On finite time delay dependent stability of linear discrete delay systems: Numerical solution approach

U ovom radu razmatra se jedno moguće rešenje bazične nelinearne kvadratne matrične jednačine. To rešenje ima krucijelni značaj u formulisanju posebnog kriterijuma, zavisnog od iznosa čisto vremenskog kašnjenja, za stabilnost na konačnom vremenskom intervalu posebne klase sistema sa kašnjenjem, opisane svojim matričnim modelom x(k+1)=A0(k) + A1x(k-h). U tom smislu izveden je i odgovarajući kriterijum stabilnosti koji uključuje i iznos čisto vremenskog kašnjenja. Mimo toga, posebno je apostrofiran značaj nelinearnog diskretnog matričnog polinoma u stabilnosti sistema. Koristeći matematički formalizam, baziran na Traub-ovom i Bernuli-jevom algoritmu, zaključeno je da sračunavanje dominantnog solventa matričnog polinoma, ne garantuje potrebnu konvergenciju u svim slučajevima, kao sto je slučaj u tradicionalnim numeričkim procedurama. U ovom radu, prezentuje se jedno posebno i jedno opste rešenje, koje važi za slučaj kada se matrični polinom može prikazati u faktorizovanom obliku. Numeričkim primerom ilustrovana je opravdanost predložene procedure.In this paper, a possible solution of the basic nonlinear quadratic matrix equation was proposed. The solution is crucial in the formulation of the particular criteria for the delay-dependent finite time stability of discrete time delay systems represented as x(k+1)=A0(k)+A1x(k-h). The time delay-dependent criteria have been derived. In addition, the significance of the nonlinear discrete polynomial matrix equation is explained. With the use of the mathematical formalism based on the Traub and Bernoulli's algorithms, it was concluded that the computation of the dominant solvent of the matrix polynomial equation does not guarantee a necessary convergence in all cases, unlike in the traditional numerical procedures. In this paper, we presented one particular and one general solution valid in the case when the discrete matrix equation was presented in its factorial form. The numerical computations are performed to illustrate the suggested results

On finite time delay dependent stability of linear discrete delay systems: Numerical solution approach

U ovom radu razmatra se jedno moguće rešenje bazične nelinearne kvadratne matrične jednačine. To rešenje ima krucijelni značaj u formulisanju posebnog kriterijuma, zavisnog od iznosa čisto vremenskog kašnjenja, za stabilnost na konačnom vremenskom intervalu posebne klase sistema sa kašnjenjem, opisane svojim matričnim modelom x(k+1)=A0(k) + A1x(k-h). U tom smislu izveden je i odgovarajući kriterijum stabilnosti koji uključuje i iznos čisto vremenskog kašnjenja. Mimo toga, posebno je apostrofiran značaj nelinearnog diskretnog matričnog polinoma u stabilnosti sistema. Koristeći matematički formalizam, baziran na Traub-ovom i Bernuli-jevom algoritmu, zaključeno je da sračunavanje dominantnog solventa matričnog polinoma, ne garantuje potrebnu konvergenciju u svim slučajevima, kao sto je slučaj u tradicionalnim numeričkim procedurama. U ovom radu, prezentuje se jedno posebno i jedno opste rešenje, koje važi za slučaj kada se matrični polinom može prikazati u faktorizovanom obliku. Numeričkim primerom ilustrovana je opravdanost predložene procedure.In this paper, a possible solution of the basic nonlinear quadratic matrix equation was proposed. The solution is crucial in the formulation of the particular criteria for the delay-dependent finite time stability of discrete time delay systems represented as x(k+1)=A0(k)+A1x(k-h). The time delay-dependent criteria have been derived. In addition, the significance of the nonlinear discrete polynomial matrix equation is explained. With the use of the mathematical formalism based on the Traub and Bernoulli's algorithms, it was concluded that the computation of the dominant solvent of the matrix polynomial equation does not guarantee a necessary convergence in all cases, unlike in the traditional numerical procedures. In this paper, we presented one particular and one general solution valid in the case when the discrete matrix equation was presented in its factorial form. The numerical computations are performed to illustrate the suggested results

