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Youngjin Choi For Jinhwa and Kiseo, my family. Though it must be a widely accepted controller for mechanical control systems, it is still short of theoretical bases, e. These subjects will be thoroughly discussed in this book. There are two neuberechhnung to solve a class of HJ equations: Neuberrechnung, there are two control methods according to the objective: This book is focusing on an inverse optimization method and the trajectory tracking control system.

Second, the PID control is shown to be neubfrechnung optimal one for a performance index found inversely from a class of HJ equations. It is proved that an inverse optimal PID control exists if and only if the mechanical system is extended disturbance input-to-state stable ISS. Also, selection guidelines for gains are suggested using the time derivative of Lyapunov function.

VIII Preface Third, the performance tuning methods of PID control are derived from performance limitations expressed by a state vector and composite error, respectively. They are suggested for bikubischee tracking models of mechanical systems.

To begin with, two simple performance tuning rules are suggested for an inverse optimal PID controller: And then, the compound performance tuning rule is suggested unifying both square tuning and linear one. Fourth, an automatic performance tuning method of PID control is derived from the direct adaptive control scheme, because above three tuning methods are all passive in that the control performance can not be directly estimated by them because they are composed of proportional relations.

Also, it is proved that the control performance is enhanced by applying an auto-tuning law to the PID controller. The experimental results show the validity of auto-tuning law for PID control.

Finally, PID state observer is suggested to accomplish output feedback PID controller, similar to the design method for high-gain observer. The observer gains of PID state neuberedhnung are found from the robustness against disturbances.

Also, it is proved that the output feedback PID control system including a PID state observer can be disturbance input-to-state stable under the condition for observer bikbuische. State-Space Description of Lagrangian Neubsrechnung.

Square and Linear Rules. Performance Estimation bikubisdhe Optimality. Desired Trajectory of Robot Manipulator System. Target performance and non-tuning region. Performance of auto-tuning PID controller.

Composite error and auto-tuned gains. PD plus linear state observer.

### PID trajectory tracking control for mechanical systems – PDF Free Download

Global Asymptotic Stability Hamiltonian system: Hamiltonian control system HJ: Lagrangian control system LQG: Uniform Ultimate Boundedness 1 Introduction 1. In PID control, the P-control Proportional control is the present effort making a present state into desired state, I-control Integral control is the accumulated effort using the experience information of bygone state and D-control Derivative control is the predictive effort reflecting the tendency information for ongoing state.

Also, PID control has been utilized irrespective of linear system or nonlinear one, electrical system or mechanical one, time-invariant system or time-varying one. Though the PID control has been widely accepted in industry, PID control itself is still short of the theoretical basis, e.

For electrical control systems, there have been a lot of control methods because the linear characteristics is dominant near operating region, e. On the contrary, since most mechanical systems show nonlinear characteristics generally, there has been only a few control methods for mechanical systems.

The representative control method for nonlinear mechanical systems is a feedback linearization method such as computed-torque one. The feedback linearization method is applicable to the case that the mechanical parameters such as moment of Inertia, center of mass and a variety of friction parameters are all identified exactly.

If they are not possible to be identified all, then the feedback linearization with an adaptive Y. Springer-Verlag Berlin Heidelberg 4 1 Introduction control scheme in [23, 30, 56] can be applied to nonlinear systems. Though the adaptive control is useful for uncertain mechanical systems, it has brought tremendous complexity in controller. According to the survey reported in [35], since n-link robot manipulators have 10n parameters to be identified by adaptation, a great deal of algebraic calculations are required to make use of an adaptive control according to the increase of the number of linkage.

The simple control is preferable to the complex control at least in industry, if the performance enhanced by using complex control does not stand in different levels. Inspired by this idea, the author tried to extend the theoretical basis of PID control to both robust control theory and direct adaptive ones including several issues such as optimality, performance limitation, performance tuning rule, auto-tuning law, and observer design.

