^{1}

Self-triggered control is a control method that the control input and the sampling period are computed simultaneously in sampled-data control systems and is extensively studied in the field of control theory of networked systems and cyber-physical systems. In this paper, a new approach for self-triggered control is proposed from the viewpoint of model predictive control (MPC). First, the difficulty of self-triggered MPC is explained. To overcome this difficulty, two problems, that is, (i) the one-step input-constrained problem and (ii) the

In recent years, analysis and synthesis of networked control systems (NCSs) have been extensively studied [

In this paper, the periodic paradigm is focused as one of the technical issues in NCSs. The periodic paradigm is that the controller is periodically executed at a given unit of time. The period is chosen based on CPU processing time, communication bandwidth, and so on. However, in NCSs, communication should occur, when there exists important information, which must be transmitted from the controller to the actuator and/or from the sensor to the controller. In this sense, the periodic paradigm is not necessarily suitable, and it is important to consider a new approach for the design of NCSs. One of the methods to overcome this drawback of the periodic paradigm, self-triggered control has been proposed so far (see, e.g., [

In this paper, we propose two methods for self-triggered model predictive control (MPC) using optimization with horizon one. First, the optimal control problem with horizon one is formulated. However, in constrained systems, one-step prediction may be insufficient, and a longer time interval, in which the input constraint is imposed, is required. Focusing on this fact, another problem in which the time interval with input constraints is enlarged is also formulated. In the former problem, the first sampling period and the first control input are optimized. In the latter problem, the first sampling period and the control input sequence are optimized. Next, an iterative solution method for the latter problem and an approximate solution method for the former problem are proposed. In the iterative solution method, a QP problem is repeatedly solved. In the approximate solution method, the problem is approximated by one QP problem. The obtained QP problem is in general not convex, and we discuss the convexity. By solving either problem according to the receding horizon policy, self-triggered MPC can be realized. Finally, the effectiveness of the proposed approach is shown by a numerical example. The proposed approach provides us a basic result for self-triggered optimal control.

Consider the following continuous-time linear system:

First, for the system (

Suppose that for the system (

Next, we present a procedure of MPC based on the self-triggered strategy.

Note here that in this procedure, the timing (i.e., the sampling time) to measure the state and to recompute the control input is computed. In this sense, self-triggered control is realized.

In the above procedure, Problem

In self-triggered MPC, only

One-step input-constrained problem.

In the former problem, the constraint is imposed for only

Before these problems are formally given, some preparations are given. In the time interval

Suppose that for the system (

In [

In constrained systems, when control is started, the control input is frequently saturated. It is important to determine the time interval of input saturation. Then, one-step prediction may be insufficient, and a longer time interval in which the input constraint is imposed is required. From this viewpoint, another problem, that is, the

First, suppose that the input constraint is imposed in the time interval

Under the above preparation, consider the following

Suppose that for the system (

In Problem

Hereafter, in Section

The parameter

First, for a fixed

Suppose that for the system (

From the conventional result on sampled-data control theory, Problem

Suppose that the initial time

Next, consider reducing Problem

Under the above preparation, Problem

Consider

A QP problem can be solved by using a suitable solver such as MATLAB and IBM ILOG CPLEX [

Third, by using the obtained QP problem, we propose an algorithm for solving Problem

In a numerical example (Section

Finally, we discuss the stabilization issue. For Problem

In this section, first we derive a solution method for the one-step input-constrained problem (Problem

First, noting that the control input is piecewise constant, from (

The one-step input-constrained problem of Problem

Consider

By rewriting

By solving Problem B, we can obtain suboptimal

In Theorem

The cost function in Problem B is in general nonconvex. In other words, the matrix

The matrix

Consider approximating the cost function in Problem B, that is,

First, we show an example of the iterative solution method proposed in Section

Consider the following system:

We show the computational result on self-triggered MPC with the

State trajectory.

Control input.

Next, compare two cases. In these cases, the obtained state trajectories are almost the same. The difference between two cases is as follows. In Figures

In the case of

In addition, we discuss the effect of changing

Finally, we discuss the computation time for solving the

Next, we show an example of the approximate solution method proposed in Section

Consider the system (

We show the computational result on self-triggered MPC with the transformed Problem B. The initial state is given as

State trajectory.

Control input.

Finally, we discuss the computation time for solving the transformed Problem B. The transformed Problem B with the different initial state is solved 17 times. Then, the mean computation time was 0.01 [sec], where we used IBM ILOG CPLEX 11.0 as the QP solver. Since in this example the number of decision variables is only two, computation is very fast.

In this paper, we discussed self-triggered MPC of linear systems. Since it is difficult to solve the original problem (Problem

In the future works, first, it is important to develop a more efficient method for solving Problem

The authors declare that they have no conflict of interests.

This work was partially supported by Grant-in-Aid for Young Scientists (B) 23760387.