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Search Methods
Published in Yogesh Jaluria, Design and Optimization of Thermal Systems, 2019
Several methods have been developed with this general approach to solve constrained optimization problems. These include the constrained steepest descent (CSD) method, the method of feasible directions, the gradient projection method, and the generalized reduced gradient (GRG) method. Many efficient algorithms have been developed to obtain the optimum with the least number of trials or iterations. Some of these are available in the public domain, while others are available commercially. The difference between all these methods lies in deciding on the direction of the move and the scheme used to return to the constraint. The major problem remains the calculation of the gradients. Linearization of the nonlinear optimization problem is also carried out in some cases, and linear programming techniques can then be used for the solution. For details on these and other methods, see Arora (2004), Bertsekas (2016), and the various other references mentioned earlier.
Phenomenological Creep Models of Fibrous Composites (Probabilistic Approach)
Published in Leo Razdolsky, Phenomenological Creep Models of Composites and Nanomaterials, 2019
Tom Caughey was a pioneer in the development of the stochastic equivalent linearization procedure for estimating the mean and variance of a non-linear system to random variables. The stochastic equivalent linearization procedure or statistical linearization procedure was almost simultaneously introduced more than fifty years ago by three independent investigators: Booton [9], Kazakov [10], and Caughey [11]. In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamic systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. Linearization is an effective method for approximating the output of a function y = f(x) at any x = a based on the value and slope of the function at x = b, given that y = f(x) is continuous on [a,b] (or [b,a]) and that a is close to b. In, short, linearization approximates the output of a function near x = a. The concept of local linearity applies to the most of points arbitrarily close to x = a, and the slope M should be, most accurately, the slope of the tangent line at x = a.
Modeling of Dynamic Systems
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
The most popular and general approach employed in modeling nonlinear systems is small signal linearization. It is applicable for a broad range of systems and is extensively used in industry. Linearization is the process of finding a linear model that approximates a nonlinear one. If a small signal linear model is valid near an equilibrium and is stable, then there is a region (usually very small) containing the equilibrium within which the nonlinear system is stable. Thus one can make a linear model and design a linear control for it such that, in the neighborhood of the equilibrium, the design will be stable. Since a very important role of feedback control is to maintain the process variables near equilibrium, such small-signal linear models are a frequent starting point for control models.
Nonlinear Optimal Control for the Translational Oscillator with Rotational Actuator
Published in Cybernetics and Systems, 2021
The dynamic model of the TORA undergoes linearization around the temporary operating point where is the present value of the system’s state vector and is the most recent control input that was applied to it. The linearization relies on Taylor series expansion and on the computation of the system’s Jacobian matrices. The initial state-space model of the system is given in Eq. (12). By applying approximate linearization, one arrives at the equivalent state-space description where is the cumulative disturbances vector while the Jacobian matrices A and B are given by
Stealth identification strategy for closed loop system structure
Published in International Journal of Systems Science, 2020
Hong Wang-jian, Ricardo A. Ramirez-Mendoza
Since linearisation is an approximation in the neighbourhood of an operating point, it can only predict the local behaviour of the nonlinear system in the vicinity of that point. Consider the problem of linear approximation to that nonlinear system , firstly we need to solve one equilibrium point or nominal operating point to satisfy . As the linear approximation to the nonlinear system is valid at most in a region or vicinity close to the normal operating point , so the existence and uniqueness of the normal operating point is not guaranteed unless some restrictions are placed on the nature of that nonlinear system .
Non-linear optimal control for four-wheel omnidirectional mobile robots
Published in Cyber-Physical Systems, 2020
G. Rigatos, K. Busawon, M. Abbaszadeh, P. Wira
The state-space model of the omnidirectional mobile robot undergoes approximate linearisation around the temporary operating point , where is the present value of the system’s state vector and is the last sampled value of the control inputs vector. The linearisation point is updated at each time-step of the control algorithm. The linearisation relies on Taylor series expansion and on the computation of the associated Jacobian matrices. The approximately linearised model is rewritten as: