We propose computational techniques for model predictive control of large-scale systems with both continuous-valued control inputs and discrete-valued control inputs, which are a class of hybrid systems. In the proposed method, we introduce the notion of virtual control inputs, which are obtained by relaxing discrete-valued control inputs to continuous variables. In online computation, first, we find continuous-valued control inputs and virtual control inputs minimizing a cost function. Next, using the obtained virtual control inputs, only discrete-valued control inputs at the current time are computed in each subsystem. In addition, we also discuss the effect of quantization errors. Finally, the effectiveness of the proposed method is shown by a numerical example. The proposed method enables us to reduce and decentralize the computation load.

Control of large-scale systems is one of the fundamental problems in control theory and has been extensively studied so far (see, e.g., [

In large-scale systems, many kinds of actuators are included. For example, in air-conditioning systems, the output of air conditioners can be regarded as a continuous-valued control input, and ON/OFF switches of ceiling fans can be regarded as a discrete-valued control input. Thus, it is appropriate to consider large-scale systems with both continuous-valued control inputs and discrete-valued control inputs. However, to our knowledge, such a large-scale system has not been directly considered so far. Furthermore, such a large-scale system is a class of hybrid systems, and the finite-time optimal control problem is in general reduced to a mixed integer quadratic programming (MIQP) problem. For large-scale systems, the computation time for solving the MIQP problem is too long, and it is difficult to realize the model predictive control (MPC) method in which the MIQP problem is solved at each time.

We, thus, propose computational techniques for model predictive control of large-scale systems with both continuous-valued control inputs and discrete-valued control inputs. In the proposed method, we introduce the notion of virtual control inputs, which are obtained by relaxing discrete-valued control inputs to continuous variables. This term has been used in, for example, [

In online computation, first, continuous-valued control inputs and virtual control inputs minimizing a cost function are found in a centralized controller. Next, using the obtained virtual control inputs, only discrete-valued control inputs at the current time are computed in each decentralized controller. In other words, a sequence of discrete-valued control inputs is not computed because in MPC only the control input at the current time is applied to the plant. In a centralized controller, the QP problem is solved at each time. In each decentralized controller, the problem of finding the discrete-valued control input is solved at each time. This problem is reduced to an integer programming problem, or can be solved by the look-up table method in which a look-up table is generated off-line. Thus, the computation load is reduced and decentralized by the proposed method.

In addition, we also discuss the effect of quantization errors based on our previous result [

This paper is organized as follows. In Section

Consider the discrete-time large-scale system consisting of

We show an example of air-conditioning systems.

Suppose that the dynamics of temperature in room

Next, consider transforming (

Finally, we consider the effect of other rooms. Assume that airflow is expressed as a directed graph. In this example, suppose that airflow is given as in Figure

Illustration of multiple rooms.

Directed graph expressing airflow.

For the large-scale system consisting of

Suppose that for the large-scale system consisting of subsystems

By assigning a binary variable to each element of

Set

Solve Problem

Apply only

Set

In the above procedure, Problem

In this section, first, the outline of the proposed method is explained. Next, the notion of virtual control inputs is proposed. Finally, using virtual control inputs, a solution method for Problem

For large-scale systems, it is in general difficult to solve Problem

Illustration of state trajectory.

Furthermore, on implementation of the proposed procedure of MPC, we consider both centralized and decentralized controllers (see Figure

Control system considered in this paper.

Hereafter in this section, first, the notion of the virtual control input will be formally defined. Next, an approximate solution method of Problem

The matrix

Hereafter,

We show one example.

Suppose that

Many actuators are included in each subsystem. In the example of air-conditioning systems, we can consider ceiling fans, local heaters, and so on. In many cases, these correspond to discrete-valued control inputs, and the number of these may be greater than the dimension of the state. Thus on the derivation of virtual control inputs, we reduce

First, define

Suppose that for the large-scale system consisting of subsystems

Since the decision variable in Problem

By solving Problem

Find binary variables

By a simple calculation, Problem

The proposed solution method for Problem

In addition, the fact that Problem

Finally, we summarize the procedure of MPC combining centralized control with decentralized control.

Set

In the centralized controller, find both a continuous-valued control input and a virtual control input by solving Problem

Send the optimal values of both a continuous-valued control input and a virtual control input from the centralized controller to each decentralized controller.

In each decentralized controller, solve Problem

Apply only the control input at

Set

In the previous section, we do not consider quantization errors and focus on the lower bound of the cost function. In this section, we discuss a method for considering quantization errors.

First, the virtual control input is introduced. In this section,

We show one example.

Suppose that

Next, consider transforming the system (

In this section, the dimension of the virtual control input is

As a numerical example, consider the large-scale system consisting of 9 subsystems given by (

Directed graph expressing couplings.

For this system, we consider solving Problem

We show the computation result. Figures

Trajectories of the first element of the state.

Trajectories of the second element of the state.

Next, we discuss how to derive the discrete-valued control input from the virtual control input. In this example, the optimal virtual control input at

Finally, we explain the computation time for solving Problem

In this paper, we have proposed computational techniques for model predictive control of large-scale systems with both continuous-valued and discrete-valued control inputs. By introducing the notion of virtual control inputs, the computation load is reduced and decentralized. The effectiveness of the proposed method has been shown by a numerical example. The proposed method is useful for solving the control problem of large-scale system with both continuous-valued and discrete-valued control inputs.

One of the future works is to apply our approach to several practical systems such as air-conditioning systems. On the other hand, a large-scale system studied in this paper is a class of hybrid systems. It is also important to extend the proposed method to general hybrid systems such as piecewise affine systems and MLD systems.

First, by using a binary variable, the element

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