Time-varying Subsystems Research Articles

Stabilization of Periodic Piecewise Time-Varying Systems With Time-Varying Delay Under Multiple Cyber Attacks: An Augmented Lyapunov Functional Approach.

This article investigates the asymptotic stabilization of periodic piecewise time-varying systems with time-varying delay under various cyber attacks, particularly deception and acrlong DoS attacks. The addressed system is reformed into a number of time-varying subsystems based on the time interval for each period. Following that, a state-feedback controller with periodic time-varying gain parameters is developed to solve the stabilization problem. The control design depicts the possibility of the aforementioned cyber attacks with two mutually exclusive stochastic Bernoulli distributed parameters. Then, an augmented Lyapunov-Krasovskii functional with periodically varying matrices is used to determine the conditions for designing the proposed controller that ensures the mean-square asymptotic stability of the addressed system. The results of numerical examples support the conclusion that the proposed method is effective and superior, regardless of the cyber attacks involved.

Deterministic Learning-Based Adaptive Neural Control for Nonlinear Full-State Constrained Systems.

In this article, an adaptive neural learning method is introduced for a category of nonlinear strict-feedback systems with time-varying full-state constraints. The two challenging problems of state constraints and learning capability are investigated and solved in a unified framework. To obtain the learning of unknown functions and satisfy full-state constraints, three main steps are considered. First, an adaptive dynamic surface controller (DSC) based on barrier Lyapunov functions (BLFs) is structured to implement that the closed-loop systems signals are bounded and full-state variables remain within the prescribed time-varying intervals. Moreover, the radial basis function neural networks (RBF NNs) are used to identify unknown functions. The output of the first-order filter, instead of virtual control derivatives, is used to simplify the complexity of the RBF NN input variables. Second, the state transformation is used to obtain a class of linear time-varying subsystems with small perturbations such that the recurrence of the RBF NN input variables and the partial persistent excitation condition are actualized. Therefore, the unknown functions can be accurately approximated, and the learned knowledge is kept as constant NN weights. Third, the obtained constant weights are borrowed into an adaptive learning scheme to achieve the batter control performance. Finally, simulation studies illustrate the advantage of the reported adaptive learning method on higher tracking accuracy, faster convergence rate, and lower computational expense by reusing learned knowledge.

Comparison of two polynomial approaches in performance analysis for periodic piecewise polynomial systems

In this paper, the theory and effectiveness of two polynomial approaches are compared in the analysis of L2-L∞ and H∞ performance for a type of periodic piecewise polynomial systems, where the time-varying subsystems can be characterized in Bernstein polynomials. Using the Bernstein polynomial-based lemma and the existing lemma concerning the negativity/positivity of matrix polynomial functions, sufficient conditions are established in tractable forms aimed at the global asymptotic stability and performance analysis. Four cases of optimization constraints are considered based on the proposed conditions. The performance indices obtained via the four cases are compared through a numerical example, and the lower conservatism achieved by the proposed Bernstein polynomial approach is demonstrated.

A Novel Scheme of Nonfragile Controller Design for Periodic Piecewise LTV Systems

In this article, a novel nonfragile controller design scheme is developed for a class of periodic piecewise systems with linear time-varying subsystems. Two types of norm-bounded controller perturbations, including additive and multiplicative ones, are considered and partially characterized by periodic piecewise time-varying parameters. Using a new matrix polynomial lemma, the problems of nonfragile controller synthesis for periodic piecewise time-varying systems (PPTVSs) are made amenable to convex optimization based on the favorable property of a class of matrix polynomials. Depending on selectable divisions of subintervals, sufficient conditions of the stability and nonfragile controller design are proposed for PPTVSs. Case studies based on a multi-input multi-output PPTVS and a mass-spring-damper system show that the proposed control schemes can effectively guarantee the close-loop stability and accelerate the convergence under controller perturbations, with more flexible periodic time-varying controller gains than those obtained by the existing methods.

