**4. The performance goals for the optimization of H∞**

The 2-DOF for controller can be tuned by utilizing the approach of H infinity optimization. The open loop gain being a critical indicator of feedback loop behavior, gain of open loop should be more than one in control bandwidth to confirm better DR (Disturbance Rejection) and it must be lower than one outer of control bandwidth to confirm insensitivity to measurement noise of unmodelled dynamics. The ideal performance terms can be displayed regarding execution objectives. To accomplish a decent disturbance rejection and tracking, three execution objectives/limitations are forced on the tuning of controller gain (see discussion in previous section), which is as follows:


The CGs (Controller-gains) must be tuned with constraint i.e., function cost, connected with every specification subjected to minimization in H-infinity framework.

### **4.1 Tracking as an execution objective**

The frequency domain specification for monitoring between output and input is described in this performance target. This frequency domain constraint indicates the most extreme relative error as a FF (frequency function). The ME (maximum error) is given by:

$$error\_{\max} = \frac{(error\_{peak})s + o\_{\varepsilon}(error\_{dc})}{s + o\_{\varepsilon}} \tag{7}$$

where, *ω<sup>c</sup>* denotes cut-off frequency.

The scalar function *f(x)* describes the tracking goal, where *x* is denoted as a tuneable vector of entire parameters in the system. The target optimization (TO) is to modify the parameter function *f(x)*, which is optimized. The scalar function for tracking case is described using *f(x)*:

$$f(\mathbf{x}) = \left\| \frac{\mathbf{1}}{\operatorname{error}\_{\max}} (T(\mathbf{s}, \mathbf{x}) - I) \right\|\_{\infty} \tag{8}$$

where,*T(s, x)* is symbolized as closed loop transfer function between input and output.

### **4.2 Min-LG (minimum) as execution limitation**

The minimal gain on the open loop frequency response at specified frequencies is limited by this performance goal. The frequency dependent minimum gain constraint in turn gives the inverse sensitivity function of minimum gain limit. The min constraint gain characterizes the capacity of scalar function *f(x)* whereas the advancement procedure attempts to drive lower value of *f(x)*. The capacity of scalar *f(x)* is defined as:

$$f(\mathfrak{x}) = \left\|{\mathcal{W}\_S(D^{-1}\mathbb{S}D)}\right\|\_{\mathfrak{so}} \tag{9}$$

where, *WS* is denoted as Min-LG profile and *S* is defined as sensitivity function.

### **4.3 Max loop gain as execution limitation**

This execution goal aims at highest gain of open loop at the determined frequencies in the given framework. The Max-LG can be characterized as the frequency domain element. This type of constraint restrains upper limit on the corresponding sensitivity. The maximum loop gain determines the scalar function *f(x)* given by:

$$f(\mathbf{x}) = \left\|{\mathcal{W}\_T(\mathbf{D}^{-1}\mathbf{T}\mathbf{D})}\right\|\_{\ast} \tag{10}$$

where, *WT* is symbolized as the reciprocal profile of the Max-LG. *T* is symbolized as the function of complementary sensitivity.
