Stability Of Continuous-time Systems Research Articles

Safety Embedded Differential Dynamic Programming Using Discrete Barrier States

Certified safe control is a growing challenge in robotics, especially when performance and safety objectives must be concurrently achieved. In this work, we extend the barrier state (BaS) concept, recently proposed for safe stabilization of continuous time systems, to safety embedded trajectory optimization for discrete time systems using discrete barrier states (DBaS). The constructed DBaS is embedded into the discrete model of the safety-critical system integrating safety objectives into the system's dynamics and performance objectives. Thereby, the control policy is directly supplied by safety-critical information through the barrier state. This allows us to employ the DBaS with differential dynamic programming (DDP) to plan and execute safe optimal trajectories. The proposed algorithm is leveraged on various safety-critical control and planning problems including a differential wheeled robot safe navigation in randomized and complex environments and on a quadrotor to safely perform reaching and tracking tasks. The DBaS-based DDP (DBaS-DDP) is shown to consistently outperform penalty methods commonly used to approximate constrained DDP problems as well as CBF-based safety filters.

Symmetric Properties of Routh–Hurwitz and Schur–Cohn Stability Criteria

It is often noticed in the literature that some key results on the stability of discrete-time systems of difference equations are obtained from their corresponding results on the stability of continuous-time systems of differential equations using suitable conformal mappings or bilinear transformations. Such observations lead to the search for a unified approach to the study of root distribution for real and complex polynomials, with respect to the left-half plane for continuous-time systems (Routh–Hurwitz stability) and with respect to the unit disc for discrete-time systems (Schur–Cohn stability). This paper is a further contribution toward this objective. We present, in a systematic way, the similarities, and yet, the differences between these two types of stability, and we highlight the symmetry that exists between them. We also illustrate how results on the stability of continuous-time systems are conveyed to the stability of discrete-time systems through the proposed techniques. It should be mentioned that the results on Schur–Cohn stability are known to be harder to obtain than Routh–Hurwitz stability ones, giving more credibility to the proposed approach.

Input-to-state stability of continuous-time systems via finite-time Lyapunov functions

In this paper, input-to-state stability (ISS) of continuous-time systems is analyzed via finite-time Lyapunov functions. ISS of a continuous-time system is first proved via finite-time robust Lyapunov functions for an introduced auxiliary system of the considered system. It is then obtained that the existence of a finite-time ISS Lyapunov function implies that the continuous-time system is ISS. The converse finite-time ISS Lyapunov theorem is proposed. Furthermore, we explore the properties of finite-time ISS Lyapunov functions for the continuous-time system on a bounded and compact set without a small neighborhood of the origin. The effectiveness of our results is illustrated by four examples.

<i>D</i>-stability of parameter-dependent linear systems including discretisation by Taylor series expansion and search in a scalar parameter

This work addresses the allocation of closed-loop poles of a discretised system from a continuous-time one with varying parameters, aiming at its control through a computer. It proposes a sufficient condition for an allocating state feedback with parameter-dependent gain. The condition is verified through a feasibility test of a set of linear matrix inequalities (LMIs), based on the existence of a homogeneous polynomially parameter-dependent Lyapunov function of arbitrary degree. The main contribution of this work is the guarantee of the continuous-time system's stability and simultaneously the allocation of the closed-loop poles of the discretised system in a D-stable region. In order to allow this, the discretised model is formed by homogeneous polynomial matrices of arbitrary degree, augmented by an additive norm-bounded term, which represents the discretisation residual error. Numerical examples show a larger number of feasible cases associated with the proposed condition in comparison with the condition based on a parameter-independent gain.

On positive para-odd and complex discrete reactance functions

Positive para-odd functions have been extensively used in the stability analysis of continuous-time systems of differential equations. Their discrete-time counterparts, called complex discrete reactance functions are much less famous. In the present work, some of the major results on stability of continuous-time systems are recalled. They are presented in terms of special partial fraction and continued fraction expansions of a positive para-odd function associated with the system. By means of suitable conformal mappings, these results are converted to discrete-time systems of difference equations. We establish a characterization of complex discrete reactance functions in terms of Schur polynomials, and we develop a continued fraction expansion for complex discrete reactance functions associated with systems which are Schur stable.

Predictive Feedback Control Method for Stabilization of Continuous Time Systems

Recent researchers have studied the chaotic phenomenon and introduced methods to control chaos. One of these methods is the Predictive Feedback Control (PFC). PFC method has been used to stabilize only the discrete chaotic maps. In this paper, we generalize the PFC method by extending it to stabilize continuous time systems. The numerical simulations are carried out to solve the system using Euler method for its simplicity, and then the PFC method is applied to the discretized system. The controlled trajectory converges to an unstable equilibrium point via small control action. The stability analysis is shown compared to that of the continuous system in details. Also the choice of the controlling input of the PFC method and its properties are discussed. Lorenz system and R\ossler system are the well known chosen chaotic examples. The PFC method is applied to each of them, showing stability at each of their equilibrium point. With a very small control, the behaviour of the system is completely changed from chaos to stable.

