A Novel PDF Shape Control Approach for Nonlinear Stochastic Systems

In this paper, a self-tuning hierarchical controller in which a Fuzzy logic controller supervises the control actions of a conventional PID has been proposed, implemented and presented. The controller has been applied to a control study of Fluid Catalytic Cracking (FCC) unit riser temperature, and regenerator temperature respectively. Performance comparison of the proposed Fuzzy-PID controller and the conventional PID was made in simulation studies of regulatory and servo performances of the two controller types. Six performance measures: Percent overshoot (OS), settling time (ST), integral absolute error (IAE), integral square error (ISE), integral time absolute error (ITAE) and integral time square error (ITSE) were employed as the tools for performance comparison between the conventional PID and the Fuzzy-PID controller. For the tracking of riser temperature with a set point at 524oC, the performance indicators under PID control gave the following results overshoot (14.5%); settling time (40 seconds) Integral absolute error (8.246), integral square error (3.3); integral time absolute error(1762);integral time square error (43.77) while for the same indicators under Fuzzy-PID control the following values: overshoot (3.3%); settling time (40 seconds) ;Integral absolute error (8.811); integral square error (14.5); integral time absolute error(280),;integral time square error (31.11) .The results allude to the superiority of the fuzzy-PID scheme over the PID scheme in tracking the optimal set point of riser temperature. More so, for tracking the regenerator set point temperature of 746oC, comparative study of step response under the two schemes gave the following results in six performance indicators: overshoot (PID(12.6%)/Fuzzy-PID(6%)); settling time (PID(80 seconds)/Fuzzy-PID(20seconds));Integral absolute error(PID(14.29)/Fuzzy-PID(8.63)); integral square error(PID(6.713).Fuzzy-PID(4.506)); integral time absolute error(PID(2858)/Fuzzy-PID(305.9)), integral time square error (PID(77.55)/Fuzzy-PID(33.05)).Moreover, the fuzzy-PID controller also showed superior performance over the conventional PID controller in terms disturbance rejection (regulatory response) of both riser and regenerator temperature. The results from this study suggest that the application of fuzzy-PID scheme to temperature control offers good promise of improved fluid catalytic cracking unit (FCCU) operations.

In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochastic differential equations (SDE), therefore the Fokker-Planck-Kolmogorov (FPK) equation is expressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of δ-correlated excitations, the existence of multiplicative excitations, and the dimension of SDE or FPK equation. Examples are given and numerical results are provided for comparing with known exact solution to show the effectiveness of the method.

Theory of marginally unstable stochastic systems, recently developed for linear systems with lumped parameters, is applied here to a nonlinear single-degree-of-freedom system. The name “marginal instability” is used for nominally stable systems with potential short-term instability due to temporal random variations of parameter(s) with brief excursions out of the stability domain. Prediction of the corresponding brief outbreaks in response is based on parabolic approximation of the parameter variation during excursion (asymptotic analysis of the Slepian model). The theory provides probabilistic description of the response (such as response probability density function (PDF) or solution to the first-passage problem) in terms of that of the parameter(s) variations. Systems with nonlinear restoring forces are considered in this Note with the use of quasiconservative version of averaging over response period. Presented results illustrate influence of nonlinearities on the response characteristics which are important for reliability evaluation.

Sufficient conditions for the controllability of nonlinear stochastic fractional boundary control systems are established. The equivalent integral equations are derived for both linear and nonlinear systems, and the control function is given in terms of the pseudoinverse operator. The Banach contraction mapping theorem is used to obtain the result. A controllability result for nonlinear stochastic fractional integrodifferential systems is also attained. Examples are included to illustrate the theory.