Improved results on finite time stability of time delay systems: Jensen's inequality-based approach

In this study, finite-time stability of the linear continuous time-delay systems was investigated. A novel formulation of the Lyapunov-like function was used to develop a new sufficient delay-dependent condition for finite-time stability. The proposed function does not need to be positive-definite in the whole state space, and it does not need to have negative derivatives along the system trajectories. The proposed method was compared with the previously developed and reported methodologies. It was concluded that the stability investigation using the novel condition for stability investigation was less complicated for numerical calculations. Furthermore, it gives results in comparison with the ones obtained with other analyzed conditions, and it provides superior results for these class of systems

Delay Dependent Finite Time Stability of Discrete Time Delay Systems: Towards the New Solution

In this paper, a possible solution of the basic nonlinear quadratic matrix equation was proposed. The solution is crucial in the formulation of the particular criteria for the delay-dependent finite time stability of discrete time delay systems represented as x(k+1) = A0(k) + A1x(k-h). The time delay-dependent criteria have been derived. In addition, the significance of the nonlinear discrete polynomial matrix equation is explained. With the use of the mathematical formalism based on the Traub and Bernoulli's algorithms, it was concluded that the computation of the dominant solvent of the matrix polynomial equation does not guarantee a necessary convergence in all cases, unlike in the traditional numerical procedures. In this paper, we presented one particular solution valid in the case when the discrete matrix equation was presented in its factorial form. The numerical computations are performed to illustrate the suggested results

Dinamika sistema opisanih algebro-diferencijalnim jednačinama sa pomerenim argumentom, II deo

Okosnica ove monografije počiva na savremenoj teoriji upravljanja i postavlja i razrešava niz veoma složenih pitanja dinamike, posebnih klasa, sistema automatskog upravljanja, opisanih sistemom diferencijalnih (diferencnih) jednačina sa pomerenim argumentom. U tom smislu, izložen je ograničen broj radova eminentnih naučnika i osvedočenih autoriteta iz ove oblasti, kao i manji broj radova samih autora iz ove, uvek aktuelne i više nego provokativne, oblasti savremene teorije upravljanja i stabilnosti. Poseban doprinos monografije daje pregled upravljačkih struktura sa unutrašnjim modelima. Opisana su tri globalna prilaza sinteze sistema sa unutrašnjim modelima koja se odnose na korišćenje: principa unutrašnjeg modela IMP (Internal Model Principle), IMC (Internal Model Control) strukture i Tsypkinove IMPACT (Internal Model Principle and Control Together) strukture. Ovi prilazi su nastali potpuno nezavisno jedni od drugih, u različitim vremenskim razdobljima, kao plod rada mnoštva istraživača, angažovanih na rešavanju različitih upravljačkih problema. U ovim upravljačkim strukturama ugrađuju se elementi čiji karakter eksplicitno zavisi od modela poremećaja i/ili modela objekta upravljanja. Zavisno od primarnih upravljačkih ciljeva, razvijeni su različiti koncepti primene unutrašnjih modela u strukturnoj sintezi sistema. U monografiji su opisana svojstva struktura sa unutrašnjim modelima. Svi do sada razvijeni postupci strukturne sinteze upravljanih sporih industrijskih procesa mogu se svrstati u pomenute tri kategorije u zavisnosti od toga kako se i na kom principu unutrašnji modeli unose u upravljački deo strukture sistema automatskog upravljanja. Posebno poglavlje je posvećeno novoj upravljačkoj strukturi za upravljanje sporih industrijskih procesa sa transportnim kašenjenjem. Odlikuje je mali broj podešljivih parametara sa jasnim fizičkim značenjem i eksplicitnim uticajem na dinamičke i robustne performanse sistema sa zatvorenom povratnom spregom. Sistematizacija i strukturne i parametarske sinteze IMPACT strukture, i njenih osnovnih varijacija, je data u poslednjem poglavlju

Delay Dependent Finite Time Stability of Discrete Time Delay Systems: Towards the New Solution