Analytical solutions to three-dimensional hypersonic gliding trajectory over rotating Earth

In this paper, high-precision analytical solutions are developed for 3-dimensional (3-D) hypersonic gliding trajectory over the rotating and spherical Earth using the regular perturbation method. First, an Auxiliary Geocentric Rotating (AGR) frame fixed on the rotating Earth is created based on a generalized equator, which is defined as the great circle in the plane containing the initial position and velocity vectors of the hypersonic vehicle, and then a new entry dynamics model is established based on the generalized longitude and latitude. As the rotational angular velocity of Earth is no longer just along the z-axis of the AGR frame, the new model becomes much more complex than the conventional one. However, the benefit of this model is that the AGR frame caters to the trajectory characteristic that the downrange is much greater than the crossrange. Thus the model is conducive to being linearized. By simplifying the model skillfully, a reduced-order system is obtained with energy as independent variable and capable of predicting the downrange, crossrange and heading angle of 3-D trajectory. However, in this system, the crossrange and heading angle are still coupled seriously. To cope with this, the system is decomposed into a Taylor series by the regular perturbation method and approximated by truncating the series after the first two terms. The remaining two terms are linear time-varying (LTV) sub-systems and can be solved by the spectral-decomposition-based method. In the derivation process, we find and prove that some complex formulae related to Earth's rotation keep almost constant since the glider flies near the generalized equator. The simulation results demonstrate that the new analytical solutions have high accuracy and require a small amount of calculation.

Radiation Efficiency Limits in Direct Antenna Modulation Transmitters

Relative bounds on radiation efficiency are established for time-modulated antenna systems where radiation is generated by broadband conduction currents impressed using idealized non-radiating time-varying subsystems, such as those found in direct antenna modulation (DAM) transmitters. Analytical and numerical examples demonstrate that the condition of quasi-resonance, common in nearly all practical direct antenna modulation transmitters, imposes severe restrictions on the otherwise unbounded gains in effective efficiency theoretically achievable by this class of time-modulated transmitters.

Stability and $L_2$ Synthesis of a Class of Periodic Piecewise Time-Varying Systems

In this paper, the stability, stabilization, and $L_{2}$ -gain problems are investigated for periodic piecewise systems with time-varying subsystems. Continuous Lyapunov function with time-varying Lyapunov matrix is adopted. A condition guaranteeing the negative definiteness of a matrix polynomial, deriving from the Lyapunov derivative, is first obtained. Based on such a condition, an exponential stability condition is provided. Moreover, a state-feedback controller with time-varying gain is developed to stabilize the unstable periodic piecewise time-varying system. The $L_{2}$ -gain criterion for periodic piecewise time-varying system is also studied. Numerical examples are given to show the validity of the proposed techniques.

H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function

In this paper, the H∞ control problem of periodic piecewise systems with polynomial time-varying subsystems is addressed. Based on a periodic Lyapunov function with a continuous time-dependent Lyapunov matrix polynomial, the H∞ performance is studied. The result can be easily reduced to the conditions for periodic piecewise systems with constant subsystems or linear time-varying systems based on a common Lyapunov function or a linear time-varying Lyapunov matrix. Moreover, an H∞ controller with time-varying polynomial controller gain is proposed as well, which could be directly solved with the linear matrix inequalities. A numerical example is presented to demonstrate the effectiveness of the proposed method.

A novel H∞ tracking control scheme for periodic piecewise time-varying systems

In this work, a novel H∞ tracking control scheme is designed for a class of periodic piecewise systems with time-varying subsystems. The reference model is a periodic piecewise linear system that allows the existence of unstable subsystems. Based on periodically time-varying Lyapunov functions, a sufficient condition is proposed to guarantee the exponential stability of augmented tracking system in the H∞ sense. Taking advantage of the negative definiteness property of a type of matrix polynomials, an H∞ state tracking controller with periodically time-varying gains is provided via a series of continuous matrix functions, which can be efficiently online computed by the solutions of straightforward convex optimization. Numerical simulations are presented to demonstrate the effectiveness of the proposed scheme.

Dynamic output feedback control based on virtual feedback control law for planar switched nonlinear systems with time-varying delays and multiple subsystems

This paper addresses the dynamic output feedback control problem for a class of discrete-time planar switched nonlinear systems with time-varying delays and multiple subsystems. First, the virtual state feedback control law is designed based on adding a power integrator approach. Secondly, the nonlinear reduced-order compensator is designed for the subsystems of the planar switched nonlinear systems under arbitrary switchings. Then, the dynamic output feedback controller is constructed based on the virtual state feedback control law and the nonlinear reduced-order compensator. By introducing the new discrete Lyapunov–Krasovskii functional, it can be seen that the solutions of the closed-loop system converge to an adjustable bounded region. Compared with the previous works, the subsystems of this planar switched system may contain the uncontrollable/unobservable Jacobian linearizations. In addition, the obtained results are further extended to the general nonlinear case and one-link manipulator case with the motor dynamics. Finally, two simulation examples are performed to show the effectiveness of the proposed method.

