- Udari Madhushani - Google 学术搜索引用
- Quadrotor trajectory tracking using PID cascade control
- PID Trajectory Tracking Control for Mechanical Systems
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Assume that there exists a Lyapunov matrix P x, t 2. Since the differential Riccati equation 2. Also, we should notice that the auxiliary controller is the PD one.

## Udari Madhushani - Google 学术搜索引用

Finally, the modified computed-torque controller 2. Lemma 4. As a matter of fact, the Lagrangian system 2. From now on, we are to obtain the differential Riccati equation from HJI one 2. First, we simplify the HJI equation to a matrix differential Riccati one.

## Quadrotor trajectory tracking using PID cascade control

For arbitrarily given weighting matrices Q x, t and R x, t , the Lyapunov matrix 2. Similarly to Theorem 1 in previous section, the weighting matrices Q and R will be inversely found from the differential Riccati equation. Theorem 2.

There is also a vector partial differential equation satisfied by an optimal return function. Bellman[] has generalized the Hamilton-Jacobi theory to include multi-variable systems and combinatorial problems and he calls this overall theory dynamic programming. Also, Bellman theory is associated with finding the first-order nonlinear partial differential equation HJB equation. In certain areas of game theory, one can find problems similar to our general control problem, but with the important exception that the behavior of the target as a function of time is not known to us in advance - there is an opponent capable of influencing the target, who is trying to keep us away from hitting the target.

In other game theory problems, the opponent might be able to directly affect our state and control input also, see [41]. Isaacs[] showed how differential game theory can be applied to control theory. Isaacs theory is also associated with finding the first-order nonlinear partial differential equation HJI equation : for more detail, see [29, 65] derived from the two player one player corresponds to the control input, the other player to the exogenous disturbance input differential game theory.

There are many types of PID controllers, e. The wide acceptance of the PID controller in industry is based on the following advantages: it is easy to use, each term in the PID controller has clear physical meanings present, past and predictive , and it can be used irrespective of the system dynamics. Most industrial mechanical systems can be described by the Lagrangian equation of motion. Conventional PID trajectory tracking controllers are used because they provide very effective position control in Lagrangian systems. Unfortunately, they lack an asymptotic stability proof.

Under some conditions with PID gains, the global or semi-global asymptotic stability of a PID set-point regulation controller was proved by [2, 32, 46, 51, 60] for robotic manipulator systems without external disturbances. However, they did not deal with a PID trajectory tracking controller, but set-point regulation one. On the other hand, the robust stability of decentralized PID control for mechanical systems have been tried to prove it by using either Kharitonov theorem or Lyapunov method in [43, 52, 61]. The basic control law theories are found in two papers[29, 64]: one describing the full state feedY.

This is not a trivial problem. There have been several attempts to solve the HJI equation. The approximation method was used by [12] to obtain an approximate solution to the Hamilton-Jacobi HJ equation for Lagrangian systems. The concept of extended disturbances, including system error dynamics, was developed by [18, 47, 48, 49] to solve the HJI equation. This chapter is organized as follows. Next section deals with the statespace representation used for trajectory tracking in a Lagrangian system. In section 3.

Also, an inverse optimal PID control law is suggested with the necessary and sufficient condition for its existence in section 3. Disturbances exerted on the system can be caused by the friction nonlinearity, parameter perturbation, etc. If the extended disturbance 3. If the state vector is defined for the tracking system model 3. This characteristics offers the clue to solve the inverse optimal problem for above Lagrangian system 3.

Remark 1. If the controller stabilizes the trajectory tracking system model 3. Remark 2. For the set-point regulation control, the system model 3. On the other hand, for the trajectory tracking control, we obtained the system model 3. Here, above two system models 3. When there exist unknown bounded inputs such as perturbations and external disturbances acting on systems, the behavior of the system should remain bounded.

Also, when the set of inputs including the control, perturbation and disturbance go to zero, the behavior of system tends toward the equilibrium point. This ISS notion is helpful to understand the effect of inputs on system states. Moreover, Krstic et al. The basic characteristics and properties on the ISS are summarized in the followings. Especially, the ISS becomes available by using Lyapunov function. For the system 3. However, we do not know whether the extended disturbance w satisfies the condition or not, hence, only ISS is proved. Above properties on ISS will be utilized in following sections.

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However, there can be many control laws satisfying the 3. Hence, we need the definition which can bring the Lyapunov function and a unique control input under the assumption on the unknown disturbance. If the Lagrangian system 3.

## PID Trajectory Tracking Control for Mechanical Systems

First, we show that the Lyapunov matrix P 3. Hence, we obtain the control law 3. Also, since the infimum among the control inputs that satisfy 3. An important characteristics of controller 3. Strictly speaking, the controller 3. To overcome this difficulty of a direct optimization, Krstic et al showed in [36] that the inverse optimal problem is solvable if the system is disturbance input-to-state stable.

The HJI equation for the Lagrangian system and its analytic solution were suggested by [47], but it dealt with the modified computed torque controller form, not a PID controller type. Above HJI equation 3.

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Theorem 4. For a given Lagrangian system 3. If the PID control law 3. First, we show that the matrix Q x of 3.

Let us obtain the state weighting matrix Q x using 3. This was proved by [48] for the first time. This fact was proved by [36] for the first time. Therefore, we conclude that the PID control law 3. In a viewpoint of an optimal control theory, the magnitude of a state weighting Q has the relation with system errors, e. This property will be shown through experimental results later. Therefore, if we are to reduce the error of system, we should enlarge the magnitude of a matrix K in the state weighting. However, it reduces the magnitude of a control input weighting matrix R and produces the bigger control effort.

Conversely, if we reduce the magnitude of K, then the smaller control effort is required and the bigger error is generated. The common K in the state weighting and the control input weighting has the trade-off characteristics between the system performance and control effort.

Here, we define the inverse optimal PID controller using the static gain one in Remark 3 and summarize its design conditions in following Theorem. Theorem 5. If the inverse optimal PID controller 3. Therefore, along the solution trajectory of 3. Here, if above equation is rearranged by using the HJI 3.

The inverse optimal PID controller of 3. This fact brings the selection guidelines for gains of an inverse optimal PID control. The selection conditions of gains are suggested using the time derivative of Lyapunov function. First of all, let us reconsider the extended disturbance of 3. Also, we can obtain the upper bounds of gains kP of 3. Hence, used for small Inertia systems and vice versa.

If we manipulate 3. Hence, the value of kI should be chosen inversely proportional to m and directly proportional to kP. Remark 4. Assume that the Lagrangian system 3. Here, if above inequality is inserted in 3. Also, if the applied controller stabilizes the closed system, then the magnitude of extended disturbance is mainly affected by that of inverse dynamics h in 3.