An Interval Approach for Robust Parameterization of Controllers for Electric Drives

Abstract

Uncertain models, e.g., due to component variations and measurement errors during system identification, in combination with the desire to be able to provide guarantees regarding system performances a priori, represent major challenges for users when designing controllers on the basis of models. The aim of this publication is to mitigate this obstacle in the design. For this purpose, criteria of classical control engineering are combined with interval arithmetic in general and the SIVIA algorithm in particular. Thereby, the approach is valid for linear dead time systems. The algorithm evaluates performance criteria for closed interval-bounded sets of parameter values in a single iteration. When overestimation prevents a definite result, the size of the parameter set is recursively reduced by bisecting the evaluated set. The application of the methodology to the controller design for an electromechanical linear actuator shows that the approach not only contributes to theory but can also simplify the practical controller design, especially in the area of electric drives.

1. Introduction

When designing control algorithms (for electric drives in particular), one is inevitably exposed to uncertainties regarding the knowledge of the controlled system [1,2]. The sources of these uncertainties start with the manufacturing tolerances and end with the environmental conditions at the customer’s site. A well-known example of an effect that is highly dependent on ambient conditions is friction [3].

At the same time, customers expect to meet certain performance criteria under a wide range of conditions. In addition, there is the desire for a short downtime in the case of a failure, which goes hand in hand with the requirement that components can be replaced without time-consuming re-parameterization of the controller.

One common approach in the literature to deal with unknown (but bounded) uncertainties are interval arithmetic methods. In engineering, and especially in systems theory, they are already used to solve problems along the entire life cycle of a product. They are used for modeling [4], system identification [5], state estimation [6,7], predictive control [8], and fault diagnosis [9].

Moreover, approaches that robustly parameterize controllers using interval arithmetic approaches have emerged in recent years (see e.g., [10,11]). The aforementioned approaches are based on the proof from [12]. Two linear dynamical systems with interval coefficients are considered, both having matching numerator and denominator orders. It is shown that the system performance of the first system is included in that of the second system if and only if all parameters of the first system are subsets of their corresponding parameters of the second system. This inclusion relation applies to the system performance in the time and frequency domain. In order to use this property in the controller design, the interval-valued transfer function of the closed control loop is first set up as a function of the controller parameters. In addition, a desired interval-valued reference transfer function is defined, which includes all permissible system dynamics. The authors then use the Set Inverter Via Interval Analysis (SIVIA) algorithm [13] to determine the set of controller parameters for which the closed-loop transfer function is included in the desired transfer function. However, this approach raises the question of how a desired interval-valued reference transfer function can be specified. In the previously mentioned publications [10,11], it was proposed to use a first-order model that has an interval-valued gain and an interval-valued time constant. This reference model is supplemented with interval-valued poles and zeros far left from the imaginary axis to match the numerator and denominator order of the closed-loop transfer function. The basic idea behind this is that a stable first-order interval system that is supplemented with stable interval-valued poles and zeros is transformed into a stable system of higher order. However, this assumption is invalid due to the interval nature of the system. This statement can be supported by [14] where it is shown that the evaluation of four polynomials is necessary to investigate the stability of a linear differential equation system with interval parameters. The definition of the interval reference model in the proposed approach is de facto only based on two polynomials though. This finally leads to the fact that the proposed approach cannot find any practical application.

The aim of this publication is to further pursue the general idea of using the SIVIA algorithm for model-based controller parameterization that was presented in [10,11]. The problem of this approach is fixed by developing an alternative method to bound the desired behavior of a linear dynamic system with interval parameters. This is achieved by combining classical criteria from control engineering with interval arithmetic functions. Subsequently, this alternative method is combined with the SIVIA algorithm.

At the same time, the considered system class is extended in comparison to the mentioned publications in the sense that the linear systems may also exhibit dead-time behavior. For such systems, the parameters determined guarantee the stability of the closed control loop. Furthermore, factors for the damping specification are included in order to be able to guarantee not only stability but also dynamic performance.

To summarize, this publication aims to provide a method for model-based controller parameterization of uncertain systems with dead time that generates a set of controller parameters leading to guaranteed system properties. It thereby offers a deeper understanding of the controller parameter space and enables parameterization of uncertain systems without a trial-and-error approach.

In addition to the theoretical contribution of this work, a practical contribution is also made in the form that the approach is applied to the controller parameterization of an electromagnetic linear actuator.

The presentation of the results is as follows. To introduce the theory, the mathematical basics of interval arithmetic are presented first. Then the model of the electromechanical linear actuator is presented, followed by the definition of performance criteria for interval systems. Based on these fundamentals, the combination of the criteria with the SIVIA algorithm is then presented and simulated results for the linear actuator are shown. Finally, the results are analyzed and possible extensions are discussed.

2. Mathematical Fundamentals

At this point, the mathematical basics of interval arithmetic, which are necessary for understanding the algorithms presented below, are introduced. For more in-depth descriptions such as the calculation rules of interval arithmetic, the reader is referred to [13].

Definition 1.

A real interval $\left[x\right]$ is a connected subset of the set of real numbers $\mathbb{R}$. For the limits of the interval applies $\left[\underline{x},\overline{x}\right]=\left\{x\in \mathbb{R}\mid \underline{x}\le x\le \overline{x}\right\}$.

The set of all closed intervals is denoted as $\mathbb{IR}$.

Definition 2.

A real interval vector $\left[\mathit{x}\right]$ is defined as the Cartesian product of i intervals: $\left[\mathit{x}\right]=\left[x_1\right]\times \left[x_2\right]\times \dots \times \left[x_i\right]$ with $\left[x_i\right]=\left[{\underline{x}}_i,{\overline{x}}_i\right]$.

Definition 3.

If $\left[\mathit{f}\right]:{\mathbb{IR}}^n\to {\mathbb{IR}}^m$ is an inclusion function of a vector-valued function $\mathit{f}:{\mathbb{R}}^n\to {\mathbb{R}}^m$, then $\left[\mathit{f}\right]\left(\left[\mathit{x}\right]\right)$ defines a superset to the function values of $\mathit{f}\left(\left[\mathit{x}\right]\right)$ in the form of a box, which encloses the values of the vector-valued function:

$$\forall \left[\mathit{x}\right]\in {\mathbb{IR}}^n,\mathit{f}\left(\left[\mathit{x}\right]\right)\subset \left[\mathit{f}\right]\left(\left[\mathit{x}\right]\right).$$

(1)

Definition 4.

A Boolean interval $\left[b\right]$ is a connected subset of the set of Boolean values $\mathbb{B}\triangleq \{0,1\}$. For the limits of the interval applies $\left[\underline{b},\overline{b}\right]=\left\{b\in \mathbb{B}\mid \underline{b}\le b\le \overline{b}\right\}$.

The set of all closed Boolean intervals is denoted as $\mathbb{IB}$. Because $\mathbb{B}$ is finite, so is $\mathbb{IB}$. The definition allows for only three Boolean intervals to exist in $\mathbb{IB}$. These are named as follows:

$$\begin{array}{cc}\hfill [1,1]& :=\left[\mathrm{TRUE}\right],\hfill \\ \hfill [0,1]& :=\left[\mathrm{INDETERMINATE}\right],\hfill \\ \hfill [0,0]& :=\left[\mathrm{FALSE}\right].\hfill \end{array}$$

Definition 5.

The result $\mathcal{R}$ of the inclusion comparison of two real intervals $\left[x_1\right]$ and $\left[x_2\right]$, i.e., $\left[x_1\right]\in \left[x_2\right]$ is a Boolean interval:

$$\begin{array}{cccc}\hfill \mathcal{R}& :=\left[\mathrm{TRUE}\right]\hfill & \hfill & \mathrm{if}\left[x_1\right]\subseteq \left[x_2\right],\hfill \\ \hfill \mathcal{R}& :=\left[\mathrm{FALSE}\right]\hfill & \hfill & \mathrm{if}\left[x_1\right]\cap \left[x_2\right]=\varnothing ,\hfill \\ \hfill \mathcal{R}& :=\left[\mathrm{INDETERMINATE}\right]\hfill & \hfill & \mathrm{else}.\hfill \end{array}$$

Definition 6.

The interval hull $\left[I\right]$ of multiple intervals $\left[x_1\right]=\left[{\underline{x}}_1,{\overline{x}}_1\right]$, $\left[x_2\right]=\left[{\underline{x}}_2,{\overline{x}}_2\right]$, …, $\left[x_i\right]=\left[{\underline{x}}_i,{\overline{x}}_i\right]$ is defined as

$$\left[I\right]=\left[\left[x_1\right],\left[x_2\right],\dots ,\left[x_i\right]\right]=\left[\min \left({\underline{x}}_1,{\underline{x}}_2,\dots ,{\underline{x}}_i\right),\max \left({\overline{x}}_1,{\overline{x}}_2,\dots ,{\overline{x}}_i\right)\right].$$

(2)

Definition 7.

The comparison operator < is extended to real intervals by

$$\begin{array}{cccc}\hfill [{\underline{x}}_1,{\overline{x}}_1]<[{\underline{x}}_2,{\overline{x}}_2]& =\left[\mathrm{TRUE}\right]\hfill & & \mathrm{if}{\overline{x}}_1 <{\underline{x}}_2,\hfill \\ \hfill [{\underline{x}}_1,{\overline{x}}_1]<[{\underline{x}}_2,{\overline{x}}_2]& =\left[\mathrm{FALSE}\right]\hfill & & \mathrm{if}{\underline{x}}_1>{\overline{x}}_2,\hfill \\ \hfill [{\underline{x}}_1,{\overline{x}}_1]<[{\underline{x}}_2,{\overline{x}}_2]& =\left[\mathrm{INDETERMINATE}\right]\hfill & & \mathrm{else}.\hfill \end{array}$$

Definition 8.

