Nonlinear control systems - A brief overview of historical and recent advances <s> Introduction <s> 1. Background and Preview. 2. Highlights of Classical Control Theory. 3. State Variables and the State Space Description of Dynamic Systems. 4. Fundamentals of Matrix Algebra. 5. Vectors and Linear Vector Spaces. 6. Simultaneous Linear Equations. 7. Eigenvalues and Eigenvectors. 8. Functions of Square Matrices and the Cayley-Hamilton Theorem. 9. Analysis of Continuous and Discrete Time State Equations. 10. Stability. 11. Controllability and Observability for Linear Systems. 12. The Relationship between State Variable and Transfer Function Descriptions of Systems. 13. Design of Linear Feedback Control Systems. 14. An Introduction to Optimal Control Theory. 15. An Introduction to Nonlinear Control Systems. <s> BIB001 </s> Nonlinear control systems - A brief overview of historical and recent advances <s> Introduction <s> Covers in a progressive fashion a number of analysis tools and design techniques directly applicable to nonlinear control problems in high performance systems (in aerospace, robotics and automotive areas). <s> BIB002 </s> Nonlinear control systems - A brief overview of historical and recent advances <s> Introduction <s> Pressing demands of productivity and accuracy in today’s robotic applications have highlighted an urge to replace classical control strategies with their modern control counterparts. This recent trend is further justified by the fact that the robotic manipulators have complex nonlinear dynamic structure with uncertain parameters. Highlighting the authors’ research achievements in the domain of manipulator design and control, this paper presents a systematic and comprehensive review of the state-of-the-art control techniques that find enormous potential in controlling manipulators to execute cuttingedge applications. In particular, three kinds of strategies, i.e., intelligent proportional-integral-derivative (PID) scheme, robust control and adaptation based approaches, are reviewed. Future trend in the subject area is commented. Open-source simulators to facilitate controller design are also tabulated. With a comprehensive list of references, it is anticipated that the review will act as a firsthand reference for researchers, engineers and industrialinterns to realize the control laws for multi-degree of freedom (DOF) manipulators. <s> BIB003
Real world systems are inherently nonlinear in nature at least when considered over wide operating range although many of the systems are assumed to 'behave' as linear in the vicinity of a certain operating point at low speeds under certain assumptions. Many physical processes are represented by nonlinear models. Examples include; coulomb friction, gravitational and electrostatic at-traction, voltage-current characteristics of most electronic systems and drag on a vehicle in motion. Recently, many researchers from such broad areas like process control, biomedical engineering, robotics, aircraft and spacecraft control have shown an active interest in the design and analysis of nonlinear control strategies. Thus, most of the real problems necessitate invariably bumping into nonlinearities BIB003 . Primary reasons behind growing interest in nonlinear control include BIB002 ; improvement of linear control systems, analysis of hard nonlinearities, need to deal with model uncertainties and design simplicity. Nonlinear strategies improve trivial approaches by taking into accounts the dynamic forces like centripetal and Coriolis forces which vary in proportion to the square of the speed. So, the linear control laws put a serious constraint on speed of motion to achieve a specified accuracy. However, simple nonlinear controller can reasonably compensate the nonlinear forces thus achieving high speed in an ample workspace. Also, hard nonlinearities like dead-zones, hysteresis, Coulomb friction, stiction, backlash and saturation do not permit linear approximation of real-world systems  . After predicting these nonlinearities, nonlinear approaches properly compensate these to achieve unmatched performance. Moreover, real systems often exhibit uncertainties in the model parameters primarily due to sudden or slow change in the values of these parameters. A nonlinear controller through robustness or adaptability can handle the consequences due to model uncertainties  . The systems that are not linearly controllable or observable may be controllable/observable in a nonlinear sense. Modern technology such as high accuracy high speed robots require fulfilling strict design requirements. The positioning of such robots is a nonlinear problem since it involves coordinate transformation matrices having sine and cosine terms. The linear control of the manipulators assumes each joint to be independent and considers the inertia 'seen' by each joint as a constant. This approximation leads to non-uniform damping throughout the workenvelope and results other undesirable effects  . Considering a 6 Degree Of Freedom (DOF) manipulator, the over performance of a nonlinear strategy particularly in the presence of a disturbance is evident from Fig. 1 where a bounded matched disturbance (Fig. 1a) is fed to the manipulator and tracking performance (Fig. 1b) of a nonlinear approach Sliding Mode Control (SMC) is compared with a linear strategy i.e. Computed Torque Control with Proportional Integral Derivative (CTC-PID). Non applicability of superposition and homogeneity in case of nonlinear systems results in major implications on the analysis and design of the control systems BIB001 . The straight forward relationship between transfer function zero(s) and pole(s) locations and time response does not hold valid in general. An unforced nonlinear system can possess limit cycles not speculated by linear theory. The controllability and observability cannot be determined simply based on rank tests. Moreover, stability of a system is no longer just a simple function of eigenvalue locations since nonlinear systems may have multiple stable/unstable equilibrium points. Owing to the pertinent importance of nonlinear control in wide range of recent applications, this paper presents a brief comment on the subject topic. Interested readers are encouraged to refer to the original literature cited for more specific details. The remaining paper is structured as follows: Section 2 briefly presents historical perspective of nonlinear control systems while recent advances and challenges are discussed in Section 3 and 4 respectively. Finally Section 5 comments on conclusion.