Volume 16 Number 6
December 2019
Article Contents
Sergey Gayvoronskiy, Tatiana Ezangina, Ivan Khozhaev and Viktor Kazmin. Determination of Vertices and Edges in a Parametric Polytope to Analyze Root Indices of Robust Control Quality. International Journal of Automation and Computing, vol. 16, no. 6, pp. 828-837, 2019. doi: 10.1007/s11633-019-1182-y
Cite as: Sergey Gayvoronskiy, Tatiana Ezangina, Ivan Khozhaev and Viktor Kazmin. Determination of Vertices and Edges in a Parametric Polytope to Analyze Root Indices of Robust Control Quality. International Journal of Automation and Computing, vol. 16, no. 6, pp. 828-837, 2019. doi: 10.1007/s11633-019-1182-y

Determination of Vertices and Edges in a Parametric Polytope to Analyze Root Indices of Robust Control Quality

Author Biography:
  • Sergey Gayvoronskiy received the Ph. D. degree in control systems engineering from the Tomsk Polytechnic University, Russia in 1990. He is presently an associated professor of the Division for Automation and Robotics at the School of Computer Science and Robotics, National Research Tomsk Polytechnic University, Russia. He was repeatedly awarded by Ministry of Education and Science of Russian Federation, Russian Union of Young Scientists and Tomsk Polytechnic University for his educational and scientific achievements. His research and teaching interests include analysis and synthesis of robust and adaptive control systems for control objects and processes with uncertain parameters. E-mail: saga@tpu.ru ORCID iD: 0000-0002-7156-2807

    Tatiana Ezangina received the Ph. D. degree in system analysis, control and data processing from Tomsk Polytechnic University, Russia in 2016. She is presently a junior researcher of Telecommunications, Electronics and Underwater Geology Laboratory, National Research Tomsk Polytechnic University, Russia. She was repeatedly awarded by the Government of Russian Federation, Ministry of Education and Science of Russian Federation, Tomsk Polytechnic University and other institutions for her scientific achievements. Her research interests include robust and adaptive control system analysis and synthesis, tethered underwater vehicles development and software development. E-mail: eza-tanya@yandex.ru ORCID iD: 0000-0002-4948-5972

    Ivan Khozhaev received the B. Sc. and M. Sc. degrees (honors) in control systems engineering from the Tomsk Polytechnic University, Russia in 2014 and 2016, accordingly. He is presently a Ph. D. student at the Division for Automation and Robotics, the School of Computer Science and Robotics, National Research Tomsk Polytechnic University, Russia. His research interests include robust and adaptive control systems synthesis and analysis, unmanned underwater vehicles development and computational fluid dynamics. E-mail: khozhaev.i@gmail.ru (Corresponding author) ORCID iD: 0000-0002-8874-0200

    Viktor Kazmin received the Ph. D. degree in control systems engineering from the Tomsk Polytechnic University, Russia in 1996. He is presently an associated professor of the Division for Automation and Robotics at the School of Computer Science and Robotics, National Research Tomsk Polytechnic University, Russia. He was repeatedly awarded by Tomsk Polytechnic University for his educational achievements. His research and teaching interests include analysis and synthesis of feedback control systems for internal combustion engines, fundamentals of control theory and automated control.E-mail: kvp@tpu.ru

  • Received: 2018-11-11
  • Accepted: 2019-04-16
  • Published Online: 2019-07-16
  • The research deals with the methodology intended to root robust quality indices in the interval control system, the parameters of which are affinely included in the coefficients of a characteristic polynomial. To determine the root quality indices we propose to depict on the root plane not all edges of the interval parametric polytope (as the edge theorem says), but its particular vertex-edge route. In order to define this route we need to know the angle sequence at which the edge branches depart from any integrated pole on the allocation area. It is revealed that the edge branches can integrate into the route both fully or partially due to intersection with other branches. The conditions which determine the intersection of one-face edge images have been proven. It is shown that the root quality indices can be determined by its ends or by any other internal point depending on a type of edge branch. The conditions which allow determining the edge branch type have been identified. On the basis of these studies we developed the algorithm intended to construct a boundary vertex-edge route on the polytope with the interval parameters of the system. As an illustration of how the algorithm can be implemented, we determined and introduced the root indices reflecting the robust quality of the system used to stabilize the position of an underwater charging station for autonomous unmanned vehicles.
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Determination of Vertices and Edges in a Parametric Polytope to Analyze Root Indices of Robust Control Quality

Abstract: The research deals with the methodology intended to root robust quality indices in the interval control system, the parameters of which are affinely included in the coefficients of a characteristic polynomial. To determine the root quality indices we propose to depict on the root plane not all edges of the interval parametric polytope (as the edge theorem says), but its particular vertex-edge route. In order to define this route we need to know the angle sequence at which the edge branches depart from any integrated pole on the allocation area. It is revealed that the edge branches can integrate into the route both fully or partially due to intersection with other branches. The conditions which determine the intersection of one-face edge images have been proven. It is shown that the root quality indices can be determined by its ends or by any other internal point depending on a type of edge branch. The conditions which allow determining the edge branch type have been identified. On the basis of these studies we developed the algorithm intended to construct a boundary vertex-edge route on the polytope with the interval parameters of the system. As an illustration of how the algorithm can be implemented, we determined and introduced the root indices reflecting the robust quality of the system used to stabilize the position of an underwater charging station for autonomous unmanned vehicles.