The logical operator ∧ (AND) is extended to Boolean intervals by

$$\begin{array}{cccc}\hfill [{\underline{b}}_1,{\overline{b}}_1]\wedge [{\underline{b}}_2,{\overline{b}}_2]& =\left[\mathrm{TRUE}\right]\hfill & & \mathrm{if}{\underline{b}}_1\wedge {\underline{b}}_2=1,\hfill \\ \hfill [{\underline{b}}_1,{\overline{b}}_1]\wedge [{\underline{b}}_2,{\overline{b}}_2]& =\left[\mathrm{FALSE}\right]\hfill & & \mathrm{if}{\overline{b}}_1\wedge {\overline{b}}_2=0,\hfill \\ \hfill [{\underline{b}}_1,{\overline{b}}_1]\wedge [{\underline{b}}_2,{\overline{b}}_2]& =\left[\mathrm{INDETERMINATE}\right]\hfill & & \mathrm{else}.\hfill \end{array}$$

If the actual shape of a set deviates from that of a hyperrectangle, the approximation by interval vectors inevitably leads to overestimations. Therefore, in the following, a representation called a subpaving is defined, which is also based on interval vectors, but allows for a more precise approximation.

Definition 9.

A set of non-overlapping interval vectors is called subpaving. With subpaving, one can either represent an inner coverage of a set (see $\underline{\mathbb{X}}$ in Figure 1) or an outer coverage (see $\overline{\mathbb{X}}$ in Figure 1).

3. Modeling

The method for controller parameterization that is presented in this publication shall be applied to a linear axis driven by a voice coil motor. The linear axis used here is a V-528 linear stage from the PI product portfolio [15]. The control loop for this system is shown in Figure 2. It consists of two cascaded control loops with an outer PID position controller (${}_{p}PID$) and an inner PID velocity controller (${}_{v}PID$). The output of the PID velocity controller is the desired current of the voice coil motor. The current controller is assumed to be ideal and is not considered further. The linear axis (i.e., the plant) is modeled by the transfer function G which outputs the position of the linear axis. The feedback loop of the ${}_{v}PID$ controller contains a differentiating element s to calculate the velocity of the linear axis from its position. $U\left(s\right)$ and $X\left(s\right)$, respectively, are the target position and the actual position of the linear axis.

First, a model of the linear axis (the plant) is created based on its known electromechanical properties. In good approximation, the force exerted on the motion platform is proportional to the current through the voice coil, which is the output variable of the PID velocity controller. Friction is neglected in order to simplify the system model as it is mainly used for demonstration purposes here. The system can now be described by the differential equation

$$\begin{array}{cc}\hfill \sum F_i=C_F·i\left(t\right)& -m\ddot{x}\left(t\right)=0\hfill \\ \hfill & \iff \hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill C_F·I\left(s\right)& -m{s}^{2}X\left(s\right)=0\hfill \end{array}$$

(4)

where $C_F$ is a force constant, m is the moving mass of the platform, $i\left(t\right)$ is the current of the voice coil motor, and $\sum F_i$ is the sum of all forces acting on the platform. Equation (4) is the Laplace transform of Equation (3) and s is the Laplace variable. From Equation (4), the generalized transfer function of the plant can now be obtained:

$$\tilde{G}\left(s\right)=\frac{X\left(s\right)}{I\left(s\right)}=\frac{C_F}{m{s}^2}=\frac{{\tilde{b}}_0}{{\tilde{a}}_2{s}^2}.$$

(5)

A comparison with measurement data shows that the real linear axis additionally shows behavior that is best modeled as a dead time element. Potential sources for this dead time could be friction, stiction, or coil impedance among other things.

Additionally, the generalized transfer function can be simplified by reducing the number of unknown parameters. Define $a_2=1,b_0={\tilde{b}}_0/{\tilde{a}}_2$, the modified transfer function is now

$$G\left(s\right)=\frac{b_0}{s^2}{\mathrm{e}}^{-T_{d}s}.$$

(6)

The system coefficient $b_0$ and the value of the dead time $T_d$ are yet to be found. To do so, the frequency response of the plant is determined through measurements.

The frequency range for determining the frequency response of the plant is limited by the maximum possible excitation frequency of the controller. When the excitation frequency nears the maximum possible excitation frequency, the measurement accuracy decreases significantly. Therefore, the measurement data is intentionally cut at 2 kHz.

The model parameters are chosen in such a way that the model matches the measured frequency response.

It may turn out to be difficult to determine point-valued model parameters that universally represent the system behavior with sufficient accuracy. For example, due to measurement inaccuracies, production tolerances, and changes in environmental conditions model parameters may vary greatly in some cases. It may then be beneficial to model the system using interval parameters instead.

A detailed description of the system identification process is omitted here, as this falls outside the scope of this publication. For the system that is considered, the following model parameters are determined:

$$\begin{array}{cc}\hfill \left[b_0\right]& =[0.95·{10}^4,1.05·{10}^4],\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill T_d& =0.7\mathrm{ms}.\hfill \end{array}$$

(8)

For the midpoint of the interval $\left[b_0\right]$ a comparison between some exemplary measurement data and the plant model is shown in Figure 3. This illustrates that the model describes the general system behavior with good accuracy.

4. Performance Criteria for Interval Systems

In this section, four different categories for analyzing the performance of a linear time-invariant (LTI) interval system are defined. For each category, criteria are developed, which allow to characterize the system’s behavior. By combining the information from all four categories, significant knowledge of the system’s performance can be obtained.

The four categories that are differentiated are static gain, stability, settling time, and damping ratio. For linear dynamical systems with point-valued coefficients, these categories are well-covered in basic control theory literature. Here, the known approaches are extended to systems with interval coefficients. Furthermore, extensive consideration is given to transfer functions that describe a plant that contributes significant amounts of dead time to the control loop. It is assumed that the controller does not contribute any further dead time.

Each criterion is used to evaluate whether the system performance conforms to a specific condition. As the system in question is an interval system, the conclusion needs to be given as an interval boolean value. Interval booleans can take three values, denoting that the condition is met fully $\left[\mathtt{TRUE}\right]$, not at all $\left[\mathtt{FALSE}\right]$, or only by some parts of the interval system $\left[\mathtt{INDETERMINATE}\right]$. The evaluation of each criterion will therefore yield a result $\mathcal{R}\in \{\left[\mathtt{TRUE}\right],\left[\mathtt{FALSE}\right],\left[\mathtt{INDETERMINATE}\right]\}$.

Any linear system consisting of a plant with optional dead-time behavior and any number of cascaded control loops is able to be analyzed using the approach discussed in this publication. For a generic example of such a system with N cascaded control loops:

$$\left[C_i\right]\left(s\right)=\frac{{\displaystyle \sum _{n=0}^{n_{C,i}}}\left[d_n\right]s^n}{{\displaystyle \sum _{m=0}^{m_{C,i}}}\left[c_m\right]s^m},i=1,\dots ,N$$

(9)

is the interval transfer function of a controller,

$$\left[R_i\right]\left(s\right)=\frac{{\displaystyle \sum _{n=0}^{n_{R,i}}}\left[q_n\right]s^n}{{\displaystyle \sum _{m=0}^{m_{R,i}}}\left[r_m\right]s^m},i=1,\dots ,N$$

(10)

is an interval transfer function in the feedback loop and

$$\left[G\right]\left(s\right)=\frac{{\displaystyle \sum _{n=0}^{n_G}}\left[b_n\right]s^n}{{\displaystyle \sum _{m=0}^{m_G}}\left[a_m\right]s^m}{\mathrm{e}}^{-T_{d}s}$$

(11)

is the interval transfer function of the plant. The structure of this control loop is shown in Figure 4.

The open-loop transfer function of such a system generally has the following form:

$$\left[O\right]\left(s\right)=\frac{\sum \left[{\gamma }_c\right]s^c·{\mathrm{e}}^{-T_{d}s}}{\sum \left[{\alpha }_a\right]s^a·{\mathrm{e}}^{-T_{d}s}+\sum \left[{\beta }_b\right]s^b}.$$

(12)

This can be easily seen when iteratively deriving and comparing the open-loop transfer functions for a system with one, two, three, or more control loops.

The closed-loop transfer function is

$$\left[H\right]\left(s\right)=\frac{\left[O\right]\left(s\right)}{1+\left[O\right]\left(s\right)}$$

(13)

and the characteristic equation of the system is

$$\left[F\right]\left(s\right)=1+\left[O\right]\left(s\right)=1+\frac{\sum \left[{\gamma }_c\right]s^c·{\mathrm{e}}^{-T_{d}s}}{\sum \left[{\alpha }_a\right]s^a·{\mathrm{e}}^{-T_{d}s}+\sum \left[{\beta }_b\right]s^b}=0.$$

(14)

Remark 1.

In general, the presented approach can be applied to other types of systems, as long as it is possible to write their characteristic equation in the general form of Equation (14).

4.1. Static Gain Criterion

For many applications, it is desirable that the static gain K of the closed-loop system is equal or close to one. For an illustrative example, consider a linear axis and its controller as the closed-loop system. The input of this system is its commanded target position and the output is the actual position of the axis. For $t\to \infty$ and a constant input value, the output value should match the input value. For example though, if the gain was $K=2.0$, the linear axis would move twice as far as commanded relative to its zero point.

The final value theorem of Laplace is applied to the closed-loop transfer to determine the value of K. If the result is independent of all parameters that still need to be determined, static gain does not need to be considered during the parameterization of the controller. Furthermore, certain control loop structures guarantee that $K=1.0$, independent of how the controller parameters are chosen.

If the structure of the control loop does not guarantee that $K=1.0$, a criterion for evaluating the static gain during the controller parameterization is required. To bound the static gain, a maximum static error $\epsilon$ is defined. The acceptable system performance is the allowed range of the static gain of the closed-loop transfer system. This can be written as $\left[K\right]=[1-\epsilon ,1+\epsilon ]$.

A result ${\mathcal{R}}_{\epsilon }=li{m}_{s\to 0}\left[H\right]\left(s\right)\in \left[K\right]$ is obtained by evaluating the final value theorem of Laplace for the closed-loop system.

Remark 2.

This criterion is applicable without any further modification to systems exhibiting dead-time behavior as ${lim}_{s\to 0}{\mathrm{e}}^{-T_{d}s}=1$, meaning that the dead time element becomes a constant value during the evaluation. Therefore, the amount of dead time that is present in the system does not influence the static gain.

4.2. Stability Criterion for Interval Systems without Dead Time

First, the case of a linear dynamical system with interval coefficients and no dead-time behavior (i.e., $T_d=0$) is considered. To prove the stability of the LTI system, it is necessary to show that all roots of the characteristic equation $\left[F\right]\left(s\right)$ have negative real parts [16].

A very important finding on the stability of interval polynomials was made by Kharitonov in [14]. Kharitonov states that the stability of any interval polynomial $\left[p\right]\left(s\right)=\left[a_0\right]+\left[a_1\right]s+\dots +\left[a_n\right]s^n,a_i\in \mathbb{R}$ can be determined by evaluating the stability of four well-chosen point-valued polynomials. With $s=j\omega$ and $\omega \ge 0$, the polynomials to be examined are $k_{kl}$ as defined in [17]:

$$\begin{array}{cc}{g}_1\left(s\right)\hfill & ={\underline{a}}_0+{\overline{a}}_2{s}^2+{\underline{a}}_4{s}^4+\dots \hfill \\ g_2\left(s\right)\hfill & ={\overline{a}}_0+{\underline{a}}_2{s}^2+{\overline{a}}_4{s}^4+\dots \hfill \\ h_1\left(s\right)\hfill & ={\underline{a}}_{1}s+{\overline{a}}_3{s}^3+{\underline{a}}_5{s}^5+\dots \hfill \\ h_2\left(s\right)\hfill & ={\overline{a}}_{1}s+{\underline{a}}_3{s}^3+{\overline{a}}_5{s}^5+\dots \hfill \\ \hfill & \\ k_{kl}\left(s\right)\hfill & =g_k\left(s\right)+h_l\left(s\right);k,l\in \{1,2\}\hfill \end{array}$$

(15)

$g_1,g_2$ are purely real ($s^i=\pm {\omega }^i;s=jw;i=0,2,4,\dots$) while $h_1,h_2$ are purely imaginary ($s^i=\pm j{\omega }^i;s=jw;i=1,3,5,\dots$).

To prove the stability of the set of LTI systems $\left[H\right]\left(s\right)$, the four Kharitonov polynomials of the characteristic equation $\left[F\right]\left(s\right)$ are calculated. $\left[H\right]\left(s\right)$ is completely stable if all four Kharitonov polynomials are Hurwitz.

Conversely, though, it is impossible to conclude that no part of the interval system is stable if none of the four Kharitonov polynomials is Hurwitz. The Kharitonov theorem only allows to prove the stability of an interval polynomial but not the instability. A stability criterion for the LTI system based on the Kharitonov theorem can therefore only give a result $\mathcal{R}\in \left\{\right[\mathtt{TRUE}],[\mathtt{INDETERMINATE}\left]\right\}$ and is of limited use.

In [18], Padmanabhan shows that the complete instability of an interval polynomial can be determined by examining point-valued polynomials at the interval boundaries, similar to Kharitonov’s theorem. It is helpful to view this problem in coefficient space, where the interval coefficients of an n-th order polynomial represent an n-dimensional box. The edges of the said box are defined by the extreme values of the interval coefficients. The interval polynomial is completely unstable if instability can be proven for all edges of the box. For this, it is necessary to test the $2^n$ point-valued polynomials obtained by varying all n coefficients at their extreme values. This set of point-valued polynomials also contains the four Kharitonov polynomials defined above. Padmanabhan further determines that it is sufficient, to test $2^{n-1}$ polynomials, by arbitrarily fixing one of the coefficients, thereby reducing the number of polynomials that need to be evaluated.

A complete interval Boolean answer on the stability of the interval system can finally be given. To do this efficiently, the following is proposed: The four Kharitonov polynomials are evaluated first to try and prove the complete stability of the interval system. If complete stability cannot be proven and none of the four Kharitonov polynomials are stable, the $2^{n-1}$ polynomials defined by Padmanabhan are tested to try and prove the complete instability of the interval system instead. If this cannot be proven, the result is indeterminate, meaning that the interval system is partially stable.

4.3. Stability Criterion for Systems with Dead Time

In this section, an approach for proving the stability and instability of LTI interval transfer systems where the plant exhibits dead-time behavior is developed. The approach shown in the previous section is not applicable, as the Hurwitz criterion and similar stability criteria, such as the Routh criterion, cannot be used. These criteria require a rational transfer function, meaning that the numerator and denominator functions of the closed-loop transfer function need to be polynomials. With the existence of a dead time term in the transfer function, the numerator and denominator of the closed-loop transfer function generally become quasi-polynomials, though. Therefore, a different approach is required.

4.3.1. A Necessary Stability Condition for Systems with Dead-Time Behavior

Qualitatively, one can describe the influence of a dead time element on the stability of a closed-loop transfer system quite easily. A dead time element has a constant unity gain and a phase angle that decreases linearly with increasing frequency [16]. This always reduces the phase margin of the system wherefore it is impossible for a dead time element to improve the stability of a closed control loop. More precisely, three types of behaviors can be differentiated in this context.

The first type of system is guaranteed to be unstable, independent of any dead-time behavior that may exist.

For the second type of system, dead time does not influence its stability, meaning that the magnitude of dead time can become arbitrarily large. This requires an infinite phase margin, which is only possible if the gain of the system never exceeds one (0 dB) at any frequency.

For the third type of system, the stability is dependent on the magnitude of dead time. In this third case, there exists a critical dead time $T_{crit}$ for which the system is critically stable. A system of the third type is stable for any $T_d$ where $0\le T_d<T_{crit}$ and unstable if $T_d>T_{crit}$ [16,19,20].

Systems of type one and type two can also be understood as edge cases of the third type with $T_{crit}=0$ and $T_{crit}\to \infty$, respectively.

The following necessary condition can now be formulated:

For any system with dead-time behavior to be stable, it is first necessary that the same system without dead time is stable as well.

This condition can be evaluated easily. By assuming a hypothetical dead time of zero to create a reduced system, the stability of this reduced system can be proven using well-known criteria, such as the Hurwitz criterion. The system with dead time can only be stable if the reduced system is stable as well.

If the reduced system in question is an interval system, the methods from Section 4.2 need to be employed to prove its stability.

4.3.2. Sufficient Conditions for the Stability of Point-Valued Systems with Dead-Time Behavior

In this section, a sufficient condition for the stability of point-valued systems with dead-time behavior is developed. The condition is based on a sufficient stability criterion that was first presented by Satche [19]. The Satche criterion itself is an extension of the commonly used Nyquist criterion [16] to systems with dead time. In this publication, the original Satche criterion is extended in two ways.

As a first step, a new approach for the evaluation of the criterion is developed. For this purpose, the original criterion is introduced for point-valued systems, as it is described by Satche. Then, the new method for evaluating the criterion is explained. The need for this extension arises because it is necessary to efficiently evaluate the criterion in a controller parameterization algorithm.

Following that, the second step is the extension of the criterion and its new evaluation method to LTI interval systems. This is required so that it is possible to apply the criterion to the interval models of uncertain systems that are considered in this publication.

For the Satche criterion to be applicable, it needs to be possible to write the characteristic equation $F\left(s\right)=0$ as $\tilde{F}\left(s\right)-(-{\mathrm{e}}^{-T_{d}s})=0$. With the generic characteristic equation defined in Equation (14), this results in

$$\tilde{F}\left(s\right)-(-{\mathrm{e}}^{-T_{d}s}-)=\frac{\sum {\beta }_b{s}^b}{\sum {\alpha }_a{s}^a+\sum {\gamma }_c{s}^c}-(-{\mathrm{e}}^{-T_{d}s})=0,$$

(16)

$$\tilde{F}\left(s\right)=\frac{\sum {\beta }_b{s}^b}{\sum {\alpha }_a{s}^a+\sum {\gamma }_c{s}^c}=\frac{\sum {\beta }_b{s}^b}{\sum {\delta }_d{s}^d},d=max(a,c).$$

(17)

According to Satche, when s follows the Nyquist curve $\mathcal{D}$, the system is stable if the total variation of the argument

$$V_{\mathcal{D}}=V\left(arg\left(\tilde{F}\left(s\right)-\left(-{\mathrm{e}}^{-T_{d}s}\right)\right),\mathcal{D}\right)=2\pi (N-P)$$

(18)

where N is the number of zeros and P is the number of poles of the characteristic equation that lie within the contour of $\mathcal{D}$.

$\mathcal{D}$ encloses the open right half-plane. Starting at the origin, it follows the positive imaginary axis to $j\infty$, enclosing the right half-plane in a semi-circle with radius $R\to \infty$ before ending in the origin again. The curve does not enclose poles of the system that lie on the imaginary axis and such poles are excluded by taking a small detour around them [16]. An example of the Nyquist curve is shown in Figure 5.

The images of $\mathcal{D}$ by $\tilde{F}\left(s\right)$ and $-{\mathrm{e}}^{-T_{d}s}$ are defined as curves in $\mathbb{C}$:

$$\begin{array}{cccc}\hfill \tilde{\mathcal{F}}& =\{\tilde{F}\left(s\right)\hfill & & |\forall s\in \mathcal{D}\},\hfill \end{array}$$

(19)

$$\begin{array}{cccc}\hfill \mathcal{E}& =\{-{\mathrm{e}}^{-T_{d}s}\hfill & & |\forall s\in \mathcal{D}\}.\hfill \end{array}$$

(20)

The two curves $\tilde{\mathcal{F}}$ and $\mathcal{E}$ are connected by a moving vector in the complex plane. The angle that this vector encloses with the positive real axis is $arg\left(\tilde{F}\left(s\right)-(-{\mathrm{e}}^{-T_{d}s})\right)$.

When s follows the Nyquist curve $\mathcal{D}$, the curve $\mathcal{E}$ is a circle with radius one centered on the origin for the part where s travels along the imaginary axis. The radius of this circle decreases towards zero, while s follows the semi-circle that encloses the positive real half-plane. If $T_d=0$, the curve $\mathcal{E}$ is reduced to a point at $(-1,0j)$ for all parts of $\mathcal{D}$.

The shape of $\tilde{\mathcal{F}}$ is far more variable than $\mathcal{E}$, but in general, it is symmetric with respect to the real axis.

The stability criterion in the sense of Satche is now modified for easier evaluation. Instead of determining the number of poles P, zeros N, and the total variation of the argument of the complex vector, an alternative approach is developed for the application in this publication.

It was already mentioned that for $T_d=0$, the curve $\mathcal{E}$ is reduced to the point $(-1,0j)$, and the tail of the complex vector is then fixed at this point. In that case, the stability criterion in the sense of Satche becomes the general Nyquist criterion. From this, it is possible to recognize that a reduced system without dead time is stable in the sense of Satche if the characteristic equation of the said reduced system is Hurwitz.

Previously, it was already explained that a dead time element cannot improve the stability of any system. Therefore, when going back to the system with dead time, it is consequentially necessary that the presence of the dead time element does not contribute toward changing the total variation of the argument $V_{\mathcal{D}}$. Any change of the total variation of the argument would result in the stability condition no longer being fulfilled, see Equation (18).

Based on this statement, the sufficient stability condition in Equation (18) can now be redefined as a combination of the following: First, the reduced system without dead time has to be stable, and second, the existence of the dead time element must then not change the total variation of the argument of the complex vector between $\mathcal{E}$ and $\tilde{\mathcal{F}}$. The former is trivial to evaluate and conditions for the latter are introduced in the following.

First, it is helpful to understand how the total variation of the argument $V_{\mathcal{D}}$ describes the behavior of the complex vector between $\mathcal{E}$ and $\tilde{\mathcal{F}}$. To give a descriptive example for the total variation of the argument, if the complex vector rotates $-6\pi$ (clockwise) and then $+2\pi$ (anti-clockwise), the total variation of the argument of the complex vector is $-4\pi$. I.e., the argument of the complex vector is varied by $-4\pi$. Notice that the total variation of the argument is proportional to the winding number of the vector with a proportionality factor of $2\pi$.

Because $\mathcal{D}$ is a closed curve, the complex vector between $\mathcal{E}$ and $\tilde{\mathcal{F}}$ starts and ends in the exact same position with the same orientation. $V_{\mathcal{D}}$ can therefore only be an integer multiple of $2\pi$. This can also be seen when taking a look at the stability condition in Equation (18). The stability of the system is therefore dependent on the number of full windings of the complex vector while s takes one tour around $\mathcal{D}$.

Thus, the requirement that the presence of the dead time element may not change the total variation of the argument of the complex vector is now easier to understand. The condition simply states that no additional winding of the complex vector may take place when dead time is non-zero.

To analyze if the dead time element changes the total variation of the argument $V_{\mathcal{D}}$, three distinct cases need to be differentiated. These cases are visualized in Figure 6. For each case, exemplary behavior of the curve $\tilde{\mathcal{F}}$ is shown. To simplify the examples, all three exemplary systems are chosen in such a way that $V_{\mathcal{D}}=0$ if $T_d=0$, i.e., the tip of the complex vector does not wind around its tail if the dead time is zero. To recognize this, one can imagine that the tail of the complex vector is fixed in $(-1,0j)$ and then imagine that the tip of the vector moves along the path of $\tilde{\mathcal{F}}$. It can be seen that the vector never performs a full rotation and, thus, $V_{\mathcal{D}}=0$. It is then necessary that this holds true when the tail of the complex vector is allowed to circle on $\mathcal{E}$ so that the system is stable when the dead time is non-zero.

• Case 1:stable, dead time independent

If $\tilde{\mathcal{F}}$ starts outside the unit circle and never enters it, as shown in Figure 6a, the system is stable independently of the magnitude of dead time.

In the example, the tail of the complex vector is always to the left of its tip. Therefore, the vector can never make a full rotation, no matter if its tail is fixed in $(-1,0j)$ or if it is moving along the curve $\mathcal{E}$.

• Conditions for Case 1: $|\tilde{F}\left(0\right)|>=1$ and $\tilde{\mathcal{F}}$ has no intersection with the unit circle.
• Example for Case 1: $O\left(s\right)=\frac{1.0}{0.2s+1.2}·{\mathrm{e}}^{-T_{d}s}$ (PT₁ element with additional dead time), shown in Figure 6a.
• Case 2: dead time dependent stability

When $\tilde{\mathcal{F}}$ starts inside the unit circle, then exits and circles around before entering from the opposite side (Figure 6b), there exists a critical dead time as defined in Section 4.3.1. It is important that $\tilde{\mathcal{F}}$ only enters the unit circle at the beginning and the end of the curve, where $\omega$ tends towards zero. This behavior can be observed with many real systems.

In this case, the tail of the complex vector can rotate around its tip if the tip of the vector stays inside the unit circle for too long. The vector then rotates around fully and additional winding is caused, thus $V_{\mathcal{D}}$ is changed and the system is unstable. Figure 7a shows the beginning of the curves $\mathcal{E}$ and $\tilde{\mathcal{F}}$ for a stable system. There, the complex vector never performs a full rotation. Contrary to that, Figure 7b shows the unstable system where the tail of the complex vector rotates around its tip, causing additional winding.

Once the tip of the complex vector exits the unit circle, the movement of the tail of the complex vector on $\mathcal{E}$ becomes irrelevant for the total variation of the argument. Consider that $\tilde{\mathcal{F}}$ extends towards infinity while $\mathcal{E}$ has a finite size. By “zooming out” far enough, $\mathcal{E}$ is effectively reduced to a single point similar to the case where $T_d=0$. The remaining minuscule movement of the tail of the complex vector at this point is inconsequential.

• Conditions for Case 2: $|\tilde{F}\left(0\right)|<1$ and $\tilde{\mathcal{F}}$ has exactly two intersections with the unit circle.
• Example for Case 2: $O\left(s\right)=\frac{4.16·{10}^{-8}{s}^3+1.132·{10}^{-5}{s}^2+1.0}{2.0·{10}^{-11}{s}^4}·{\mathrm{e}}^{-T_{d}s}$, shown in Figure 6b.
• Case 3: unstable, dead time independent

Figure 6c shows $\tilde{\mathcal{F}}$ entering the unit circle for high frequencies $\omega \to \infty$. The position of the tail of the complex vector on $\mathcal{E}$ is given by ${\mathrm{e}}^{-T_{d}j\omega }$ (see Equation (20) for reference) which is undefined for $\omega \to \infty$. Therefore, the winding of the complex vector is undefined as well and, thus, any system with this behavior must be considered unstable.

A slight variation of this occurs if $\tilde{\mathcal{F}}$ only skims through the unit circle at high frequencies $\omega$ but is outside the unit circle again for $\omega \to \infty$. This can be viewed as an edge case that is nearing the just-explained problem of the undefined position of the complex vector’s tail. In this special (and potentially only theoretical) case, stability is hypothetically possible but any stability margin will be minimal. The stability margin will become even smaller the higher the frequencies $\omega$ are, for which $\tilde{\mathcal{F}}$ skims through the unit circle, finally becoming undefined as $\omega$ nears infinity. Therefore, and especially in the context of analyzing uncertain systems, any system showing this behavior will be considered practically unstable in this publication. In short, the curve $\tilde{\mathcal{F}}$ is only allowed to enter the unit circle near $\omega =0$ (which is Case 2), else the system is considered to be unstable for any amount of dead time.

• Conditions for Case 3: $\tilde{F}\left(0\right)<1$ and $\tilde{\mathcal{F}}$ has more than two intersections with the unit circle or $|\tilde{F}\left(0\right)|>=1$ and $\tilde{\mathcal{F}}$ has any intersections with the unit circle.
• Example for Case 3: $O\left(s\right)=-\frac{0.002s^2+0.02s+1.0}{0.7}·{\mathrm{e}}^{-T_{d}s}$, shown in Figure 6c.

To evaluate stability and to determine which case is applicable, it is necessary to find all values ${\omega }_{crit,i}$ for which $\tilde{\mathcal{F}}$ intersects the unit circle.

From the conditions given above, it follows that if ${lim}_{s\to \infty }\tilde{F}\left(s\right)<=1$, the system corresponds to Case 3 and is, therefore, unstable. Accordingly, for Case 1 and Case 2, it can be assumed that ${lim}_{s\to \infty }\tilde{F}\left(s\right)>1$. Thus, all values of $\tilde{\mathcal{F}}$ lie strictly outside the unit circle while s traverses the semi-circle with infinite radius in the right half plane. Intersections of $\tilde{\mathcal{F}}$ with the unit circle can therefore only occur for purely imaginary s. The problem of finding the intersections ${\omega }_{crit,i}$ can now be expressed as

$$|\tilde{\mathcal{F}}|=\frac{|\sum {\beta }_b{s}^b|}{|\sum {\delta }_d{s}^d|}=|-{\mathrm{e}}^{-T_{d}s}|=1;s=j\omega ,\omega \in \mathbb{R}.$$

(21)

The absolute values of the complex polynomials can be written as their Euclidean norm to obtain

$$\frac{|\sum {\beta }_b{s}^b|}{|\sum {\delta }_d{s}^d|}=\frac{\sqrt{\Re {\left\{\sum {\beta }_b{s}^b\right\}}^2+\Im {\left\{\sum {\beta }_b{s}^b\right\}}^2}}{\sqrt{\Re {\left\{\sum {\delta }_d{s}^d\right\}}^2+\Im {\left\{\sum {\delta }_d{s}^d\right\}}^2}}=1.$$

(22)

The real and imaginary parts of the polynomials are defined as

$$\begin{array}{cc}\Re \{\sum {\beta }_b{s}^b\}\hfill & :=\sum _{k=0}^{n/2}{\left.\Re \left\{{\beta }_b\right\}{s}^b\right|}_{b=2k}+\sum _{k=0}^{(n-1)/2}{\left.\Im \left\{{\beta }_b\right\}{s}^b\right|}_{b=2k+1}\hfill \\ \hfill & =\Re \left\{b_0\right\}+\Im \left\{b_1\right\}s+\Re \left\{b_2\right\}{s}^2+\Im \left\{b_3\right\}{s}^3+\dots \hfill \end{array}$$

(23)

$$\begin{array}{cc}\Im \{\sum {\beta }_b{s}^b\}\hfill & :=\sum _{k=0}^{n/2}{\left.\Im \left\{{\beta }_b\right\}{s}^b\right|}_{b=2k}+\sum _{k=0}^{(n-1)/2}{\left.\Re \left\{{\beta }_b\right\}{s}^b\right|}_{b=2k+1}\hfill \\ \hfill & =\Im \left\{b_0\right\}+\Re \left\{b_1\right\}s+\Im \left\{b_2\right\}{s}^2+\Re \left\{b_3\right\}{s}^3+\dots \hfill \end{array}$$

(24)

$\Re \{\sum {\delta }_d{s}^d\}$ and $\Im \{\sum {\delta }_d{s}^d\}$ are obtained equivalently.

It is then possible to transform Equation (22) into

$$\Re {\left\{\sum {\beta }_b{s}^b\right\}}^2+\Im {\left\{\sum {\beta }_b{s}^b\right\}}^2-\Re {\left\{\sum {\delta }_d{s}^d\right\}}^2-\Im {\left\{\sum {\delta }_d{s}^d\right\}}^2=0.$$

(25)

After substituting $s=jw$, all ${\omega }_{crit,i}$ are found as the real roots of the now purely real polynomial. They appear in pairs of $\pm {\omega }_{crit,i}$ as $\tilde{F}\left(s\right)$ is symmetric with respect to the real axis.

At this point, it is possible to differentiate whether the system that is analyzed corresponds to Cases 1, 2, or 3, and it may already be possible to conclude stability or instability. If the system corresponds to Case 2, though, it is necessary to calculate its critical dead time before a conclusion is possible.

Arguing graphically, a system that corresponds to Case 2 is stable when $\tilde{\mathcal{F}}$ starts inside the unit circle and exits before $\mathcal{E}$ passes the intersection of $\tilde{\mathcal{F}}$ with the unit circle. Figure 7a shows, for a stable system, the beginning of the curves $\mathcal{E}$ and $\tilde{\mathcal{F}}$ as well as the complex vector that connects them. It can be seen that the complex vector never completes a full rotation while $\tilde{\mathcal{F}}$ is within the unit circle. Therefore, $V_{\mathcal{D}}$ is unchanged compared to when the tail of the complex vector is fixed in $(-1,0j)$ in the case of $T_d=0$.

Figure 7b on the other hand shows an unstable system. Here, the tail of the complex vector rotates around its tip before $\tilde{\mathcal{F}}$ exits the unit circle. As a result, the total variation of the argument is changed and the system is unstable.

$T_d$ defines how fast the tail of the complex vector moves along the unit circle for increasing $\omega$. Critical stability occurs when the tip and the tail of the complex vector meet at the intersection of $\tilde{\mathcal{F}}$ with the unit circle. For a system that corresponds to Case 2 (which by definition is the only case that is applicable here), there exists only one pair of $\pm {\omega }_{crit}$. Because $\tilde{\mathcal{F}}$ is symmetric with respect to the real axis, it is sufficient to calculate $T_{d,crit}$ based on the positive value ${\omega }_{crit}$.

The calculation of $T_{d,crit}$ is done by comparing the definition of the dead time element with the definition of a generic unit vector in the complex plane.

With $\pi -\phi$ being the argument of the generic unit vector pointing from the origin to $-{\mathrm{e}}^{-T_{d}s}$, as shown in Figure 8, the critical dead time of the system can be calculated as shown in the following.

At the intersection of $\tilde{\mathcal{F}}$ with the unit circle, the equation $\omega ={\omega }_{crit}$ holds. With the help of the generic unit vector, it is possible to define the critical angle at this intersection:

$${\phi }_{crit}=\pi -arg\tilde{F}\left(j{\omega }_{crit}\right).$$

(26)

Next, the function that describes the dead time element is modified. s is substituted by $j\omega$ and the leading negative sign is written as an angular offset.

$$-{\mathrm{e}}^{-T_{d}s}=-{\mathrm{e}}^{-j{T}_d\omega }={\mathrm{e}}^{j(\pi -T_d\omega )}$$

(27)

After that, the relation of $T_d$ to $\phi$ and $\omega$ is found by equating the coefficient of the exponential function of the dead time element with the coefficient of the exponential function that describes the generic unit vector.

$$\begin{array}{cc}\hfill {\mathrm{e}}^{j(\pi -T_d\omega )}& ={\mathrm{e}}^{j(\pi -\phi )}\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \pi -T_d\omega & =\pi -\phi \hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill T_d& =\frac{\phi }{\omega }\hfill \end{array}$$

(30)

Finally, it is possible to calculate

$$T_{d,crit}=\frac{{\phi }_{crit}}{{\omega }_{crit}}.$$

(31)

The system is stable if

$$T_{d,crit}>T_d·(1+S),S\ge 0$$

(32)

where $T_d$ is the nominal dead time of the system and S is an additional relative stability margin.

An overview of all the various steps that are necessary to evaluate the sufficient stability condition is shown in Appendix A, Figure A1 in the form of a flow diagram.

4.3.3. Sufficient Conditions for the Stability of Interval-Valued Systems with Dead-Time Behavior

In this section, the second part of the extension of the Satche criterion is developed. The Satche criterion and the already introduced new method for its evaluation are now extended to LTI interval systems so that they apply to the type of uncertain system that is considered in this publication.

To prove the stability of an interval system with dead time, it is first again necessary that the same interval system without dead time is stable. For this, the reduced system without dead time can be analyzed with the methods described in Section 4.2. If the reduced interval system without dead time is fully stable, it is then necessary to show for all parts of the interval system that the presence of the dead time element does not change the total variation of the argument of the complex vector between the two curves. The general approach for this is equivalent to the approach that is used for point-valued systems.

The original stability condition given by Satche for point-valued systems (see Equation (18)) differs only slightly for the interval systems:

$$\left[V_{\mathcal{D}}\right]=V\left(arg\left(\left[\tilde{F}\right]\left(s\right)-\left(-{\mathrm{e}}^{-\left[T_d\right]s}\right)\right),\mathcal{D}\right)=2\pi (\left[N\right]-\left[P\right]).$$

(33)

For any value $s\in \mathbb{C}$, $\left[\tilde{F}\right]\left(s\right)$ is now a rectangular shaped set in $\mathbb{C}$, as shown in Figure 9 [17,21,22].

When analyzing whether the dead time element changes the total variation of the argument $\left[V_{\mathcal{D}}\right]$, the same three generalized cases that are described in Section 4.3.2 need to be distinguished. The described conditions for the value of $|\left[\tilde{F}\right]\left(0\right)|$ and the number of intersections of $\left[\tilde{\mathcal{F}}\right]$ with the unit circle need to be met by all four curves that are traced out by the corners of the rectangular shaped set $\left[\tilde{F}\right]\left(s\right)$ so that they hold for the entire set. This reasoning follows graphically, similar to the interpretation of Kharitonov’s theorem in [21,22].

All intersections $\left[{\omega }_{crit,i}\right]$ that $\left[\tilde{\mathcal{F}}\right]$ has with the unit circle are now found as the real interval roots of the following equation:

$$\Re {\left\{\sum \left[{\beta }_b\right]s^b\right\}}^2+\Im {\left\{\sum \left[{\beta }_b\right]s^b\right\}}^2-\Re {\left\{\sum \left[{\delta }_d\right]s^d\right\}}^2-\Im {\left\{\sum \left[{\delta }_d\right]s^d\right\}}^2=0.$$

(34)

Hansen and Walster describe an approach to sharply bound the roots of an interval polynomial in [23] with which it is straightforward to determine all $\left[{\omega }_{crit,i}\right]$. While the approach calculates sharp bounds on interval roots, the interval math operations involved in deriving Equation (34) introduce uncertainties to the coefficients of the equation. All $\left[{\omega }_{crit,i}\right]$ will, therefore, be conservative outer approximations of the intersection ranges.

Now, it is possible to differentiate between the three generalized cases, which were described in Section 4.3.2. If the interval system corresponds to Case 2, its critical dead time must be found before it is possible to determine its stability. Two values $\left[T_{d,crit,i}\right]$ should be calculated for the upper and lower bound of $\left[{\omega }_{crit}\right]$ individually to reduce uncertainties in the calculation. Equivalent to the non-interval system, only one pair $\pm \left[{\omega }_{crit}\right]$ exists for a system that corresponds to Case 2, and it is sufficient to calculate $T_{d,crit}$ based on the positive value ${\omega }_{crit}$.

$$\begin{array}{cccc}\hfill \left[T_{d,crit,1}\right]& =\frac{\left[{\phi }_{crit,1}\right]}{{\underline{\omega }}_{crit}}\hfill & & =\frac{\pi -arg\left[\tilde{F}\right]\left(j{\underline{\omega }}_{crit}\right)}{{\underline{\omega }}_{crit}}\hfill \end{array}$$

(35)

$$\begin{array}{cccc}\hfill {}_{d,crit,2}]& =\frac{\left[{\phi }_{crit,2}\right]}{{\overline{\omega }}_{crit}}\hfill & & =\frac{\pi -arg\left[\tilde{F}\right]\left(j{\overline{\omega }}_{crit}\right)}{{\overline{\omega }}_{crit}}\hfill \end{array}$$

(36)

After that, $\left[T_{d,crit}\right]$ is obtained as the interval hull of the two intervals $\left[T_{d,crit,i}\right]$:

$$\left[T_{d,crit}\right]=[\left[T_{d,crit,1}\right],\left[T_{d,crit,2}\right]].$$

(37)

In Section 4.3.2, stability related to dead time was only discussed for systems with a point-valued amount of dead time. This is now extended to systems with an uncertain amount of dead time. When the analyzed system corresponds to Case 2 and its dead time is given by the interval $\left[T_d\right]$, the following result is obtained for the stability of the system:

$$\begin{array}{cccc}& {\mathcal{R}}_s=\left[\mathrm{TRUE}\right]\hfill & & \mathrm{if}\left[T_{d,crit}\right]>\left[T_d\right]·(1+S),\hfill \end{array}$$

(38)

$$\begin{array}{cccc}& {\mathcal{R}}_s=\left[\mathrm{FALSE}\right]\hfill & & \mathrm{if}\left[T_{d,crit}\right]<\left[T_d\right]·(1+S),\hfill \end{array}$$

(39)

$$\begin{array}{lll}{\mathcal{R}}_s=\left[\mathrm{INDETERMINATE}\right]\hfill & & \mathrm{else}\hfill \end{array}$$

(40)

where S is again a relative stability margin ($S\ge 0$).