An intensity-modulated Gaussian process {x(t)} is defined as the product of a non-negative stationary random process {σx(t)} and a stationary Gaussian process {z(t)} that is statistically independent of {σx(t)}. When such an intensity-modulated process is the excitation to a linear time-invariant system, the response process {y(t)}, conditioned on the excitation-modulating function σx(t), is a generally nonstationary Gaussian process. This property is used to develop an approximation to the probability density function (PDF) of the system response by assuming that the time-varying conditional response variance σ2y(t), conditioned on the excitation-modulating function σx(t), is governed by the gamma PDF. The resulting response process PDF is characterized by two parameters: the mean and variance of the conditional response variance σ2y(t). An alternative approximation to the response PDF, characterized by these same two parameters, is developed by using a Taylor’s series representation in the variable σ2y of the response PDF conditioned on σ2y. This latter PDF approximation is shown to be identical to the Gram–Charlier series of type A through terms involving the Hermite polynomial of degree 6. The two response PDF approximations are shown to give essentially identical results for values of the coefficient of variation of σ2y of one-half or less. Simple, easily evaluated expressions for the mean and variance of σ2y are derived in terms of measurable quantities by utilizing the assumption that fluctuations in σx(t) occur slowly in comparison with those of z(t).

Nonlinear stochastic seismic dynamic response analysis of submerged floating tunnel subjected to non-stationary ground motion

Adaptive fault-tolerant shape control for nonlinear Lipschitz stochastic distribution systems

Random vibration of pretensioned rectangular membrane structures under heavy rainfall excitation

A generalized method for the stationary probabilistic response of nonlinear dynamical system

An efficient method for solving high-dimension stationary FPK equation of strongly nonlinear systems under additive and/or multiplicative white noise

The operation of dynamic systems is often accompanied by abrupt and random changes in their configurations, which will dramatically change the stationary probability density function of their response. In this article, an effective procedure is proposed to reshape the stationary probability density function of nonlinear stochastic systems against abrupt changes. Based on the Markov jump theory, such a system is formulated as a continuous system with discrete Markov jump parameters. The limiting averaging principle is then applied to suppress the rapidly varying Markov jump process to generate a probability-weighted system. Then, the approximate expression of the stationary probability density function of the system is obtained, based on which the reshaping control law can be designed, which has two parts: (i) the first part (conservative part) is designed to make the reshaped system and the undisturbed system have the same Hamiltonian; (ii) the second (dissipative part) is designed so that the stationary probability density function of the reshaped system is the same as that of undisturbed system. The proposed law is exactly analytical and no online measurement is required. The application and effectiveness of the proposed procedure are demonstrated by using an example of three degrees-of-freedom nonlinear stochastic system subjected to abrupt changes.

An approximation theory of optimal control for nonlinear stochastic dynamic systems has been established. Based on the generalized Hamilton-Jacobi-Bellman equation of the cost function of nonlinear stochastic systems, general iterative procedures for approximating the optimal control are developed by successively improving the performance of a feedback control law until a satisfactory suboptimal solution is achieved. A successive design scheme using upper and lower bounds of the exact cost function has been developed for the infinite-time stochastic regulator problem. The determination of the upper and lower bounds requires the solution of a partial differential inequality instead of equality. Therefore it provides a degree of flexibility in the design method over the exact design method. Stability of the infinite-time sub-optimal control problem was established under not very restrictive conditions, and stable sequences of controllers can be generated. Several examples are used to illustrate the applicati on of the proposed approximation theory to stochastic control. It has been shown that in the case of linear quadratic Gaussian problems, the approximation theory leads to the exact solution of optimal control.

Adaptive fuzzy decentralized control for stochastic large-scale nonlinear systems with unknown dead-zone and unmodeled dynamics

Higher order nonlinear systems with prevailing delay have been very challenging in significance with stability and process performance. This paper investigates the performance of a type-2 Mamdani intelligent controller implemented a delay dominant system which is modelled using black box approach identified to be a second order nonlinear model for a dual spherical tank liquid level system (DSTLLS) under LabVIEW environment. The adaptive approach of the intelligent Mamdani-based fuzzy controller proves itself to be very competent compared with the earlier experimented methods that already exist. The performance indices like integrated absolute error (IAE) and integrated squared error (ISE) are also calculated for different varying set point changes of DSTLLS. The response and error reduction efficiency of this method of adaptive type-2 intelligent fuzzy (ATIF) controller has been experimented for different flow configurations of this DSTLLS, like multiple input single output (MISO), multiple input multiple output (MIMO) and single input single output (SISO).

Constructing transient response probability density of non-linear system through complex fractional moments