Time-delay systems : Lyapunov functionals and matrices by Vladimir Kharitonov
By Vladimir Kharitonov
Half 1. Retarded kind structures -- basic concept -- unmarried hold up case -- a number of hold up case -- structures with disbursed hold up -- half 2. impartial style platforms -- basic concept -- Linear platforms -- disbursed hold up case
Read Online or Download Time-delay systems : Lyapunov functionals and matrices PDF
Similar control systems books
Strength trade is a big beginning of the dynamics of actual platforms, and, therefore, within the learn of advanced multi-domain structures, methodologies that explicitly describe the topology of power exchanges are instrumental in structuring the modeling and the computation of the system's dynamics and its regulate.
In an period of in depth festival the place plant working efficiencies needs to be maximized, downtime as a result of equipment failure has turn into extra high priced. to chop working expenditures and elevate sales, industries have an pressing have to are expecting fault development and closing lifespan of commercial machines, techniques, and platforms.
That includes a model-based method of fault detection and analysis in engineering structures, this ebook includes updated, useful details on fighting product deterioration, functionality degradation and significant equipment harm. ;College or college bookstores may perhaps order 5 or extra copies at a distinct pupil cost.
- Cyber-security of SCADA and Other Industrial Control Systems (Advances in Information Security)
- System Identification and Adaptive Control: Theory and Applications of the Neurofuzzy and Fuzzy Cognitive Network Models (Advances in Industrial Control)
- Adaptive Voltage Control in Power Systems: Modeling, Design and Applications (Advances in Industrial Control)
- IP Network-based Multi-agent Systems for Industrial Automation: Information Management, Condition Monitoring and Control of Power Systems
Extra resources for Time-delay systems : Lyapunov functionals and matrices
Proof. 16). Since the matrix U(τ ) satisfies the properties, the functional v0 (ϕ ) satisfies Eq. 5) as well. 5 justifies the following definition. 5. 16). On the one hand, the new definition makes it possible to overcome the first limitation of the original definition of the Lyapunov matrices – the exponential stability assumption. 11)? The following statement provides an affirmative answer to this question. 6. 1) be exponentially stable. 11) is the unique solution of Eq. 16). Proof. 11) satisfies Eq.
1) in terms of their initial functions. 1 Fundamental Matrix The key element needed to derive this expression is the fundamental matrix of the system. 1) as the matrix exponent does for linear delay-free systems. 1 (). 1) if it satisfies the matrix equation d K(t) = K(t)A0 + K(t − h)A1, dt t ≥ 0, and K(t) = 0n×n for t < 0, K(0) = I. Here I is the identity matrix. 1. The fundamental matrix also satisfies the matrix equation d K(t) = A0 K(t) + A1K(t − h), dt t ≥ 0. This does not mean that matrix K(t) commutes individually with the coefficient matrices Ak , k = 0, 1.
Let us define the positive-definite function v1 (x) = α1 x 2 and the positivedefinite functional v2 (ϕ ) = α1 ϕ 2h . 5. 3) is uniformly stable, and for every ε > 0 there exists δ (ε ) > 0 such that the inequality ϕ h < δ (ε ) implies x(t,t0 , ϕ ) < ε for t ≥ t0 . 4. To this end, assume that t0 ≥ 0 and ϕ ∈ PC ([−h, 0], Rn ) , ϕ h < Δ0 . The corresponding solution x(t,t0 , ϕ ) is such that x(t,t0 , ϕ ) < H, for t ≥ t0 . The second condition of the theorem implies the inequality v(t, xt (t0 , ϕ )) ≤ v(t0 , ϕ )e−2σ (t−t0 ) , t ≥ t0 .