## 2 Materials and Methods
### Robust Control Design  
There is a lack of generic theoretical methods supporting simple design of low-order fixed-structure controllers, which are dominantly used in industrial motion control systems. Generally accepted methods of modern control theory usually lead to high-order controllers (the controller order is typically higher or equal to the order of the controlled plant), which are inherently difficult to tune and implement in practice. The usual approach to the fixed-structure controller design is to perform an order reduction either for the plant model or for the derived high-order controller (Figure 6). This inevitably leads to some performance loss resulting from the performed approximation. Direct methods usually rely on numerical non-convex optimisation resulting in no guarantee of convergence and global optimality of the derived results [31]. A generic approach to design of PID controllers, that are prevalent in industrial motion control hardware, is still missing. A new design method specifically tailored to PID controllers was developed at our workplace recently in an attempt to fill this gap between the academic research and practice [32]. It is primarily intended for PI(D) controllers and simple fixed-structure feedback control algorithms with two or three parameters (lead-lag compensators, low-order static feedback etc.). The basic features of this approach are summarised as follows:
* Formulation of the design specifications in the frequency domain by imposing arbitrary closed-loop weighted sensitivity inequalities;
* Possibility of automatic calculations for the auto-tuning purposes;
* Generic method for an arbitrary LTI system described by a rational transfer function + time delay;
* Suitable for robust controller design using structured or unstructured uncertainty models;
* Analytical method for the computation of the admissible set of controllers, no performance losses due to approximations, model reduction or non-convex numerical optimisation. A brief review of the main results is added in the following section for the sake of compactness. Examples of employment of this approach in motion control system design problems are given in references [13; 33]. H-Infinity Loop-Shaping Design of a Fixed-Structure Controller  
The starting point is a model of a generic controlled process \(P(s)\) without the poles on the imaginary axis (can be relaxed by adjusting the controller derivation procedure). The feedback compensator \(C(s)\) is considered in the standard PI controller form  
\[C(s,k)=k_{p}+\frac{k_{i}}{s}, \tag{33}\]  
where \(k\) denotes the parameters vector \([k_{p},k_{i}]\). An arbitrary number of design constraints can be formulated in the frequency domain as loop-shaping inequalities in the form of  
\[||H(s,k)||_{\infty}<\gamma;\;H(s,k)\stackrel{{\Delta}}{{=}}W(s) S_{*}(s), \tag{34}\]  
where \(S_{*}(s)\) denotes one of the closed-loop sensitivity functions (sensitivity, complementary-, input-, and controller-sensitivity, respectively--see Figure 7 for the physical meaning in terms of the loop input and output signals), \(W(s)\) introduces an arbitrary user-defined frequency-dependent scaling, and \(\gamma\) is a scalar design parameter. Figure 6: Model reduction workflow allowing implementation of motion systems using industrial-grade hardware and fixed-structure low-order control schemes. The goal is to find a controller \(C(s,k)\), which, together with the given \(H(s)\), fulfils the following three conditions  
1. \(C(s,k)\) internally stabilises the closed-loop;
2. \(H(s,k)\) used in the design criterion is stable;
3. The H-infinity condition \(||H(s,k)||_{\infty}<\gamma\) holds. This controller is called the \(H_{\infty}\)_controller_. Typically, there is a whole set of the admissible controllers fulfilling the above-mentioned conditions, which can be represented directly in the parametric plane \([k_{i},k_{p}]\) (Figure 8). Figure 8: \(H_{\infty}\) controller-admissible set of controller gains fulfilling the formulated loophsaping inequality constraint in the parametric plane.