Brad Lehman

Vibrational control of nonlinear time lag systems with bounded delay: averaging theory, stabilizability, and transient behavior

This paper develops the theory of vibrational control of nonlinear time lag systems with arbitrarily large but bounded delay. Averaging theory for fast oscillating, differential delay equations is presented and then applied to vibrational control. Conditions are given which ensure the existence of parametric vibrations that stabilize nonlinear time lag systems. Transient behavior is also discussed. Illustrative examples are given which show 1) the feasibility of the theory to important applications and 2) the differences in the theory presented and the existing known theory for vibrational control of ordinary differential equations. >

Delay independent stability conditions and decay estimates for time-varying functional differential equations

This paper presents sufficient delay independent conditions that guarantee stability of nonlinear time varying functional differential equations (FDEs). Estimates on the rate of decay of solutions are also obtained. >

Stability of a continuous stirred reactor with delay in the recycle stream

The authors discuss the local stability properties of a first-order exothermic irreversible reaction carried out in a well-mixed continuously stirred tank reactor (CSTR) with a delayed recycle stream. Surprising results are obtained which suggest that, in a first-order, irreversible, exothermic reaction A to B carried out in a CSTR, the delay in the recycle stream does not affect stability of steady states. The delay still may cause increased oscillation and, perhaps, a slower response time; however, local stability is unchanged. This may allow the engineer the ability to ignore the recycle delay in the model when developing a control scheme.<<ETX>>

Nonlinear control designs for systems with bifurcations with applications to stabilization and control of compressors

This paper studies the effects of applying both nonlinear feedback and time-varying control algorithms in the form of inlet flow disturbances by introducing variations in the shut-off head of a single stage axial compressor. The closed loop controller utilizes nonlinear feedback in order to stabilize rotating stall. Using bifurcation analysis, it is possible to analytically guarantee stability of local bifurcated solutions near the stall point. Physically, this means that the system will no longer jump to a large amplitude oscillatory mode when bifurcating at the stall point. Thus the possible operating region of the axial compressor is enlarged. The open loop control strategy provides oscillatory input to the shut-off head coefficient. For some model systems, this type of strategy is shown to be effective in enhancing stability margins and improving efficiency. Methods of implementing both types of control laws using variations inlet guide vanes and bleed valves and by blowing air into the compressor are under investigation.

Vibrational control of linear time lag systems with arbitrarily large but bounded delays

It is shown that vibrational stabilization can be effective for linear systems with large bounded delays. Theorems are given that define the procedures for the search of the stabilizing vibrations. Robust oscillatory stabilization insensitive to the delay size is also shown to take place for some classes of systems. >

Stability of chemical reactions in a CSTR with delayed recycle stream

This paper shows how recycling delays affect the dynamics of certain types of exothermic reactions in continuous stirred tank reactors (CSTR). We only consider a special type of recycle in which all the chemical specie at the output are recycled at an identical rate.

Extensions of classical averaging techniques to delay differential equations

This paper extends the method of averaging due to Krylov and Bogoliubov (1947) and Bogoliubov and Mitropolskii (1961) to delay differential equations. Near identity change of variables are used to transform time varying delay differential equations into autonomous delay differential equations plus small perturbations. Then Lyapunov functionals are used to relate the autonomous averaged delay differential equation to the original time varying delay differential equation.<<ETX>>

Vibrational control of nonlinear time lag systems with arbitrarily large but bounded delay: averaging theory, stabilizability, and transient behavior

The authors develop the theory of vibrational control of nonlinear time lag systems with arbitrarily large but bounded delay. The averaging theory for fast oscillating differential delay equations is presented and then applied to vibrational control. Conditions are given which ensure the existence of parametric vibrations that stabilize nonlinear time lag systems. Transient behavior is also discussed. Illustrative examples are given which show (1) the feasibility of the theory for important applications and (2) the differences in the theory presented and the known theory for vibrational control of ordinary differential equations.<<ETX>>

Vibrational stabilization and calculation formulas for nonlinear time delay systems: linear multiplicative vibrations

Abstract This paper shows that vibrational control is an effective means of stabilization of nonlinear time lag systems with arbitrarily large but bounded delays. Conditions for stabilizing are formulated and procedures for the synthesis of the corresponding stabilizing vibrations are proposed.

Application of vibrational control to chemical reactions in a CSTR with delayed recycle stream

Vibrational control is a method of modification of dynamic properties obtained by introducing zero mean oscillations into a system parameters. This paper shows that by introducing oscillations into the flow rates of a continuous stirred tank reactor (CSTR) with delayed recycle stream, it is possible to operate reactions in the vicinity of previously unstable steady states.