Multi-Objective Optimal Control Design For Nonlinear Dynamical Systems

Dynamical systems are mathematical structures whose aim is to describe the evolu-tion of an arbitrary deterministic system through time.In general,dynamical system can be divided into linear and nonlinear.Linear systems obey the superposition principle,the output of the device is propor-tional to its input.Systems with this property are governed by linear differential equations.A major subclass is systems which in addition have parameters which do not change with time,called linear time invariant?LTI?systems.These systems are amenable to power-ful frequency domain mathematical techniques of great generality,such as the Laplace transform,Fourier transform,and Z transform,root-locus,Bode plot,and Nyquist sta-bility criterion.These lead to a description of the system using terms like bandwidth,frequency response,eigenvalues,gain,resonant frequencies,poles,and zeros,which give solutions for system response and design techniques to most problems of interest.For linear dynamical systems with control,there are a lot of control methods designed in time domain can let the system to run along the desired trajectories,such as proportional-integral-derivative?PID?,Linear Quadratic Regulator?LQR?and full state feedback and so on.Nonlinear systems are often governed by nonlinear differential equations.Nonlinear control theory covers a wider class of systems that do not obey the superposition principle.The mathematical techniques which have been developed to handle them are more rigor-ous and much less general,often applying only to narrow categories of systems.These include limit cycle theory,Poincar?e maps,Lyapunov stability theory,and describing func-tions.If only solutions near a stable point are of interest,nonlinear systems can often be linearized by approximating them by a linear system obtained by expanding the nonlinear solution in a series,and then linear techniques can be used.Nonlinear systems are often analyzed using numerical methods on computers,for example by simulating their oper-ation using a simulation language.Even if the plant is linear,a nonlinear controller can often have attractive features such as simpler implementation,faster speed,more accura-cy,or reduced control energy,which justify the more difficult design procedure.Sliding mode control?SMC?is a powerful nonlinear control technique.The basic idea of SMC is to drive the system to a sliding surface,a manifold in the state space,and let the system move along the sliding surface towards the desired steady state.Uncer-tainties in dynamical and control systems are unavoidable.Uncertainty can be divided into parameter uncertainties and uncertainties due to un-modeled dynamics.Robust con-trols represent a class of methods to handle uncertainties.The sliding mode control is a powerful robust nonlinear control technique for systems with uncertainties.The multi-objective optimization of SMC design usually aims at determining several free parameters of the controller to meet the performance requirements in time or frequen-cy domain.These performance requirements are often conflicting.Also,the optimization should consider the hardware capability such as saturation and parameter uncertainty,which makes the multi-objective optimal SMC design more challenging and interesting.The multi-objective optimal control design problems usually form a set in the pa-rameter space,called Pareto set.Many algorithms for solving multi-objective optimiza-tion problems?MOPs?have been developed.There are biologically inspired optimization algorithms such as Genetic Algorithm?GA?,Ant Colony Optimization?ACO?,Immune Algorithm?IA?and Particle Swarm Optimization?PSO?.Converting a MOP to a single-objective optimization problem?SOP?with the steepest descend search is another way to solve MOPs.The cell mapping methods originally developed by Hsu in the 1980s for global analysis of nonlinear dynamical systems are found to be highly effective in discov-ering the global structure of the Pareto set,the solution of the multi-objective optimization problems?MOPs?.The cell mapping methods have been successfully applied to low and moderate dimensional problems and the multi-objective optimal PID control design for linear and nonlinear dynamical systems.Furthermore,cell mapping methods are naturally parallelization in computing.The parallel simple cell mapping method is highly effective in obtaining the global solutions of high-dimensional MOPs.In particular,we apply the method to the multi-objective optimal design problem of the sliding mode control.We consider the post-processing of the Pareto set of the multi-objective optimal control designs by using the improved K-means cluster analysis method.We then propose a switching strategy to select a control from the identified clusters.For the first time in the literature,we apply these two seemingly unrelated techniques to develop a post-processing and implementation strategy of the Pareto optimal controls designed by the method of multi-objective optimization.There is a growing interest in analysis,control and applications of dynamical sys-tems with time delay.Time delay can be part of the system property.It can also arise in the control system when there is a signal transport time lag or the delay due to the hardware of the digital controller.Many engineering applications involve time delay.The semi-discretization?SD?is a well stablished method in the literature and used widely in structural and fluid mechanics.The method has been applied to delayed dynamical sys-tems by Insperger,and later extended to study delayed feedback controls.The continuous time approximation?CTA?method is an extension of the method of semi-discretization and provides an alternative to handle multiple independent time delays.The Chebyshev CTA method provides the most accurate solution of time-delayed dynamical systems.Sliding mode control?SMC?has also been applied to time-delayed systems.We investigate two control design methods:the optimal feedback gain with the semi-discretization method and a high-order control design.Both simulations and experiments are carried out to demonstrate the utility of the control.The semi-discretization method offers an optimal control without increasing the dimension of the gain vector,while the high-order control involves an increased number of gains.The disadvantages and advan-tages of both the methods are discussed with the support of simulation and experimental results.The paper highlights the fact that the high-order control is determined by an N^th order filter where N is the discretization level.We have also found that the performance of the high-order control appears to be insensitive to N.We also present the simulation and experimental studies for the nonlinear time-delayed dynamical systems with uncer-tainties.The system is assumed to have constant delay time and uncertain parameters with known upper and lower bounds.We design an optimal sliding surface for the sliding mode control.Simulations and experiments are carried out to demonstrate the utility of the control method.