The controller,

$C(s)$ in this case, is a classic PID controller, which is one of the most distributed control solutions. The PID-controller transfer function is

where

$D$,

$K$ and

$I$ are the coefficients of the PID controller. The plant

$G(s)$ is represented with the second-order transfer function

with poles

$s=\alpha \pm j\omega $ and gain

${K}_{P}$. In cases of plants with initially real poles, parameter

$\omega $ is supposed to be complex with a null real part.

At the first stage of the research, a stationary LTI system was examined. The task was to figure out the rule for PID-controller coefficients, providing a non-overshoot step response. In previous research [

14], one of the necessary conditions for this was that the poles of the transfer function are strictly real.

#### 2.1. Providing Real Closed-Loop Pole Configuration

Regarding control systems structure, plant and controller transfer functions, the closed loop transfer function is

Conducting changing

$D{K}_{P}={K}^{\prime}$,

$K{K}_{P}={K}^{\prime}$, and

$I{K}_{P}={I}^{\prime}$, a characteristic equation of (2) can be represented as:

According to the theorem in [

14], it is necessary for transfer function poles to be exclusively real. Since the roots type is defined with a discriminant sign, let us write a discriminant for (3) in the following form [

25]

Substituting arbitrary real positive values of $\alpha $, $\omega $, and ${K}^{\prime}$ in (4), one can plot the surface of ${\mathsf{\Delta}}_{CE}\left({D}^{\prime},\text{}{I}^{\prime}\right)$. Since only positive values of ${\mathsf{\Delta}}_{CE}$ are of interest, in the same coordinates, let us plot plane ${\mathsf{\Delta}}_{CE}=0$ to visualize constraints of the region of interest.

Figure 2 illustrates that region where

${\mathsf{\Delta}}_{CE}>0$, denoted as

$\mathsf{\Omega}$, is bounded with two curves (white dashed lines). Expressing

${I}^{\prime}$ from (4) yields two functions describing bounding curves for

$\mathsf{\Omega}$:

For further investigation, let us plot region

$\mathsf{\Omega}$ in the

${D}^{\prime}-{I}^{\prime}$ plane as it is shown in

Figure 3.

Lines of constraints for region $\mathsf{\Omega}$ have an intersection point $\xi \left({{D}_{\xi}}^{\prime},{{I}_{\xi}}^{\prime}\right)$. Point $\xi \left({{D}_{\xi}}^{\prime},{{I}_{\xi}}^{\prime}\right)$ defines peak values for parameters ${I}^{\prime}$ and ${D}^{\prime}$.

For generalization purposes, let us get analytical expressions for

${{I}_{\xi}}^{\prime}$ and

${{D}_{\xi}}^{\prime}$ using (5)

In the same way, substitution of (6) in (5) gives the expression [

25]

Let us investigate region

$\mathsf{\Omega}$ in more detail. It is known that for the third-order polynomial, its roots can be analytically calculated with the Cardano formula. Regarding the form of (2), the dominating pole can be calculated as follows

where:

Investigating derivatives

$\frac{\partial {S}_{DOM}}{\partial {I}^{\prime}}$ and

$\frac{\partial {S}_{DOM}}{\partial {D}^{\prime}}$ in the field of real values states that (8) is a monotone, non-increasing function and thus, the stability degree

$\eta \left(\alpha ,\omega ,{K}^{\prime},{I}^{\prime},{D}^{\prime}\right)=\left|\mathrm{Re}\left({S}_{DOM}\left(\alpha ,\omega ,{K}^{\prime},{D}^{\prime},{D}^{\prime}\right)\right)\right|$ will increase according to

${I}^{\prime}$. Regarding the

${I}^{\prime}$ range of values, one can assume that the maximum stability degree can be reached in

$\xi $ point. Substituting (5) and (7) in (8) gives the following expression

Function (9) defines the maximum stability degree that can be reached for a given plant, with the chosen PID-controller proportional coefficient with strictly real poles for a closed-loop transfer function. In addition, it should be noticed that in $\xi $ point, there is a triple pole ${T}_{1}={T}_{2}={T}_{3}=T$.