New results on robust control for a class of uncertain systems and its applications to Chua’s oscillator

This paper addresses the problems of robust stability analysis and stabilization for continuous-time systems with state-dependent uncertainties via constructing a new parameter-dependent Lyapunov function. On the basis of such a parameter-dependent Lyapunov function, the stability conditions for open-loop uncertain systems are first presented. Then a relaxed stability analysis approach that utilizes a property of the time derivatives of uncertain parameters is obtained. To take full advantage of the parameter-dependent Lyapunov function, a model-dependent state-feedback stabilization scheme is proposed, which can provide more flexibilities in controller synthesis. The static state-feedback controller can be seen as a special case of the model-dependent state-feedback controller. A numerical example is provided to illustrate the effectiveness of the proposed approaches. Finally, the developed controller design methodology is applied to stabilization and synchronization of Chua’s oscillator, which has wide applications in secure communication systems and power electronic systems.

Improved Pade-Pole Clustering Approach using Genetic Algorithm for Model Order Reduction

Reducing the order of higher order systems by mixed approach is Improved Pade-Pole clustering based method to derive a reduced order approximation for a stable continuous time system is presented. In this method, the denominator polynomial of the reduced order model is derive by improved pole-clustering approach and the numerator polynomial are obtain through Pade approximation technique and by parameter optimization by minimizing the mean square error between the time responses of the original and reduced system element through genetic algorithm. The reduced order model so obtained by improved clustering algorithm guaranteed the stability in the reduced model.

Improved integral inequality approach on stabilization for continuous-time systems with time-varying input delay

This study is concerned with the issue of output delay-dependent stabilization criteria for continuous-time systems with time-varying input delays. By introducing an augment Lyapunov–Krasovskii functional and a new integral inequality, a new reduced conservative condition is obtained in terms of linear matrix inequalities (LMIs). At last, numerical examples are also designated to demonstrate the effectiveness and reduced conservatism of the developed results.

Robust Input-to-State Stability for Hybrid Systems

This work addresses input-to-state stability (ISS) for hybrid dynamical systems, which combine continuous-time dynamics on a flow set and discrete-time dynamics on a jump set. The main result entails equivalent characterizations of ISS when the right-hand side of the differential equation for the continuous-time dynamics is locally Lipschitz but the set of admissible derivative values, generated by considering all possible input values, is not necessarily convex. Under some mild assumptions, we show that existence of an ISS-Lyapunov function is equivalent to various types of robust ISS for these hybrid systems. As an application, we demonstrate how a hybrid system framework can be used to study some stability properties for continuous-time systems; in particular, we use the derived equivalent characterization results of ISS for hybrid systems to recover and generalize Lyapunov and asymptotic characterizations of input-output-to-state stability for continuous-time systems.

New stability criteria for continuous-time systems with interval time-varying delay

The authors investigate the stability for continuous-time systems with interval time-varying delay. To deal with the non-linear time-varying coefficients derived from Jensen's integral inequality, a well-known feature of the sum of these coefficients is utilised. Combining with a delay partition method, the upper bound of the derivative of the Lyapunov functional can be estimated more tightly and expressed as a convex combination with respect to the reciprocal of the delay rather than the delay. New less conservative stability criteria are derived in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of the proposed results.

Control of sampled-data systems with variable sampling rate

This paper addresses stability and performance of sampled-data systems with variable sampling rate, where the change between sampling rates is decided by a scheduler. A motivational example is presented, where a stable continuous time system is controlled with two sampling rates. It is shown that the resulting system could be unstable when the sampling changes between these two rates, although each individual closed-loop system is stable under the designed controller that minimizes the same continuous loss function. Two solutions are presented in this paper. The first solution is to impose restrictions on switching sequences such that only stable sequences are chosen. The second solution presented is more general, where a piecewise constant state feedback control law is designed which guarantees stability for all possible variations of sampling rate. Furthermore, the performance defined by a continuous time quadratic cost function for the sampled-data system with variable sampling rate can be optimized using the proposed synthesis method.

PWM型制御入力に基づく安定化制御

This paper deals with stabilization of continuous-time systems whose control inputs take two distinct values. We consider to generate control inputs based on pulse-width-modulation (PWM) and derive a discrete-time system which governs the state transitions of the continuous-time system at sampling instants. Since PWM control inputs keep pulsating, stability of fixed points for the discrete-time system implies stability of limit cycles for the continuous-time system. Based on the discrete-time system, we show analysis results for the existence of the fixed point, local controllability and local stabilizability. Furthermore, we will propose methods to synthesize locally stabilizing state and output feedback controllers based on LMIs.