Further, ${\mathcal{R}}_s=\left[\mathrm{TRUE}\right]$ if the analyzed system instead corresponds to Case 1 and ${\mathcal{R}}_s=\left[\mathrm{FALSE}\right]$ if the system corresponds to Case 3.

Remark 3.

When calculating $\left[{\phi }_{crit,1}\right]=\pi -arg\left[\tilde{F}\right]\left(j{\underline{\omega }}_{crit}\right)$ and $\left[{\phi }_{crit,2}\right]=\pi -arg\left[\tilde{F}\right]\left(j{\overline{\omega }}_{crit}\right)$, additional (conservative) uncertainties are introduced. While ${\underline{\omega }}_{crit}$ and ${\overline{\omega }}_{crit}$ were calculated as the points where $\left[\tilde{F}\right]\left(j{\omega }_{crit}\right)$ first and last intersects the unit circle for $\omega \in [0,\infty )$, inserting these values into $\left[\tilde{F}\right]\left(j\omega \right)$ will yield an interval result. This is graphically shown in Figure 10a.

It is possible to slightly reduce the uncertainty involved in calculating each $\left[{\phi }_{crit,i}\right]$, though. Per definition, $|\tilde{F}\left(j{\omega }_{crit}\right)|=1$, therefore only the values in the set $\{x\in \left[\tilde{F}\right]\left(j\left[{\omega }_{crit}\right]\right)|\left|x\right|=1\}$ are valid solutions. The extrema of this set are given by the intersections of the boundaries of the box $\left[\tilde{F}\right]\left(j{\omega }_{crit}\right)$ with the unit circle and the set itself is an arc segment on the unit circle as shown in Figure 10b. Based on within which quadrant(s) the rectangle is contained and based on which corners lie within the unit circle, it is possible to determine which sides of the rectangle are intersected by the unit circle. Using simple trigonometry, the exact intersection points can then be calculated. To give an example, assume that the rectangle is fully located within the second quadrant. Furthermore, only its lower right corner lies within the unit circle. It is obvious that the unit circle must then intersect the right and the bottom line segment of the rectangle (compare with Figure 10b, the rectangle at the top). The lower and upper bounds of $\left[{\phi }_{crit}\right]$ are then given by $sin{\underline{\phi }}_{crit}=\underline{\Im }\left\{\left[\tilde{F}\right]\left(j{\omega }_{crit}\right)\right\}$ and $cos{\overline{\phi }}_{crit}=-\overline{\Re }\left\{\left[\tilde{F}\right]\left(j{\omega }_{crit}\right)\right\}$.

Figure 10. Graphical representation of the calculation of $\left[{\phi }_{crit,i}\right]$. (a) Simple calculation of $\left[{\phi }_{crit,i}\right]$ as the argument of $\left[\tilde{F}\right]\left(j{\omega }_{crit}\right)$. (b) Calculation of $\left[{\phi }_{crit,i}\right]$ with reduced uncertainty based on the intersections of $\left[\tilde{F}\right]\left(j{\omega }_{crit}\right)$ with the unit circle.

Figure 10. Graphical representation of the calculation of $\left[{\phi }_{crit,i}\right]$. (a) Simple calculation of $\left[{\phi }_{crit,i}\right]$ as the argument of $\left[\tilde{F}\right]\left(j{\omega }_{crit}\right)$. (b) Calculation of $\left[{\phi }_{crit,i}\right]$ with reduced uncertainty based on the intersections of $\left[\tilde{F}\right]\left(j{\omega }_{crit}\right)$ with the unit circle.

4.4. Dynamics Criteria

The approach that has been shown so far is capable of proving the stability of an LTI interval system with dead time as well as bounding its static gain. Stability alone though is usually not a sufficient condition for designing and parameterizing controllers. It is often desirable to prescribe certain minimum requirements for the dynamic behavior of a system. Therefore, the introduced approach shall now be extended so that it becomes possible to define requirements for the dynamic system behavior.

The influence of the location of poles on the dynamic behavior of a system is well-known in linear control theory. Poles far left from the imaginary axis contribute to a faster system response. The ratio between the real and imaginary parts of a pole location affects how a system is damped. For underdamped second-order systems specifically, the quantitative definition of settling time and damping ratio is possible. Such exact definitions of the damping ratio and settling time generally do not exist for higher-order systems but the notion holds qualitatively. In case a so-called dominant pole pair exists, the approximation becomes fairly exact, even [16].

For simplification, the terms damping ratio and settling time are used loosely for n-th order systems in the following. The qualitative properties of the system are referred to.

By applying this knowledge about the influence of closed-loop pole locations, it is possible to transform the problem of bounding the dynamic system behavior into a stability question. To guarantee the desired behavior, it is necessary that all closed-loop poles of the system lie within a specific region of the complex plane. When this is represented as a transformation of the complex plane, the problem is reduced to proving the stability of the transformed system. It is then possible again, to use the approach described in the previous sections of this publication to test whether an interval system as a whole conforms to these requirements.

Two separate transformations of the complex plane are required to bound settling time and damping ratio, as is shown in Figure 11. These are discussed in more detail in the following.

The approach of defining a specific region in the complex plane to bound the dynamic behavior of a system is already well-known [13,16,17]. This section will, therefore, mainly focus on the specific details that are encountered for the interval system in the context of the approach that is described in this publication.

4.4.1. Bounding Settling Time

To bound the settling time of a system, all poles are restricted to the area on the left side of a vertical line at $-\sigma$. The settling time of the system can then be approximated as $T_{s,2\%}\approx 4/\sigma$ [16,17].

This constraint can be represented by shifting the complex plane along the real axis. The transformed coordinate system is

$$\check{s}=s+\sigma =j\omega +\sigma ,\sigma >0.$$

(41)

For any interval or non-interval polynomial, this transformation of the coordinate system can be represented as a transformation of the polynomial coefficients.

$$\begin{array}{ccccc}P\left(\check{s}\right)\hfill & =p_0\hfill & +p_1\check{s}\hfill & +p_2{\check{s}}^2\hfill & +\dots \hfill \\ \hfill & =p_0\hfill & +p_1(s+\sigma )\hfill & +p_2{(s+\sigma )}^2\hfill & +\dots \hfill \\ \hfill & =p_0\hfill & +p_{1}s+p_1\sigma \hfill & +p_2{s}^2+2p_{2}s\sigma +p_2{\sigma }^2\hfill & +\dots \hfill \\ \hfill & =(p_0+p_1\sigma +p_2{\sigma }^2+\dots )\hfill & +(p_1+2p_2\sigma +\dots )s\hfill & +(p_2+\dots )s^2\hfill & +\dots \hfill \\ \hfill & ={\check{p}}_0\hfill & +{\check{p}}_{1}s\hfill & +{\check{p}}_2{s}^2\hfill & +\dots \hfill \end{array}$$

(42)

Applying this coordinate system transformation to the exponential function of the dead time element results in

$${\mathrm{e}}^{-T_d\check{s}}={\mathrm{e}}^{-T_d(j\omega +\sigma )}={\mathrm{e}}^{-T_d\sigma }·{\mathrm{e}}^{-T_{d}s}=C{\mathrm{e}}^{-T_{d}s}$$

(43)

with $C={\mathrm{e}}^{-T_d\sigma }\le 1$ being a constant factor. Therefore, the transformation simply scales the radius of the dead time element representation as a circle in the complex plane.

The transformation of Equation (21) leads to

$$|{\tilde{\mathcal{F}}}_{\check{s}}|=\frac{|\sum \left[{\check{\beta }}_b\right]s^b|}{|\sum \left[{\check{\delta }}_d\right]s^d|}=|-{\mathrm{e}}^{-T_d\check{s}}|={\mathrm{e}}^{-T_d\sigma }=C;s=j\omega ,\omega \in \mathbb{R}.$$

(44)

The evaluation of the stability of the transformed system is similar to the untransformed system discussed previously. First, the system without dead time needs to be stable. After that, all $\pm {\omega }_{crit}$ are now found as the real roots of

$$\Re {\left(\sum \left[{\check{\beta }}_b\right]s^b\right)}^2+\Im {\left(\sum \left[{\check{\beta }}_b\right]s^b\right)}^2-C·\left(\Re {\left(\sum \left[{\check{\delta }}_d\right]s^d\right)}^2+\Im {\left(\sum \left[{\check{\delta }}_d\right]s^d\right)}^2\right)=0,$$

(45)

which is again a purely real polynomial. The same three cases of the generalized system behavior need to be distinguished as before. If applicable, $\left[T_{d,crit}\right]$ can be calculated as explained previously and needs to adhere to the same conditions.

A result ${\mathcal{R}}_{\sigma }\in \{\left[\mathrm{TRUE}\right],\left[\mathrm{FALSE}\right],\left[\mathrm{INDETERMINATE}\right]\}$ is obtained, depending on the specific result of the stability analysis of the transformed system.

Remark 4.