#### 2.2. Non-Overshoot Step Response Condition

Regarding special point

$\xi $, let us check non-overshooting conditions that were formulated in [

15]. According to [

15], it is necessary and sufficient that if at least one of the following conditions holds, then system step response has no overshoot.

For a more convenient robust controller synthesis procedure, let us find such a

${K}^{\prime}$ value that guarantees a non-overshoot step response for any

$p\left({{D}^{\prime}}^{*},{{I}^{\prime}}^{*}\right)\in \mathsf{\Omega}$. Investigating (10) and (11), it is clear that expression

where

${T}_{i},i\in \mathbb{N},i\in \overline{1,3}$, is a part of each of the conditions (10) and (11). For generalization purposes, let us solve (12) with respect to

${T}_{i}$. The solution is an expression that can be written as follows

Regarding the highest coefficient sign and (13), one can infer that

Since

${K}^{\prime}$,

${D}^{\prime}$, and

${I}^{\prime}$ are strictly positive, regarding (14), one can assume that the condition

$\phi \left({T}_{i},{D}^{\prime},{K}^{\prime},{I}^{\prime}\right)<0$ will always hold, and it is a monotonic decreasing function of

${T}_{i}$. Next, substitution of (6, 7, 9) into (11) yields that

Regarding the case of real poles of plant transfer function and, thus, complex value of

$\omega $ with a null real part, the aforementioned substitution and expressing

${K}^{\prime}$ gives:

In other words, a proper choice of ${K}^{\prime}$ provides a non-overshooting step response for a closed-loop system within every point $p\left({{D}^{\prime}}^{*},{{I}^{\prime}}^{*}\right)\in \mathsf{\Omega}$.

#### 2.3. Plant with Interval-Given Parameters

Regarding practical applications, the exact parameters values are unknown and, basically, can be represented as a confidence limit. Moreover, parameters tend to vary due to temperature, humidity, pressure, and mechanical deterioration. In addition, in the field of outdoor mobile robotics, conditions of functioning are highly heterogeneous and demand special approaches regarding varying parameters [

26,

27,

28]. Thus, transfer function parameters basically have to be represented as an interval value. Since further mathematical representation of the plant contains interval-given values, the controller synthesis should be conducted accordingly.

Let us consider second-order transfer function with parametric uncertainty as a plant model

where

$a\in \left[\underset{\_}{a},\text{}\overline{a}\right]$,

$b\in \left[\underset{\_}{b},\text{}\overline{b}\right]$,

$c\in \left[\underset{\_}{c},\text{}\overline{c}\right]$, and

${K}_{P}\in \left[\underset{\_}{{K}_{P}},\text{}\overline{{K}_{P}}\right]$ are given intervals.

According to [

14,

15], a system with interval-given parameters is basically a set of stationary LTI systems, and its poles’ location can be represented as a region with borders defined by ranges of interval parameters. For the purpose of analysis and control design for systems with interval-given parameters, it is sufficient only to consider external borders of poles localization region (see

Figure 4). A typical well-known representation of the poles localization region is multiparametric interval root locus (MIRL). Since possible plant pole configurations are real poles or a complex–conjugate pair, MIRL is always symmetric with respect to the x-axis. The symmetry property of MIRL allows us to simplify further research, i.e., for the second-order transfer function, it is sufficient to investigate only one half of the MIRL. In other words, each symmetrical pair of points that belongs to MIRL is a poles pair for the same LTI system. In addition, since only half of MIRL is sufficient, computational load is reduced.