Input-to-state stable finite horizon MPC for neutrally stable linear discrete-time systems with input constraints

MPC or model predictive control is representative of control methods which are able to handle inequality constraints. Closed-loop stability can therefore be ensured only locally in the presence of constraints of this type. However, if the system is neutrally stable, and if the constraints are imposed only on the input, global asymptotic stability can be obtained; until recently, use of infinite horizons was thought to be inevitable in this case. A globally stabilizing finite-horizon MPC has lately been suggested for neutrally stable continuous-time systems using a non-quadratic terminal cost which consists of cubic as well as quadratic functions of the state. The idea originates from the so-called small gain control, where the global stability is proven using a non-quadratic Lyapunov function. The newly developed finite-horizon MPC employs the same form of Lyapunov function as the terminal cost, thereby leading to global asymptotic stability. A discrete-time version of this finite-horizon MPC is presented here. Furthermore, it is proved that the closed-loop system resulting from the proposed MPC is ISS (Input-to-State Stable), provided that the external disturbance is sufficiently small. The proposed MPC algorithm is also coded using an SQP (Sequential Quadratic Programming) algorithm, and simulation results are given to show the effectiveness of the method.

Examples of GES Systems That can be Driven to Infinity by Arbitrarily Small Additive Decaying Exponentials

Examples are given of nonlinear, time-invariant systems in continuous-time, discrete-time, and of hybrid type, that have linear sector growth, the origin globally exponentially stable, and that can be driven to infinity by arbitrarily small additive decaying exponentials. Resulting observations about additive cascades and Lyapunov functions are discussed. These observations extend those derived from an example, which recently appeared in the literature, of a globally asymptotically stable continuous-time system destabilized by a piecewise constant, integrable disturbance.

ROBUST ADAPTIVE CONTROL WITH MULTIPLE ESTIMATION MODELS FOR STABILIZATION OF A CLASS OF NON‐INVERSELY STABLE TIME‐VARYING PLANTS

ABSTRACTThis paper presents an indirect adaptive control scheme for nominally stabilizable non‐necessarily inversely stable continuous‐time systems with unmodelled dynamics. The control objective is the adaptive stabilization of the closed‐loop system with the achievement of a bounded tracking‐error between the system output and a reference signal given by a stable filter. The adaptive control scheme includes several estimation algorithms and a supervisor which selects the appropriate estimator at every certain time and keeping it operating for at least a minimum period of residence time. This selection is based on a performance criterion related to a measure of the estimation errors obtained with each estimator. In this way, the performance of the output signal is improved with regard to the performance achieved with a unique estimation algorithm. All the estimators are either of the least‐squares type or gradient type. However, any well‐posed estimation algorithm is potentially valid for application. These estimators include relative dead‐zones for robustness purposes and parameter ‘a posteriori’ modifications to ensure the controllability of the estimated models of the plant, which is crucial for proving the stabilizability of the plant via adaptive pole‐placement designs.

Further conditions on the stability of continuous time systems with saturation

The global asymptotic stability of the origin for systems described by x/spl dot/=/spl sigma/~(Ax), where /spl sigma/~ is the saturation function, is studied. The case x(t)/spl isin//spl Rscr//sup 2/ is treated in detail and conditions which are necessary and sufficient are obtained. Examples are presented to illustrate the ideas developed in the paper.

Comments on "Quadratic stabilization of continuous time systems with state-delay and norm-bounded time-varying uncertainties"

In the above mentioned paper by Mahmoud-Muthairi (ibid. vol.39 (1994)) a sufficient condition for memoryless stabilization of a class of uncertain linear systems with a variable-state delay and norm-bounded time-varying uncertainties is derived in terms of an algebraic Riccati equation. This Riccati equation depends on several free matrix variables, and a subsequent result in the paper, Theorem 2, states that failure or success of the stabilization algorithm is independent of the selection of these matrix variables. In this paper, we give a counterexample to this Theorem 2 as well as providing a fix.

A Note on the Stabilization of Continuous-Time Systems Using Discrete-Time Compensators

Control systems are often designed on a continuous-time basis and, in practice, implemented in a discrete fashion. Feedback design guaranteeing the stability of the continuous-time closed-loop system may be lost when implementation is made in discrete form. This note makes some observations on the stability of continuous-time systems when use is made of discrete-time compensator state feedback. Exact and approximate conditions are given for system stability.

Quadratic stabilization of continuous time systems with state-delay and norm-bounded time-varying uncertainties

This paper develops a method for designing a feedback control law to stabilize a class of uncertain linear systems. The class of systems under consideration is described by a continuous-time model with variable-state delay and depends on time-varying unknown-but-bounded uncertain parameters. It is shown that the construction of the stabilizing controller involves solving a certain algebraic Riccati equation. Several previous results are derived as interesting special cases. Finally, a simulation example is presented to illustrate the analytical developments. >