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Recently, high speed and high precision are required for machine tools to improve their performance. However, the high speed causes unintended deterioration of accuracy and durability of them due to harmful thermal displacement¹⁾. Therefore, high-precision machine tools need a cooler to eliminate the unnecessary thermal load quickly. Some capacity control techniques for controlling target temperature in the cooler have been applied^1-2). One approach of the capacity control is a variable speed compressor (VSC) manner. Actually, this manner is more complicated and expensive than hot-gas bypass manner, because it deals with a multi-input and multi-output (MIMO) system and it requires an additional variable speed drive to regulate compressor speed. Nevertheless, it is regarded as an alternative approach in the future because of superior energy saving ability under partial load states, which is highly valued for sustainable development.^1,3-5).

Many different control methods for the VSC such as typical PID⁷⁾, PI-feedforward^6-7), linear quadratic regulator (LQR)^8-11), linear quadratic Gaussian (LQG)¹²⁾, fuzzy logic^13-14), and artificial neural network have been suggested. Comparing to conventional PID control based on a transfer function model, optimum control based on a state space model has some merits such as flexibility of optimum control design between input energy and control error which are trade-off relation, expendability for the MIMO system, and easiness for securing robustness against modelling error and disturbances. As the refrigeration cycle included VSC is basically MIMO and high-order nonlinear system, viability for MIMO control and robustness to cope with modelling errors including disturbance are very important factors to accomplish high efficiency and high precision control of the cooler system.

To date, a number of researchers have looked at optimum control of the vapour compression refrigeration system^8-12). Some of them used LQR^9-11) and another used LQG¹²⁾ as a controller. However, robustness of the control system was not fully discussed in the previous studies. Therefore, this paper describes an optimum robust control in a state space model for a cooler included VSC system for machine tools. First of all, we will focus on reference tracking to eliminate steady-state error about stepwise input and robustness against modelling error and disturbance.

At first, a state space model for designing the optimum controller of the cooler was derived from empirical transfer function model obtained experiments. Next, the optimum robust controller was designed based on the state space model to achieve the following control purposes; ① asymptotical stability of control system, ② reference tracking despite stepwise disturbances, ③ robustness against modelling errors. Finally, some experiments were conducted to verify the validity of the suggested controller. Especially, experimental results about the effects of two weight matrices in a cost function are analysed in detail.

To achieve the control purposes, an optimum robust control design approach^10~11) is applied. An evaluation function is generally given by Eq. (5).

Where, Q and R express weight matrix about state variable and input energy respectively. The main process of this design is divided into two steps. The first step is to choose weight matrix Q and R which is trade-off relation considering the design purpose. The second step is to decide the control input U(t) including state feedback gain matrix K, U(t) = -KX(t), to minimize J and also to be stable closed-loop control system asymptotically.

In servo system, Eq. (5) can not be minimized because the input has limt→∞⁡Ut≠0 even if the output reaches to limt→∞⁡Yt=R in a steady-state. Therefore, a new input V(t), Vt=U˙t, was introduced to resolve this problem, then the system equation can be formulated as Eq. (6) and Eq. (7) from Eq. (2) and Eq. (3).

Where, X_e(t) = [X(t) U(t)]^T,

Ae=A B0 0, Be=0Im, De=In0, Ce=C 0. It is very well known that if the system (A, B) is stable, then the system (A_e, B_e) is also stable. And if (C, A) is observable and rank of Z described Eq. (8) satisfies Z = n + m, then the system of (A_e, B_e) is also observable. Here, n means the number of a state variable, m indicates the number of an input of the system. It was confirmed that all of these conditions were fully satisfied from Eq. (4).

Considering small variation of the state variable and input variable in a steady-state, δX(t) = X(t) - X_s, δU(t) = U(t) - U_s, and let δX_e(t) as δX_e(t) = [δX(t) δU(t)]^T, then Eq. (9) and Eq. (10) are derived from Eq. (6) and Eq. (7).

Note that the stepwise disturbance D(t) in Eq. (6) was eliminated in Eq. (9) and if the control of δX_e(t) converges to 0, then it makes Y(t) = R in Eq. (10). Consequently, the control purpose ② is accomplished by this control. The cost function, Eq. (5) is newly expressed as Eq. (11) using δX_e(t), and it can be minimized about J_e.

Here, Q_e is a semi-positive finite symmetric matrix, and R_e is positive finite symmetric matrix. These weight matrices are decided considering control specifications and their maximum limit values. If the weight Q_e is designed as Qe=CeTCe, the weight matrix of energy term, R_e, is only considered in Eq. (11).

The input which is the optimum solution to minimize Eq. (11) can be obtained as follows;

Here, P_e is a positive finite matrix, and it is a solution of the Riccati algebraic equation (14).

As the values X_s and U_s cannot be known in general, the state feedback of δX_e(t) in Eq. (12) is impossible in reality. Therefore, δX_e(t) is replaced by terms of state variable X(t) and control error Y_e(t) = R- Y(t) as Eq. (15) by substituting the definition of δX_e(t) to Eq. (12).

Here, the term of K_eZ^-1 means a control gain matrix K. Finally, a control input U(t) is given as;

Eq. (17) shows augmented matrices A_e, B_e, and C_e.

Eq. (18) gives weight matrices Q_e and R_e. Thus, controller gain K is calculated as Eq. (19).

Fig. 4 illustrates the optimum robust control system to realize control law of Eq. (16) under an initial condition of X(0) = 0.

Fig. 4 
Optimum robust servo control system

The optimum robust control based on a state space model obtained empirical transfer function model was investigated to control target temperature and superheat simultaneously for an oilcooler system in machine tools. The servo system augmented from a basic state space model was utilized to cope with stepwise reference tracking and to reject stepwise disturbance.

The robustness of the control system against parameter variations was confirmed through some real experiments and computer simulations considering some modelling errors and external disturbances such as change of thermal load and ambient temperature. Especially, the effect of two weight matrices in a cost function was analysed in detail using experimental data to clarify their trade-off relation. The experimental results showed that the suggested robust optimum controller had fairly good control performances such as ±0.1℃ and ±1.5℃ error range in steady-state for oil outlet and superheat respectively in spite of working condition changes. Consequently, it was clear that the optimum robust control can achieve precise reference tracking about stepwise reference despite external disturbances and model errors, and optimum control performances between energy consumption and control errors.