Since every point that belongs to MIRL is a single LTI system that has its own region

$\mathsf{\Omega}$ with

$\xi $ point, the region of poles localization forms a corresponding region of

$\xi $ points on

${D}^{\prime}-{I}^{\prime}$ plane. With accordance to (6) and (7),

${{I}_{\xi}}^{\prime}$ and

${{D}_{\xi}}^{\prime}$ values are defined by

${K}^{\prime}=K{K}_{P}$, and due to the interval nature of

${K}_{P}$, for every

${K}_{P}\in \left[\underset{\_}{{K}_{P}},\overline{{K}_{P}}\right]$ mapping,

$\mathrm{M}\left({{K}_{P}}^{*}\right),\text{}{{K}_{P}}^{*}\in \left[\underset{\_}{{K}_{P}},\overline{{K}_{P}}\right]$ can be obtained (see

Figure 5). According to (5), regions of positive discriminant values

$\mathsf{\Omega}$ can be obtained for

$\forall \xi \in \mathrm{M}\left({{K}_{P}}^{*}\right),\text{}{{K}_{P}}^{*}\in \left[\underset{\_}{{K}_{P}},\overline{{K}_{P}}\right]$. The main aim of the research is to obtain PID-controller coefficient values such that a closed-loop system has a non-overshooting step response for any plant parameter value variations within given ranges. Since every point with coordinates

$\left({D}^{\prime},{I}^{\prime}\right)\in \mathsf{\Omega}$ provides strictly real closed-loop poles for (2) for corresponding point

$\xi $, then some pair of controller coefficient values

$\left({{D}^{\prime}}^{*},{{I}^{\prime}}^{*}\right)$ gives

$\left\{s\in \mathbb{R},\text{}s0\text{}|\forall {{\xi}^{i}}_{{K}_{p}}\in \mathrm{M}\left({{K}_{P}}^{*}\right),{{K}_{P}}^{*}\in \left[\underset{\_}{{K}_{P}},\overline{{K}_{P}}\right]\right\}$ in case

$\left({{D}^{\prime}}^{*},{{I}^{\prime}}^{*}\right)\in {\mathsf{\Omega}}^{*}$, such that

${\mathsf{\Omega}}^{*}={{\displaystyle \cap {\mathsf{\Omega}}^{i}}}_{{K}_{p}}$ for every

${{\xi}^{i}}_{{K}_{pi}}$.

Similarly to each $\mathsf{\Omega}$ region, for resultant region ${\mathsf{\Omega}}^{*}$, constraints are defined with (5) for particular values of $\xi $. Thus, in order to choose $\left({{D}^{\prime}}^{*},{{I}^{\prime}}^{*}\right)\text{}$ properly, it is sufficient to define points ${\xi}_{1}$ and ${\xi}_{2}$, for which (5) forms constraints for the desired set ${\mathsf{\Omega}}^{*}$, which contains ${I}^{\prime}$ and ${D}^{\prime}$ values that provide a non-overshooting step response under parameter variation.

#### 2.4. Constraints Clarification for ${\mathsf{\Omega}}^{*}$

The next problem in the research is to generalize constraints for

${\mathsf{\Omega}}^{*}$. It can be suggested that

${\mathsf{\Omega}}^{*}$ is constrained with (5) for arguments providing the

${{I}^{\prime}}_{\xi}$ value to be minimum according to (6) maximum

${{D}^{\prime}}_{{I}_{0}}$—the value of

${D}^{\prime}$ that turns

${{I}^{\prime}}_{2}(\alpha ,\omega ,{k}^{\prime},{D}^{\prime})$ into zero. The value of

${{D}^{\prime}}_{{I}_{0}}$ is defined as follows:

It should be noted that desired points

${\xi}_{1}$ and

${\xi}_{2}$ can either belong to vertices or edges mapping of MIRL. For further convenience, (7) and (18) can be rewritten as functions of plant model parameters. Since (1) and (17) are equivalent representations of the plant transfer function, (7) and (18) can be represented as follows:

Regarding the assumption that desired points ${\xi}_{1}$ and ${\xi}_{2}$ belong to edges of MIRL mapping $M\left({K}_{{P}_{i}}\right)$, it should be noted that the only possible way of edge location for ${\xi}_{1}$ and ${\xi}_{2}$ is a nonmonotonic behavior of (19) or (20). It is known that each edge of MIRL and, consequently, its mapping $\mathrm{M}\left({K}_{P}\right)$ are formed by the varying of single interval parameters with others fixed in their limit values, and it turns (19) and (20) into one-variable functions forming each edge.

