practical approach for verifying worst-case tracking performance of neuro-adaptive systems in presence of bounded uncertainties. Boundedness of the tracking error vector within an a-priori specified compact domain is obtained by applying robust invariant set analysis to the uncertain linear plant where the uncertainty and Neural Network (NN) reconstruction error are considered as norm bounded persistent uncertainties. In this framework it was possible to specify worst-case tracking error requirements via a set of LMIs and to systematically verify the specifications using a numerical LMI solver. The presented method was applied to the performance verification of an adaptive augmentation controller for the short term dynamics of an UAV model.