Feedback tracking control

The point of departure for the constructive method was the consideration of a passive control structure in the light of a suitable definition of finite-time motion stability. Then, the related dynamical inverse yielded the controller and a recursive algorithm to design the nominal batch motion. The underlying solvability conditions were identifying. The combination of the controller with an observer with a compatible structure yielded the design of the output feedback tracking controller. 

In Eq. (10-5), 1/Gp is the set point tracking controller. This is what we need if we install only a feedforward controller, which in reality, we seldom do.4 Under most circumstances, the change in set point is handled by a feedback control loop, and we only need to implement the second term of (10-5). The transfer function -GL/Gp is the feedforward controller (or the disturbance rejection... 

The set point tracking controller not only becomes redundant as soon as we add feedback control, but it also unnecessarily ties the feedforward controller into the closed-loop characteristic equation. 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

T.-C. Lee, K. Tai, Tracking Control of Unicycle-Modeled Mobile robots Using a Saturation Feedback Controller. IEEE Trans. Control Syst. Technol. 9(2), 305-318 (2001)... 

Servosystem The mechanism, which through feedback and control, keeps the laser beam on track and in focus on the disk—no easy task on a disk spinning at 4000 rpm. 

The constructive method, which is considered as a major breakthrough in control theory, was developed in the last decade. As it stands, the method is intended for feedback control design, and its application to the batch motion case requires the nominal output to be tracked and a suitable definition of finite-time batch motion stability. In a more applied eontext, the inverse optimality idea has been applied to design the nominal motion of homo [11] and copolymer [12] reactor, obtaining results that are similar to the ones drawn from direct optimization [4]. The motion was obtained from the recursive application of the process dynamical inverse [13], and the inverse yielded a nonlinear SF controller [9, 10] that was in turn used to specify a conventional feedforward-feedback industrial control scheme. However, the issues of motion stability and systematized search were not formally addressed. 

Setpoint changes can also be made, particularly in batch processes or in changing from one operating condition to another in a continuous process. These setpoint changes also act as disturbances to the dosedloop system. The function of the feedback controller is to drive the controlled variable to match the new setpoint. The dosedloop response to a setpoint disturbance is called the servo response (from the early applications of feedback control in mechanical servomechanism tracking systems). 

Centralized control can be also designed based on disturbance rejection or robustness requirements. In this case, the controller is not a static linear feedback law, as (45), but a dynamic feedback controller is obtained. Additionally, two degree of freedom controllers allow for a better control behavior in tracking and regulation. All these alternatives are beyond the scope of this introductory local control design treatment and are the subject of specialized references (see, for instance, [19]). 

In the following, the model-based controller-observer adaptive scheme in [15] is presented. Namely, an observer is designed to estimate the effect of the heat released by the reaction on the reactor temperature dynamics then, this estimate is used by a cascade temperature control scheme, based on the closure of two temperature feedback loops, where the output of the reactor temperature controller becomes the setpoint of the cooling jacket temperature controller. Model-free variants of this control scheme are developed as well. The convergence of the overall controller-observer scheme, in terms of observer estimation errors and controller tracking errors, is proven via a Lyapunov-like argument. Noticeably, the scheme is developed for the general class of irreversible nonchain reactions presented in Sect. 2.5. 

The ability to accurately model die pattern evolution as discussed in this paper provides a solution applicable to the ran by run control of multi-product patterned wafers [13]. As shown in Fig. 10, a feedback control loop incorporating the integrated density and step-height pattern dependent model was developed. For each device type, an appropriate set of model parameters (including effective blanket rate BR and planarization length) were determined. The model for the effective blanket rate includes a term Delta(n) that is updated on each run n to track the tool drift in rate over time due to pad and consumable wear ... 

Challenges in real-time process optimization mainly arise from the inability to build and adapt accurate models for complex physico-chemical processes. This paper surveys different ways of using measurements to compensate for model uncertainty in the context of process optimization. A distinction is made between model-adaptation methods that use the measurements to update the parameters of the process model before repeating the optimization, modifier-adaptation methods that adapt constraint and gradient modifiers, and direct-input-adaptation methods that convert the optimization problem into a feedback control problem. This paper argues in favor of modifier-adaptation methods, since it uses a model parameterization, measurements, and an update criterion that are tailored to the tracking of the necessary conditions of optimality. 

The proposed strategies for stabilization of gas-lifted oil wells are offline methods which are unable to track online dynamic changes of the system. However, system parameters such as flow rate of injected gas and also noise characteristic are not constant with respect to time. An adaptive Linear Quadratic Gaussian (LQG) approach is presented in this paper in which the state estimation is performed using an Adaptive Unscented Kalman Filter (AUKF) to deal with unknown time-varying noise statistics. State-feedback gain is adaptively calculated based on Linear Quadratic Regulator (LQR). Finally, the proposed control scheme is evaluated on a simulation case study. 

The dynamics of the molding process are determined through control of different but related machine elements such as motors, heaters, servovalves, etc. These machine elements are typically controlled via a hierarchical closed loop control architecture as shown in Fig. At the innermost level, only the machine elements are regulated by real time comparison of the desired machine set points with the machine feedback, such that the difference (or error) is used to correct the process. At the second level, state variables such as melt temperature and melt pressure are controlled to track prespecified profiles and provide more precise control of the state of the melt. At the outermost level, the machine inputs are adjusted by the machine operator to improve the quality of the part through specification of better set points given feedback of part quality. 

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