In order to define whether (19) and (20) are monotonic or not along each interval parameter, one can find partial derivatives for (19) and (20) with respect to

$a$,

$b$,

$c$, and

${K}^{\prime}$ for checking their monotonicity property. Partial derivation for (19) yields:

Investigating (21)–(24), it can be concluded that if $4{a}^{4}+4{K}^{\prime}{a}^{2}+4ca-{b}^{2}$ and $4{a}^{4}-2ca+{b}^{2}$ possess a non-zero value at any $a\in \left[\underset{\_}{a},\text{}\overline{a}\right]$, $b\in \left[\underset{\_}{b},\text{}\overline{b}\right]$, $c\in \left[\underset{\_}{c},\text{}\overline{c}\right]$, and ${K}^{\prime}\in \left[\underset{\_}{{K}^{\prime}},\overline{{K}^{\prime}}\right]$, then (19) has a monotonic behavior. Alternatively, for solutions for one of (21), (22), (23), or (24) with respect to changing variables, one can get the exact investigated parameter value that gives ${\xi}_{1}$.

Similarly, partial derivation for (20) gives:

Functions (26)–(28) are always non-zero, so that it is sufficient to check the behavior of (25). In addition, one has to check (25) only in the case of parameter $a$ variation.

Regarding the aforementioned results, if (21)–(28) are monotonic functions within interval parameters bounds, in special points

${\xi}_{1}\left({\alpha}_{L},\text{}{\omega}_{L}\right)$ and

${\xi}_{2}\left({\alpha}_{U},\text{}{\omega}_{U}\right)$ that define regions in which intersection gives

${\mathsf{\Omega}}^{*}$, only two members of interval family form the set containing

${I}^{\prime}$ and

${D}^{\prime}$ values that satisfies non-overshoot conditions for all the interval family. The two special members of the family are defined as follows:

In case some of (21)–(28) are non-monotonic, one has to find the exact solution with respect to varying parameters and substitute it to (29) or (30), instead of limiting the value of the parameter.

#### 2.5. PID-Controller Coefficient Choice

The final problem within the present research is to find the exact values for the PID-controller coefficients

$D$ and

$I$. One has to note that

${D}^{\prime}={K}_{P}D$ and

${I}^{\prime}={K}_{P}I$ are interval values, since

${K}_{P}$ is a given interval. Thus, it is essential to find

$D$ and

$I$ values such that:

In order to satisfy (31), one can check two points denoted as

${P}_{U}\left(D\overline{{K}_{P}},I\overline{{K}_{P}}\right)$ and

${P}_{L}\left(D\underset{\_}{{K}_{P}},I\underset{\_}{{K}_{P}}\right)$, since any point

$\left(D{{K}_{P}}^{*},I{{K}_{P}}^{*}\right),\text{}{K}_{P}\in [\underset{\_}{{K}_{P}};\overline{{K}_{P}}]$ belongs to the linear interval

${P}_{LU}$. Let us investigate the generalized condition for

${P}_{LU}$ to lay within the

${\mathsf{\Omega}}^{*}$ region. In other words, the following problem is to choose

$D$ and

$I$ in such a way that

${P}_{LU}\in {\mathsf{\Omega}}^{*}$ is as depicted in

Figure 6.

According to

Figure 6, one can observe that with a known range of gain

${K}_{P}$ and known expression for constraints (5),

$D$ and

$I$ values providing

${P}_{LU}\in {\mathsf{\Omega}}^{*}$ can be found from inequality of the form:

Regarding the limit case, i.e.,

${P}_{L}\in {{I}_{2}}^{\prime}$ and

${P}_{U}\in {{I}_{1}}^{\prime}$, the aforementioned inequality can be represented as the following equality: