scholarly journals Stability of conditionally invariant sets and controlled uncertain dynamic systems on time scales

1995 ◽  
Vol 1 (1) ◽  
pp. 1-10 ◽  
Author(s):  
V. Lakshmikantham ◽  
Z. Drici
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Invariant Sets ◽  
Continuous Systems ◽  
Stability Properties ◽  
Closed Sets ◽  
Controlled Systems ◽  
Uncertain Dynamic Systems ◽  
Imperfect Knowledge ◽  
The Stability

A basic feedback control problem is that of obtaining some desired stability property from a system which contains uncertainties due to unknown inputs into the system. Despite such imperfect knowledge in the selected mathematical model, we often seek to devise controllers that will steer the system in a certain required fashion. Various classes of controllers whose design is based on the method of Lyapunov are known for both discrete [4], [10], [15], and continuous [3–9], [11] models described by difference and differential equations, respectively. Recently, a theory for what is known as dynamic systems on time scales has been built which incorporates both continuous and discrete times, namely, time as an arbitrary closed sets of reals, and allows us to handle both systems simultaneously [1], [2], [12], [13]. This theory permits one to get some insight into and better understanding of the subtle differences between discrete and continuous systems. We shall, in this paper, utilize the framework of the theory of dynamic systems on time scales to investigate the stability properties of conditionally invariant sets which are then applied to discuss controlled systems with uncertain elements. For the notion of conditionally invariant set and its stability properties, see [14]. Our results offer a new approach to the problem in question.

Structural Decomposition Approach for the Stability of Uncertain Dynamic Systems

1988 ◽  
Vol 55 (4) ◽  
pp. 992-994 ◽  
Author(s):  
Y. H. Chen ◽  
Chieh Hsu
Keyword(s):  
Dynamic Systems ◽  
A Priori ◽  
Parameter Variation ◽  
Stability Study ◽  
Fast Time ◽  
Time Varying ◽  
Decomposition Approach ◽  
New Approach ◽  
Uncertain Dynamic Systems ◽  
The Stability

The stability property for a class of dynamic systems with uncertain parameter variation is studied. The uncertainty can be fast time-varying and unpredictable. A new approach for stability study is proposed. The only required information on the uncertain variation is its possible bound as well as structure. That is, no a priori knowledge on the realization of the variation is needed.

scholarly journals An analysis on the controllability and stability to some fractional delay dynamical systems on time scales with impulsive effects

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bakhtawar Pervaiz ◽  
Akbar Zada ◽  
Sina Etemad ◽  
Shahram Rezapour
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Fixed Point Theorems ◽  
Impulsive Effects ◽  
Qualitative Properties ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results ◽  
Fractional Delay ◽  
The Stability

AbstractIn this article, we establish a new class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. We investigate the qualitative properties of the considered systems. In fact, the article contains three segments, and the first segment is devoted to investigating the existence and uniqueness results. In the second segment, we study the stability analysis, while the third segment is devoted to investigating the controllability criterion. We use the Leray–Schauder and Banach fixed point theorems to prove our results. Moreover, the obtained results are examined with the help of an example.

Robust Control of Generalized Dynamic Systems Without the Matching Conditions

1991 ◽  
Vol 113 (4) ◽  
pp. 582-589 ◽  
Author(s):  
Zhihua Qu ◽  
John Dorsey
Keyword(s):  
Robust Control ◽  
Dynamic Systems ◽  
Control Law ◽  
General Control ◽  
Matching Conditions ◽  
Uncertain Dynamic Systems ◽  
Special Cases ◽  
The Stability ◽  
Bounded Uncertainties

A general control law and a set of conditions are proposed to guarantee the stability of dynamic systems with bounded uncertainties. The results do not require the matching conditions on the uncertainties and subsume several existing results as special cases. Moreover, it is shown that there is at least one class of uncertain dynamic systems which can always be stabilized under the proposed control law no matter what the size and the structure of the input-unrelated uncertainties.

scholarly journals Bielecki–Ulam–Hyers stability of non–linear Volterra impulsive integro–delay dynamic systems on time scales