After that Takegaki and Arimoto had proved the global asymptotic stability GAS by PD control in [60], many kinds of stability have been proved under various conditions as shown in Table 1.

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Even for the mechanical system under gravity effect, Tomei proved the GAS of PD control in [63] by using an adaptation for gravity term. On the other hand, Ortega et al.

Recently, Ramirez et al. The trajectory tracking control system is quite different from the regulation control system in that the desired configurations which are functions of time are inserted in equation of motion for mechanical system.

Hence, the trajectory tracking system has the time-varying characteristics because of desired configurations. Also, Berghuis and Nijmeijer proposed the output feedback PD control which brought the semiglobal uniform ultimate boundedness SGUUB in [7] under gravity and a bounded disturbance. Recently, Choi et al. As shown in Table 1. These characteristics will be discussed in chapter 3. Chapter 2 describes the characteristics of mechanical system.

Also, optimal control methods for mechanical system are discussed from the viewpoint of both direct and inverse optimization approaches. The direct one is an approximate method and an inverse one has been used in [18, 47, 50] based on the primitive work by Kalman. Some kinds of Hamilton-Jacobi equations, e.

In chapter 3, it is discussed that the PID control for mechanical systems can be optimal for what performance index. Relations among the chapters In chapter 4, two simple performance tuning rules are derived from the performance limitation of PID control system, e. Actually, square tuning rule and linear one have an obvious importance in an engineering sense because of their simplicity. And then, a compound performance tuning rule is suggested unifying both square rule and linear one.

Also, experimental results for a robotic manipulator show the effectiveness of those tuning rules.

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In chapter 5, an automatic performance tuning rule is proposed for mechanical systems, with an extension of the idea about above passive tuning rules derived from the performance limitation. A lot of automatic performance tuning methods for PID control system have been proposed for the chemical process control systems as we can see in [3, 66], however, they can not be directly applied to mechanical systems because most process systems show very slow responses and time-delay.

According to the direct adaptive control scheme, the auto-tuning law of PID controller and the performance improvement enhanced by using an auto-tuning law are suggested in this chapter. In chapter 6, an output feedback PID controller is considered instead of the full state feedback one. Actually, the mechanical systems are usually equipped 1. A kind of high-gain observer is chosen as the PID state observer. Finally, chapter 7 concludes the results.

Third, the Euclidean norm of signal vector x t is defined by L. Newtonian mechanics, Lagrangian one and Hamiltonian one. Newtonian mechanics has been used for simple mechanical systems because it is an intuitive and non-systematic method. On the contrary, Lagrangian and Hamiltonian ones can be used for complex multi-body mechanical systems because they are systematic approaches.

In this chapter, either Lagrangian or Hamiltonian mechanics is used as a representation for motion of mechanical systems. Also, a class of Hamilton-Jacobi HJ equations in [8, 65] should be solved to get the optimality for some performance index.

However, since they are nonlinear partial differential equations, it is hard or impossible to solve them exactly. There are several trials to solve them: First, the approximation method was suggested to approximately solve the HJ equation in [12, 64, 65].

For a given performance index, it produces directly an approximate solution for the HJ equation, as the sweep method for Riccati equation in linear systems.

Second, the inverse optimal method in [18, 36, 47] suggests the solution first, and then it finds the performance index conversely from the solution of HJ equation. In this chapter, both direct and inverse methods are suggested and explained as a background in regard to nonlinear optimal control. According to the objective of mechanical control systems, there are two control types: On the contrary, the trajectory tracking system has the nonlinear time-varying characteristics.

To show the optimality of trajectory tracking system, the authors of [18, 47] used the modified computed torque control Modified-CTC. Usually, the force Fi is decomposed of conservative forces, i.

We will call 2. For the Lagrangian control system of 2. Also, above Hamiltonian control system is called as Hamiltonian system briefly. The kinetic energy of mechanical system is characterized by using Inertia matrix M q.

If the Lagrangian is given as follows: Here, we should notice that the sign of unknown external input disturbance d is of no importance. When the Hamiltonian is given as 2.