In this paper, a possible solution of the basic nonlinear quadratic matrix equation was proposed. The solution is crucial in the formulation of the particular criteria for the delay-dependent finite time stability of discrete time delay systems represented as x(k+1) = A0(k) + A1x(k-h). The time delay-dependent criteria have been derived. In addition, the significance of the nonlinear discrete polynomial matrix equation is explained. With the use of the mathematical formalism based on the Traub and Bernoulli's algorithms, it was concluded that the computation of the dominant solvent of the matrix polynomial equation does not guarantee a necessary convergence in all cases, unlike in the traditional numerical procedures. In this paper, we presented one particular solution valid in the case when the discrete matrix equation was presented in its factorial form. The numerical computations are performed to illustrate the suggested results

NON-LYAPUNOV STABILITY OF SINGULAR SYSTEMS: CLASSICAL AND MODERN APPROACHES WITH APPLICATION TO AUTOMATIC DRUG DELIVERY

In this paper sufficient conditions for both practical and finite time stability of linear singular continuous time delay systems were introduced. The singular and singular time delay systems can be mathematically described as  and , respectively. Analyzing finite time stability, the new delay independent and delay dependent conditions were derived using the approaches based on Lyapunov-like functions and their properties on the subspace of consistent initial conditions. These functions do not need to be positive on the whole state space and to have negative derivatives along the system trajectories. When the practical stability was analyzed, the approach was combined with classical Lyapunov technique to guarantee the attractivity property of the system behavior. Furthermore, an LMI approach was applied to obtain less conservative stability conditions. The proposed methodology was applied and tested on a medical robotic system. The system was designed for different insertion tasks playing important roles in automatic drug delivery, biopsy or radioactive seeds delivery. In this paper we have summarized different techniques for adequate modeling, control and stability analysis of the medical robots. The model of the robotic system, with the tasks described above, the entire system can be decomposed to the robotic subsystem and the environment subsystem. Modeling of the system by the method mentioned has been proved to be suitable when the force appears as a result of the interaction of the two subsystems. The mathematical model of the system has a singular characteristic. The singular system theory could be applied to the case described. It is well known that all mechanical systems have some delay. In that case a theory of singular systems with delayed states may be applied, as well. For the second phase in which there is no interaction, the dynamic behavior can be analyzed by the classic theory.In this paper sufficient conditions for both practical and finite time stability of linear singular continuous time delay systems were introduced. The singular and singular time delay systems can be mathematically described as  and , respectively. Analyzing finite time stability, the new delay independent and delay dependent conditions were derived using the approaches based on Lyapunov-like functions and their properties on the subspace of consistent initial conditions. These functions do not need to be positive on the whole state space and to have negative derivatives along the system trajectories. When the practical stability was analyzed, the approach was combined with classical Lyapunov technique to guarantee the attractivity property of the system behavior. Furthermore, an LMI approach was applied to obtain less conservative stability conditions. The proposed methodology was applied and tested on a medical robotic system. The system was designed for different insertion tasks playing important roles in automatic drug delivery, biopsy or radioactive seeds delivery. In this paper we have summarized different techniques for adequate modeling, control and stability analysis of the medical robots. The model of the robotic system, with the tasks described above, the entire system can be decomposed to the robotic subsystem and the environment subsystem. Modeling of the system by the method mentioned has been proved to be suitable when the force appears as a result of the interaction of the two subsystems. The mathematical model of the system has a singular characteristic. The singular system theory could be applied to the case described. It is well known that all mechanical systems have some delay. In that case a theory of singular systems with delayed states may be applied, as well. For the second phase in which there is no interaction, the dynamic behavior can be analyzed by the classic theory

