**2. Overview of** *H*∞ **Control Theory**

The standard setup of *H*∞ control is shown in Figure 1. In this figure, *w*, *u*, *z*, and *y* are vector-valued signals: *w* is the exogenous input, typically consisting of command signals, disturbances, and sensor noises; *u* is the control signal; *z* is the performance output that is to be minimized; *y* is the measured output. *G* is the generalized plant and *K* is the controller to be designed.

**Figure 1.** Block diagram of the standard *H*∞ control.

In mixed sensitivity-based *H*∞ control, the generalized control plant, or augmen<sup>t</sup> control plant, can be formulated from feedback control. For a typical feedback control diagram shown in Figure 2a, three functions: sensitivity function *S*, complementary sensitivity function *T*, and controller sensitivity *R*, are defined, and they are calculated by

$$S = \left(1 + PK\right)^{-1}, \\ T = I - S = PK(1 + PK)^{-1}, \\ R = K(1 + PK)^{-1}.\tag{1}$$

where *P* is the transfer function of the control plant, and *K* is the to-be-designed controller. The sensitivity function *S* is the transfer function between the reference input *r* and tracking error *e*, or between the disturbance and measurement output *y*. The complementary sensitivity function *T* is the transfer function between the reference input *r* and measurement output *y*. The controller sensitivity *R* reflects the control effort, which is the transfer function between the reference *r* and controller output *u*.

**Figure 2.** Formulation of the standard *H*∞ problem. (**a**) Weighted feedback control system; (**b**) equivalent standard *H*∞ problem.

In RTHS, it is expected that the loading system can accurately reproduce the command signal and is less sensitive to external disturbances, i.e., *S*→0 and *T* →1 are demanded. To meet this requirement and consider the robustness index of additive and multiplicative uncertainties [24], performance weighting functions, namely *W*S, *W*R, and *W*T, are introduced to the feedback control loop, as shown in Figure 2a. Thus, the equivalent standard *H* ∞ block diagram can be reached, which is shown in Figure 2b. Hence, the generalized plant *G*, from ( *w*, *u*) to (*z*, *y*), is given as follows:

$$\mathbf{G}(s) = \begin{bmatrix} G\_{11}G\_{12} \\ G\_{21}G\_{22} \end{bmatrix} = \begin{bmatrix} W\_{\rm S} & -PW\_{\rm S} \\ 0 & W\_{\rm R} \\ 0 & PW\_{T} \\ 1 & -P \end{bmatrix} . \tag{2}$$

In *H* ∞ control theory [24], the controller is synthesized by optimizing the *H* ∞-norm of the cost-function, a transfer function from the exogenous input *w* to the performance output *z*, which is calculated by

$$\mathbf{T}\_{wz} = \mathbf{G}\_{11} + \mathbf{G}\_{12}\mathbf{K} (\mathbf{I} - \mathbf{G}\_{22}\mathbf{K})^{-1} \mathbf{G}\_{21} = \begin{bmatrix} W\_S(s)S(s) \\ W\_R(s)R(s) \\ W\_T(s)T(s) \end{bmatrix} \tag{3}$$

Thus, the *H* ∞ control problem can be formulated as follows: find a controller *K* that makes the closed-loop system internally stable, and make the *H* ∞ norm of Equation (3) the least (optimal), or less than a given positive constant (suboptimal) [24]. However, it is often not necessary to design an optimal controller in practice, and it is usually much cheaper to obtain controllers that are very close in the norm sense to the optimal ones. Hence, the *H* ∞ suboptimal controller was used in this study, in which the cost-function satisfies

$$\|\mathbf{T}\_{wz}\|\_{\infty} \preccurlyeq \gamma. \tag{4}$$

where *γ* is a positive number, and the minimum value is in relation to the generalized plant G [24].

Let a possible state-space realization for the generalized plant *G* be calculated by

$$\begin{array}{l}\dot{\mathbf{x}} = A\mathbf{x} + B\_{1}\mathbf{w} + B\_{2}\boldsymbol{\mu} \\ \dot{\mathbf{z}} = \mathbf{C}\_{1}\mathbf{x} + D\_{11}\mathbf{w} + D\_{12}\boldsymbol{\mu} \\ \mathbf{y} = \mathbf{C}\_{2}\mathbf{x} + D\_{21}\mathbf{w} + D\_{22}\boldsymbol{\mu} \end{array} \tag{5}$$

where *x* is the state vector, and the dimensions of *w*, *u*, *z*, and *y* are compatible with that of *x*.

Suppose G satisfies the following assumptions [24]:


• *A* − *jωI B*2 *C*1 *D*12 has full column rank for all *ω*; • *A* − *jωI B*1 *C*2*D*21 has full row rank for all *ω*.

Then, a controller can be designed employing the DGKF method by solving two Riccati equations [24].

It should be noted that the internally stable controller *K* is not unique in the suboptimal problem, and the central controller is used in general.