Any system that conforms to the settling time bound is necessarily stable. For $\sigma >0$, all system poles need to have negative real parts, which are also the general stability requirements. System stability and conformity with the settling time bound can, therefore, be evaluated simultaneously if no differentiated result is desired for the purpose of analyzing the results.

4.4.2. Bounding Damping Ratio

To bound the damping ratio of a system, all poles are restricted to the area between two sloped lines in the complex open left half-plane. The lines pass through the origin and enclose an angle $\pm \theta$ with the imaginary axis (see Figure 11) [16,17]. The damping ratio of a second-order system is defined as $\zeta =arcsin\theta$. For systems of higher order, an exact definition of the damping ratio does not exist, but an approximation can be made using the relation that is defined for second-order systems. This approximation relies on the existence of a dominant pole pair and its accuracy varies depending on the existence of such a pole pair and its dominance [16].

The damping ratio constraint can be represented by a rotation of the complex plane. In the original coordinate system, all poles are purely real or they appear as complex conjugates. The untransformed pole locations are therefore symmetric with respect to the real axis. As consequence, it is sufficient to rotate the complex plane by either $+\theta$ or $-\theta$ and evaluate the stability of the transformed system only once. The transformed coordinate system is

$$\check{s}=s{\mathrm{e}}^{-j\theta }=j\omega {\mathrm{e}}^{-j\theta }.$$

(46)

For any polynomial, it is again possible to represent the transformation of the coordinate system as a transformation of the polynomial coefficients.

$$\begin{array}{cccccccccc}P\left(\check{s}\right)\hfill & =p_0\hfill & \hfill & +p_1\check{s}\hfill & \hfill & +p_2{\check{s}}^2\hfill & \hfill & +p_3{\check{s}}^3\hfill & \hfill & +\dots \hfill \\ \hfill & =p_0\hfill & \hfill & +p_{1}s{\mathrm{e}}^{-j\theta }\hfill & \hfill & +p_2{s}^2{\mathrm{e}}^{-2j\theta }\hfill & \hfill & +p_3{s}^3{\mathrm{e}}^{-3j\theta }\hfill & \hfill & +\dots \hfill \\ \hfill & ={\check{p}}_0\hfill & \hfill & +{\check{p}}_{1}s\hfill & \hfill & +{\check{p}}_2{s}^2\hfill & \hfill & +{\check{p}}_3{s}^3\hfill & \hfill & +\dots \hfill \end{array}$$

(47)

It is important to note that after the transformation, all previously real coefficients become complex values.

The transformation of the exponential function of the dead time element results in

$${\mathrm{e}}^{-T_d\check{s}}={\mathrm{e}}^{-T_{d}s{\mathrm{e}}^{-j\theta }}={\mathrm{e}}^{-T_{d}s(cos\theta -jsin\theta )}.$$

(48)

Requiring that $s=jw,w\in \mathbb{R}$, this becomes

$${\mathrm{e}}^{-T_{d}s(cos\theta -jsin\theta )}={\mathrm{e}}^{-T_d\omega (jcos\theta +sin\theta )}={\mathrm{e}}^{-T_d\omega sin\theta }·{\mathrm{e}}^{-T_{d}j\omega cos\theta }.$$

(49)

It can be seen that the angular frequency with which ${\mathrm{e}}^{-T_d\check{s}}$ circles the origin is scaled by a constant factor $cos\theta$. Furthermore, the magnitude of ${\mathrm{e}}^{-T_d\check{s}}$ is dependent on $\omega$ through a real exponential function. The image ${\mathcal{E}}_{\check{s}}$ is, therefore, a spiral around the origin of the complex plane (as opposed to a circle before the complex plane transformation).

Again, to prove the stability of the transformed system, it is first necessary that the transformed system without dead time is stable. This is complicated by the fact that all system coefficients are complex-valued after the just applied transformation. Therefore, it is necessary to make use of a complex extension to Kharitonov’s theorem [17] when analyzing the reduced system. The extended theorem requires that eight special point-valued polynomials with complex coefficients need to be Hurwitz for the full interval polynomial to be stable. As a consequence, it is also necessary to use a complex extension to the Hurwitz criterion which was presented in [24].

After that, it is again necessary to calculate all ${\omega }_{crit,i}$. This becomes substantially more difficult to perform as well, though, compared to the original stability test. The transformation of Equation (21) now yields

$$|{\tilde{\mathcal{F}}}_{\check{s}}|=\frac{|\sum \left[{\check{\beta }}_b\right]s^b|}{|\sum \left[{\check{\delta }}_d\right]s^d|}=|-{\mathrm{e}}^{-T_d\check{s}}|={\mathrm{e}}^{-T_d\omega sin\theta },s=j\omega ,\omega \in \mathbb{R}$$

(50)

where the right side of the equation is no longer a constant value. Therefore, it is not possible anymore to transform this problem into a polynomial root-finding problem as was done previously.

Finding all intersections of a rational function with an exponential function is a mathematically complex task. It is made worse by the fact that neither the number of intersections is known a priori, nor is the range in which the intersections may occur bounded. Numeric methods to solve this problem do exist. Common root-finding algorithms, such as Newton’s method, can be used in principle. But convergence of the algorithms may be difficult to achieve. Furthermore, it is necessary that the approach guarantees that no roots are missed. This especially poses a problem for the class of equations that need to be solved here. The SIVIA algorithm potentially offers itself as a solution to this problem, albeit at the likely expense of computational efficiency. The challenge lies in the fact that a solution must be computationally efficient so that its use is feasible for solving practical problems. No final recommendation for a good approach to solving this challenge can be given here. This task especially is still a work in progress and the search for a potential simplified approach that offers an analytic solution is still ongoing.

Finally, and as a further complication, after the rotational transformation of the complex plane, the images ${\mathcal{E}}_{\check{s}}$ and ${\tilde{\mathcal{F}}}_{\check{s}}$ are no longer symmetric. Therefore, the values ${\omega }_{crit,i}$ no longer appear as pairs of positive and negative values. Still, the same three cases as described before need to be distinguished. If applicable, two independent values $T_{d,crit,i}$ must now be calculated and the more critical (lower) value determines stability.

A result ${\mathcal{R}}_{\theta }\in \{\left[\mathrm{TRUE}\right],\left[\mathrm{FALSE}\right],\left[\mathrm{INDETERMINATE}\right]\}$ is obtained, depending on the specific result of the stability analysis of the transformed system.

5. Robust Controller Parameterization

Let $\mathbb{Y}$ be an n-dimensional set that abstractly represents the control system behavior. This representation may be understood as a set of parameters, time domain trajectories, or in any other way that is suitable to define the control system behavior. Further, let $\mathbb{X}$ be an m-dimensional set of the system’s controller parameters in ${\mathbb{R}}^m$. Generally speaking, $\mathbb{Y}=\mathit{f}(\mathbb{X},\mathit{c})$, where $\mathit{c}=[c_0,c_1,\dots ,c_i]$ is a vector of fixed model parameters. The process of controller parameterization can be viewed as the inverse operation $\mathbb{X}={\mathit{f}}^{-1}(\mathbb{Y},\mathit{c})$, where $\mathbb{Y}$ is given as a specification of desired system behavior and $\mathbb{X}$ then is the set of controller parameters that satisfy this specification. The question of finding $\mathbb{X}$ for a given set $\mathbb{Y}$ is called a set inversion problem. The issue in solving this problem lies in the fact that ${\mathit{f}}^{-1}$ is usually unknown and does often not exist.

The set inversion problem can then only be solved by selectively inserting candidate values from ${\mathbb{R}}^m$ into $\mathit{f}$ and testing whether the result is included in the specified set $\mathbb{Y}$. Therefore, it is possible to express the set inversion problem as

$$\mathbb{X}=\{\mathit{x}\in {\mathbb{R}}^m|\mathit{f}(\mathit{x},\mathit{c})\in \mathbb{Y}\}={\mathit{f}}^{-1}(\mathbb{Y},\mathit{c}).$$

(51)

A Monte Carl simulation is an example of such an approach to solving the set inversion problem. It has a huge deficiency, in that it is impossible to determine $\mathbb{X}$ as a closed set when only testing point values in ${\mathbb{R}}^m$. Moreover, when the set of controller parameters $\mathbb{X}$ that fulfill the desired system behavior defined by $\mathbb{Y}$ is small, it is inevitably difficult to find $\mathbb{X}$. This is obvious when you consider that the task at hand is to find a small set $\mathbb{X}$ within the theoretically infinitely large set ${\mathbb{R}}^m$ by only testing individual points in ${\mathbb{R}}^m$. Furthermore and completely independent of this problem, such an approach makes it difficult to deal with uncertain models. Uncertainty means that the model parameters $\mathit{c}$ are not known exactly but are known to vary within a specific range. Therefore, it is necessary to test each element $\mathit{x}$ at various points of the uncertain parameter ranges, thereby increasing the extent of a Monte-Carlo-Simulation further.

To improve the set inversion process, it is suitable to apply interval arithmetic as it has been proposed in other works [10,11,25]. Using this approach, an interval vector $\left[\mathit{x}\right]$ is used for testing and finding the desired controller parameters. This method allows for testing potentially large closed subsets of ${\mathbb{R}}^m$ in a single operation. The set of controller parameters $\mathbb{X}$ can then be represented using an inner subpaving $\underline{\mathbb{X}}$ and an outer subpaving $\overline{\mathbb{X}}$. Importantly, these are closed subsets of $\mathbb{X}$.

The inner subpaving $\underline{\mathbb{X}}$ represents those controller parameters for which the system behaves as desired. The difference ${\mathbb{R}}^m\setminus \overline{\mathbb{X}}$ represents those controller parameters for which the system does not behave according to the specification. Finally, the difference $\overline{\mathbb{X}}\setminus \underline{\mathbb{X}}$ represents those parameters for which no definite answer can be given.