Quadratic Performance Analysis of Switched Affine Time–Varying Systems

Abstract We analyze quadratic performance for switched systems which are composed of a finite set of affine time-varying subsystems, where both subsystem matrices and affine vectors are switched, and no single subsystem has desired quadratic performance. The quadratic performance indexes we deal with include stability, tracking and L2 gain. We show that if a linear convex combination of subsystem matrices is uniformly Hurwitz and another convex combination of affine vectors is zero, then we can design a state-dependent switching law (state feedback) and an output-dependent switching law (output feedback) such that the entire switched affine system is quadratically stable at the origin. In the case where the convex combination of affine vectors is nonzero, we show that the tracking control problem can be posed and solved using a similar switching strategy. Finally, we consider the L2gain analysis problem for the switched affine time-varying systems under state feedback.

Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs

Abstract Necessary and sufficiently conditions are derived for the decomposition of a second order linear time- varying system into two cascade connected commutative first order linear time-varying subsystems. The explicit formulas describing these subsystems are presented. It is shown that a very small class of systems satisfies the stated conditions. The results are well verified by simulations. It is also shown that its cascade synthesis is less sensitive to numerical errors than the direct simulation of the system itself.

Extensions of weak-invariance principle for nonlinear switched systems with time-invariant and time-varying subsystems

This paper addresses the stability problem for nonlinear switched systems with both time-invariant and time-varying subsystems. Given that Lyapunov-like functions are difficult to construct for nonlinear switched systems, generalized invariance principles are established based on observer functions. For nonlinear switched systems where the subsystems share Lyapunov-like functions, the generalized invariance principles can be specialized to Lyapunov-based invariance principles, where accurate convergent region can be obtained. In addition to the above efforts, the definitions of p-limit system and limit system set are presented, under which the proposed invariance principles are extended to nonlinear switched systems with time-varying subsystems. Illustrative examples show the effectiveness of the theoretical results.

The feedback interconnection of multivariable systems: Simplifying theorems for stability

We consider the stability of the feedback interconnection of possibly unstable n-input n-output subsystems whose interconnection is described by e <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> = u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> - y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , e <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> = u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> + y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> = G <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> (e <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ), i = 1,2. We give three theorems which simplify the stability tests. Theorem 1 deals with nonlinear time-varying subsystems. It gives conditions on G <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> so that the stability of u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> ↦ y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> guarantees that of the feedback system. The other two theorems consider continuous-time linear time-invariant subsystems. It is noted that in the multivariable case, the stablity of u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ↦ y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , i = 1,2 is not sufficient to guarantee the stability of the feedback system, and Theorem 2 specifies some additional requited conditions. Theorem 3 shows that if G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> and G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> (I + G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> are in some special stable classes, so is the transfer function of the feedback system. In both theorems, corollaries specialize the results to lumped and single-input single-output cases. The paper ends by showing how these results can be translated for the discrete-time case.

On Stability of Large-Scale Systems under Structural Perturbations

A large-scale system isconsidered asasystem constituted ofsubsystems whichmay beconnected ordisconnected fromeach otherduring operation. A newconcept ofconnective stability isintro- ducedbywhich alarge-scale system isregarded asstable ifitremains stable (in thesense ofLyapunov) understructural perturbations produced bytheon-off participation ofthesubsystems. Algebraic conditions are developed thatguarantee exponential connective stability oflarge-scale systems whichmaybecomposed oflinear andnonlinear time-varying subsystems coupled bylinear ornonlinear connections.

Stability of Large-Scale Systems under Structural Perturbations

A large-scale system is considered as a system constituted of subsystems which may be connected or disconnected from each other during operation. A new concept of connective stability is introduced by which a large-scale system is regarded as stable if it remains stable (in the sense of Lyapunov) under structural perturbations produced by the on-off participation of the subsystems. Algebraic conditions are developed that guarantee exponential connective stability of large-scale systems which may be composed of linear and nonlinear time-varying subsystems coupled by linear or nonlinear connections.

On the stability of feedback systems containing linear non-differential time-varying subsystems

On the stability of feedback systems containing linear non-differential time-varying subsystems