2021 ◽  
pp. 339-349
Author(s):  
Syed Omar Shah ◽  
Akbar Zada ◽  
Cemil Tunc ◽  
Asad Ali
Keyword(s):  
Fixed Point ◽  
Time Scales ◽  
Lipschitz Condition ◽  
Dynamic Systems ◽  
Fixed Point Approach ◽  
Non Linear ◽  
The Stability

In this manuscript, the stability in terms of Bielecki–Ulam– Hyers and stability in terms of Bielecki–Ulam–Hyers–Rassias of non– linear Volterra impulsive integro–delay dynamic systems on time scales are obtained using the fixed point approach along with Gronwall inequal- ¨ ity and Lipschitz condition.

Act and Wait Concept in Force Controlled Systems With Discrete Delayed Feedback

2005 ◽  
Author(s):  
Tama´s Insperger ◽  
Ga´bor Ste´pa´n
Keyword(s):  
Sampling Period ◽  
Delayed Feedback ◽  
Feedback Gain ◽  
Stability Properties ◽  
Controlled Systems ◽  
Gain Variation ◽  
Delay Effects ◽  
Control Concept ◽  
The Stability ◽  
Special Case

A 1 DOF model of force control is considered with discrete delayed feedback. The act and wait control concept is introduced as a special case of periodic control: the feedback gain is constant for a sampling period (act), then it is zero for a certain number of periods (wait), then it is constant again, etc. It is shown that by applying the act and wait concept, the stability properties of the system improve and the force error caused by the Coulomb friction decreases. If the period of gain variation is larger than the feedback delay, then the system performance changes radically: the stability properties improve significantly, and the optimal control parameters can provide dead beat control. This way, by using the act and wait concept, the delay effects are eliminated from the system.

Convergence Rates of a Class of Uncertain Dynamic Systems

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Juan Ignacio Mulero-Martínez
Keyword(s):  
Dynamic Systems ◽  
Uncertain Systems ◽  
Convergence Rates ◽  
Initial Conditions ◽  
Compact Set ◽  
Robust Nonlinear Control ◽  
Riccati Differential Equation ◽  
Uncertain Dynamic Systems ◽  
Complete Treatment ◽  
The Stability

The problem of stabilization of uncertain systems plays a broad and fundamental role in robust control theory. The paper examines a boundedness theorem for a class of uncertain systems characterized as having a decreasing Lyapunov function in a ringlike region. It is a systematic study on stability that embraces both the transient and steady analysis, covering such aspects as the maximum overshoot of the system state, the stability region and the exponential convergence rate. The emphasis throughout is on deriving dominant time constants and explicit time expressions for a state to reach an invariant set. The central theorem provides a complete treatment of the time evolution of trajectories depending on the specific compact set of initial conditions. Toward this end, the comparison lemma along with a particular Riccati differential equation are essential and conclusive. The scope of questions addressed in the paper, the uniformity of their treatment, the novelty of the proposed theorem, and the obtained results make it very useful with respect to other works on the problem of robust nonlinear control.

scholarly journals Deductive Stability Proofs for Ordinary Differential Equations

2021 ◽  
pp. 181-199
Author(s):  
Yong Kiam Tan ◽  
André Platzer
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real World ◽  
Formal Analysis ◽  
Dynamic Logic ◽  
Logical Structure ◽  
Theorem Prover ◽  
Continuous Systems ◽  
Stability Properties ◽  
Controlled Systems

AbstractStability is required for real world controlled systems as it ensures that those systems can tolerate small, real world perturbations around their desired operating states. This paper shows how stability for continuous systems modeled by ordinary differential equations (ODEs) can be formally verified in differential dynamic logic (). The key insight is to specify ODE stability by suitably nesting the dynamic modalities of with first-order logic quantifiers. Elucidating the logical structure of stability properties in this way has three key benefits: i) it provides a flexible means of formally specifying various stability properties of interest, ii) it yields rigorous proofs of those stability properties from ’s axioms with ’s ODE safety and liveness proof principles, and iii) it enables formal analysis of the relationships between various stability properties which, in turn, inform proofs of those properties. These benefits are put into practice through an implementation of stability proofs for several examples in KeYmaera X, a hybrid systems theorem prover based on .

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