With an interval approach in general and especially when dealing with an uncertain model, the function that maps $\mathbb{X}$ to $\mathbb{Y}$ will now become $\left[\mathit{f}\right]\left(\right[\mathit{x}],[\mathit{c}\left]\right)$ as an inclusion function of $\mathit{f}$ and will deliver a conservative outer approximation. Uncertain model parameters can now be directly included as interval constants. The overestimation introduced by $\left[\mathit{f}\right]\left(\right[\mathit{x}\left]\right)$ depends on the width of $\left[\mathit{x}\right]$ as well as on the width (uncertainty) of the fixed model parameters $\left[\mathit{c}\right]$. It is now possible to write

$$\underline{\mathbb{X}}=\{\left[\mathit{x}\right]\in {\mathbb{R}}^m|\left[\mathit{f}\right](\left[\mathit{x}\right],\left[c_i\right])\subset \mathbb{Y}\}.$$

(52)

In [26], the recursive SIVIA (set inverter via interval analysis) algorithm was introduced to enable an efficient calculation of $\underline{\mathbb{X}}$ and $\overline{\mathbb{X}}$. In the beginning, it is necessary to define the desired search space as an interval vector $\left[{\mathit{x}}_0\right]$. $\overline{\mathbb{X}}$ must be guaranteed to belong to this search space. The two subpavings $\underline{\mathbb{X}}$ and $\overline{\mathbb{X}}$ are initialized as empty sets. $\left[{\mathit{x}}_0\right]$ is then bisected into ever smaller interval vectors $\left[\mathit{x}\right]$ which are recursively tested and assigned to the solution sets or bisected further. The algorithm ends when a minimum width $\epsilon$ of the interval vectors is reached.

In the context of this publication, the set of desired system behavior is not explicitly defined. Instead, an approach using various criteria for evaluating the system behavior is described. Each criterion produces a result ${\mathcal{R}}_i$ that states whether the criterion is met or not. All criteria need to be met fully for the interval system to be part of the implicit set of desired system behavior. Therefore, the final evaluation result for each $\left[\mathit{x}\right]$ is obtained as the logical conjunction (AND) of all results that are obtained individually for $\left[\mathit{x}\right]$ with the criteria described in the previous sections: $\mathcal{R}={\mathcal{R}}_{\epsilon }\wedge {\mathcal{R}}_s\wedge {\mathcal{R}}_{\sigma }\wedge {\mathcal{R}}_{\theta }$.

This approach allows for a very flexible combination of different criteria and is therefore easily adaptable to various types of problems.

Table 1. Procedure of the SIVIA algorithm.

	Condition	Result		Action
1	$\left[\mathit{f}\right]\left(\right[\mathit{x}\left]\right)\subset \mathbb{Y}$	$\mathcal{R}=\left[\mathrm{TRUE}\right]$		$\left[\mathit{x}\right]$ is added to $\underline{\mathbb{X}}$ and $\overline{\mathbb{X}}$
2	$\left[\mathit{f}\right]\left(\right[\mathit{x}\left]\right)\cap \mathbb{Y}=\varnothing$	$\mathcal{R}=\left[\mathrm{FALSE}\right]$		$\left[\mathit{x}\right]$ is not part of the solution and discarded
3	$\left[\mathit{f}\right]\left(\right[\mathit{x}\left]\right)\cap \mathbb{Y}\ne \varnothing$	$\mathcal{R}=\left[\mathrm{INDETERMINATE}\right]$	$w\left(\right[\mathit{x}\left]\right)>\epsilon$	$\left[\mathit{x}\right]$ is bisected and SIVIA is recursively applied to each of the two resulting interval vectors
			$w\left(\right[\mathit{x}\left]\right)\le \epsilon$	$\left[\mathit{x}\right]$ is added to $\overline{\mathbb{X}}$

The conditions are tested in the order shown. Only the action associated with the first matching condition is executed.

describes how the SIVIA algorithm computes the inner and outer subpaving. The explicit conditions that determine whether $\left[\mathit{x}\right]$ is part of the set $\mathbb{X}$ are given. Additionally, the results of the criteria-based approach used here are listed as well, showing the relation between the two definitions.

6. Simulation Results

The approach for parameterizing controllers (presented in the previous sections) shall now be applied to the interval model described in Section 3. The necessary algorithms are implemented in Python using Numpy [27] for the computation, Sympy [28] for symbolic math operations and Matplotlib [29] for the visualization. Simulink is used for time domain simulations to validate the results.

For this experiment, a simple P controller is chosen for the controller in the outer control loop (${}_{p}PID$). A PI controller is chosen for the controller in the inner control loop (${}_{v}PID$). The transfer functions of these two controllers are then

$$\begin{array}{cc}\hfill {}_{p}PID& =K_{pp},\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {}_{v}PID& =K_{pv}\left(1+\frac{1}{T_{nv}s}\right).\hfill \end{array}$$

(54)

The structure of the full control loop is shown in Figure 2. $K_{pp}$ and $K_{pv}$ are the parameters that shall be determined by the controller parameterization algorithm. $T_{nv}=0.005\mathrm{s}$ will be set as a fixed constant value.

In theory, the controller parameterization algorithm would be capable of handling the parameterization of the system with a full PID controller in each control loop and without any fixed parameters. The current implementation of the algorithm does not yet provide sufficient computational speed to perform such a six-dimensional parameterization in a reasonable amount of time, though. Furthermore, existing knowledge of the system allows the reduction to these three specific parameters as it is known that a more complicated control loop will not result in noticeably better performance.

For reference, the computation of the three result sets that are shown in Figure 12 takes between 30 and 50 s each when running in six parallel processes on a notebook with an Intel Core i7-10850H CPU. Combining stable, unstable, and indeterminate solutions, each final result consists of roughly 30,000 interval vectors. The computation time approximately increases with the power of the number of dimensions when a homogeneous result is assumed, thus the algorithm has a running time of $O\left(c^n\right)$ when n is the number of dimensions. A six-dimensional parameterization is, therefore, estimated to have a computation time ranging from multiple hours up to two days on equivalent hardware. Further software optimizations are possible, to achieve higher performance, but are not implemented at this point.

Three solution sets of controller parameters are computed to show the influence that the different criteria have. These solution sets are the inner subpavings of the sets of all parameters that satisfy the given criteria. In the first instance, only stability is required and the solution set of possible controller parameters is very large then. Next, a minimum approximate damping ratio of $\zeta =0.3$ is prescribed as well. Finally, this is combined with the specification of an approximate maximum settling time of $T_{s,2\%}=0.05\mathrm{s}$.

For the given system, it is not necessary to evaluate the static gain criterion. The structure of the control loop ensures that $\epsilon =0$ for $t\to \infty$ and all values $K_{pp}$ and $K_{pv}$ that guarantee a stable system.

The initial search space is chosen as $\left[K_{pp,0}\right]=[0,800]$, $\left[K_{pv,0}\right]=[0,0.2]$. The SIVIA algorithm then yields the inner subpavings shown in Figure 12 as the solution sets of controller parameters that satisfy all criteria.

To validate that the dynamic system behavior is constrained as desired, a time domain simulation is performed. The time domain behavior of the system is visualized for each of the three-parameter solution sets at the upper and lower bound of the single model interval parameter $\left[b_0\right]$. Because the model only has one single interval parameter, this suffices to represent the whole range of possible system behavior. The control loop is modeled in Simulink, which requires a point-valued model. Therefore, individual simulations are run for each interval vector in each solution set where the midpoint of the interval vector is chosen as the point-valued parameter for the simulation. The individual simulation results are overlaid to visualize the range of possible system behavior in the time domain. These results are shown in Figure 13.

It must be noted that the visualization of simulation results is biased towards extreme results. This is caused by the fact that the resolution of each solution set becomes finer near its boundaries which can be seen clearly in Figure 12. Therefore, most interval vectors lie close to the outer edge of the parameter solution set. As one simulation is run per interval vector, these areas dominate the visualization of the simulation results. The effect is desirable though, as the edge of the result set is of particular interest when validating the controller parameterization results. This bias toward extreme solutions should be kept in mind though when analyzing the time domain simulation results.

The simulation results in Figure 13 clearly show that the controller parameterization works as intended. All solutions qualitatively conform to the specifications for their dynamic behavior. The approximations of the damping ratio and settling time work very well for the system that was evaluated here. It can be seen how the presented approach makes it possible to generate a solution set of controller parameters for which the interval system possesses very desirable characteristics in the form of a short settling time with little to no overshoot. The criteria that bound the system’s time domain behaviors are stated to be exactly valid for second-order systems but only hold in good approximation for higher-order systems. Therefore, they only impose qualitative restrictions on the results set here and no further detailed quantitative analysis of their effectiveness is possible.

Note that the experimental verification of the results is omitted. The focus of this work is the model-based parameterization of a controller. Having verified the parameterization results by simulations in combination with the system model, a subsequent experimental verification can only serve to assess the quality of the system model itself, which is not of particular interest here.

7. Discussion and Conclusions

This publication shows how controllers can be designed for linear systems that exhibit dead-time behavior, on the one hand, and uncertainties in the form of interval-valued parameters, on the other. The controllers designed in this way guarantee a priori-defined criteria as long as the modeling of the uncertainties is correct. This result is achieved by defining criteria for the control loop that are based on the classical controller design in combination with the SIVIA algorithm.

The method was developed in general for a class of systems with uncertainties. With this class of linear dead time systems, the dynamic behavior of many technical systems, such as electromagnetic drives, can be described in good approximation.

The practical contribution of this work is that the method was used for the controller design of an electromagnetic linear actuator. A simulation study based on the validated model of a real system underlines the suitability of the approach for a robust controller design.

The research question of how such an approach can be extended to non-linear effects is still open. These play a role especially in linear axes with strongly significant friction effects, which motivates future work in this direction. Furthermore, even if it could be shown in this work that the approximately valid dynamics criterion shows sufficient performance for use cases occurring in practice, it could be of theoretical interest to find an alternative to this criterion.

Finally, it can be said that, with this publication, it was shown that interval arithmetic is a suitable tool for designing a low-order controller for a system subject to uncertainty and dead time.