Original scientific papers UDK 681

Summary: In this paper sufficient conditions for both practical and finite time stability of linear singular continuous time delay systems were introduced. The singular and singular time delay systems can be mathematically described as , respectively. Analyzing finite time stability, the new delay independent and delay dependent conditions were derived using the approaches based on Lyapunov-like functions and their properties on the subspace of consistent initial conditions. These functions do not need to be positive on the whole state space and to have negative derivatives along the system trajectories. When the practical stability was analyzed, the approach was combined with classical Lyapunov technique to guarantee the attractivity property of the system behavior. Furthermore, an LMI approach was applied to obtain less conservative stability conditions. The proposed methodology was applied and tested on a medical robotic system. The system was designed for different insertion tasks playing important roles in automatic drug delivery, biopsy or radioactive seeds delivery. In this paper we have summarized different techniques for adequate modeling, control and stability analysis of the medical robots. The model of the robotic system, with the tasks described above, the entire system can be decomposed to the robotic subsystem and the environment subsystem. Modeling of the system by the method mentioned has been proved to be suitable when the force appears as a result of the interaction of the two subsystems. The mathematical model of the system has a singular characteristic. The singular system theory could be applied to the case described. It is well known that all mechanical systems have some delay. In that case a theory of singular systems with delayed states may be applied, as well. For the second phase in which there is no interaction, the dynamic behavior can be analyzed by the classic theory

Dinamika sistema opisanih algebro-diferencijalnim jednačinama sa pomerenim argumentom, II deo

Okosnica ove monografije počiva na savremenoj teoriji upravljanja i postavlja i razrešava niz veoma složenih pitanja dinamike, posebnih klasa, sistema automatskog upravljanja, opisanih sistemom diferencijalnih (diferencnih) jednačina sa pomerenim argumentom. U tom smislu, izložen je ograničen broj radova eminentnih naučnika i osvedočenih autoriteta iz ove oblasti, kao i manji broj radova samih autora iz ove, uvek aktuelne i više nego provokativne, oblasti savremene teorije upravljanja i stabilnosti. Poseban doprinos monografije daje pregled upravljačkih struktura sa unutrašnjim modelima. Opisana su tri globalna prilaza sinteze sistema sa unutrašnjim modelima koja se odnose na korišćenje: principa unutrašnjeg modela IMP (Internal Model Principle), IMC (Internal Model Control) strukture i Tsypkinove IMPACT (Internal Model Principle and Control Together) strukture. Ovi prilazi su nastali potpuno nezavisno jedni od drugih, u različitim vremenskim razdobljima, kao plod rada mnoštva istraživača, angažovanih na rešavanju različitih upravljačkih problema. U ovim upravljačkim strukturama ugrađuju se elementi čiji karakter eksplicitno zavisi od modela poremećaja i/ili modela objekta upravljanja. Zavisno od primarnih upravljačkih ciljeva, razvijeni su različiti koncepti primene unutrašnjih modela u strukturnoj sintezi sistema. U monografiji su opisana svojstva struktura sa unutrašnjim modelima. Svi do sada razvijeni postupci strukturne sinteze upravljanih sporih industrijskih procesa mogu se svrstati u pomenute tri kategorije u zavisnosti od toga kako se i na kom principu unutrašnji modeli unose u upravljački deo strukture sistema automatskog upravljanja. Posebno poglavlje je posvećeno novoj upravljačkoj strukturi za upravljanje sporih industrijskih procesa sa transportnim kašenjenjem. Odlikuje je mali broj podešljivih parametara sa jasnim fizičkim značenjem i eksplicitnim uticajem na dinamičke i robustne performanse sistema sa zatvorenom povratnom spregom. Sistematizacija i strukturne i parametarske sinteze IMPACT strukture, i njenih osnovnih varijacija, je data u poslednjem poglavlju

Finite-Time Stability Analysis of Descriptor Discrete Time-Delay Systems Using Discrete Convolution of Delayed States

This paper provides sufficient conditions for the finite time stability of linear time invariant discrete descriptor time delay systems, mathematically described as Ex(k+1) = A(0)x(k) + A(1)x(t-h). A novel method was used to derive new delay dependent conditions. Stability of the system was analyzed using both the Lyapunov-like approach and the Jensen's inequality, including convolution of delayed states. The established conditions were applied to analysis of the system stability. In this case, the aggregation functional does not have to be positive in the state space domain and does not need to have the negative derivatives along the system trajectories. The system stability conditions were applicable to investigation of the finite time stability using the novel conditions proposed in this paper. This mathematical formulation guaranteed that the states of the systems do not exceed the predefined boundaries over a finite time interval