Sergey Gayvoronskiy, Tatiana Ezangina, Ivan Khozhaev and Viktor Kazmin. Determination of Vertices and Edges in a Parametric Polytope to Analyze Root Indices of Robust Control Quality. International Journal of Automation and Computing, vol. 16, no. 6, pp. 828-837, 2019. doi: 10.1007/s11633-019-1182-y
Citation: Sergey Gayvoronskiy, Tatiana Ezangina, Ivan Khozhaev and Viktor Kazmin. Determination of Vertices and Edges in a Parametric Polytope to Analyze Root Indices of Robust Control Quality. International Journal of Automation and Computing, vol. 16, no. 6, pp. 828-837, 2019. doi: 10.1007/s11633-019-1182-y
    • Major challenge in modern industrial production is the development and design of high-quality automated control systems capable at operating when its parameters are unstable and not determined. In the real control systems the object parameters are often undetermined. It is connected with measuring errors, equipment ageing, and other disturbances impacting the object characteristics. Likewise, there are some systems, parameters of which can change in certain intervals. In both cases it is fair to apply the interval parameters approach to control systems synthesis. The systems, encompassing the control objects with interval parameters, are called interval control system (ICS)[1, 2]. Such systems can be introduced with the interval characteristic polynomials (ICP), the coefficients of which include the interval parameters of a control object. The character of how the interval parameters of ICS integrate into the ICP coefficients identifies a type of these coefficients uncertainty. There are four types of ICP coefficients uncertainty[13]: interval, affine, polylinear and polynomial.

      The presence in ICS of non-stable parameters, which vary within certain intervals, can lead to a dynamic properties change in a system and result in its instability. The first research studies devoted to solving problems related to the analysis of ICS stability were performed by S. Dezoir, L. Zadeh and S. Faedo. The fundamental outcome in the field of ICS stability analysis with the focus on ICP coefficients was achieved by V. Kharitonov[4, 5]. Among the studies addressing the analysis of the robust stability are worth mentioning follows: J. Tsypkin, I. Vyshigorodsky, Yu. Neimark, B. Polyak, P. Shcherbakov, Yu. Petrov, J. Ackermann, B. R. Barmish, J. Kogan, R. Tempo, A. Packard, J. C. Doyle and others. In studies[621], the evaluation of ICP stability is performed within the frequency approach and probability approach. The studies based on μ-analysis are conducted in [2225]. The studies based on Lyapunov functions are conducted in [2628].

      It is clear that ICS must be stable and support manipulated variables in allowable limits. Therefore, to date it stands more for the analysis of the robust quality than the analysis of the robust stability in ICS. In this field a root approach is the most illustrative one[2938], when based on the allocation areas of ICP roots we can determine the requested indices of the robust quality – the degree of robust stability and the degree of robust oscillation. The simplest approach for it is the approach based on the edge theorem with the concept on a base of vertex-edge polynomials. A good development of this approach is performed in studies[3640], where ICS is introduced with characteristic polynomials containing interval coefficients. These studies resulted in the methods according to which evaluation of the robust quality root indices requires the analysis of only those vertices of coefficients polytope, which are depicted on the border of the allocation area of ICS poles.

      It should be noted that the methods based on reduction of characteristic polynomial coefficients to an interval form (by rules of the interval analysis) leads to a conservative solution in case of ICS with the real interval parameters. Therefore, in order to increase the accuracy of ICS quality analysis it is necessary to consider the real interval physical parameters included in a certain way into the characteristic polynomial coefficients. Let us consider ICP, the coefficients of which integrate linearly into the physical parameters:

      $D(s) = \sum\limits_{i = 1}^m {[{T_i}]{A_i}(s)} + B(s)$

      (1)

      where $[{T_i}] = [\underline {{{\rm{T}}_i}} ;\,\overline {{{\rm{T}}_i}} ]$. Such ICP are called polynomials with affine coefficients uncertainty. Example on how the ICP roots with affine coefficients uncertainty are projected on the complex plane is shown in Fig. 1.

      Figure 1.  Image of a parametric polytope with affine uncertainty of ICP coefficients. (a) Projections of a parametric polytope vertex on a complex plane; (b) Projection of an inner point of a parametric polytope edge on a complex plane.

      As seen in Fig. 1, the required indices of ICS robust quality conform to the worst root indices when the interval parameters in prescribed limits are changed. In this case unlike the cases with the interval ICP coefficients uncertainty can be defined not only by the polytope vertices ${P_T}$ (Fig. 1(a)), but also by the internal points of its edges (Fig. 1(b)). However, to depict all edges is a very complicated task. Considering the fact that the borders of the allocation area for ICP roots are not the images of all polytope edges ${P_T}$, but some of them, the interest is to determine the vertices and edges ${P_T}$, comprising a boundary vertex-edge route.

      Hence, for ICS with affine ICP coefficients the task is set to develop the algorithm able to determine the robust stability and robust oscillability degree on the basis of boundary vertex-edge route.

    • ICP (1), whose coefficients include m interval parameters, form a rectangular hyper-parallelepiped ${P_T} = \{ {T_i}\left| {\underline {{T_i}} } \right. \le {T_i} \le \overline {{T_i}} ,\,i = \overline {1,m} \} $, with ${2^m}$ vertices and $m{2^{m - 1}}$ edges. Let us define the vertices of ${P_T}$ via ${V_q}$, $q = \overline {1,{2^m}} $, where $q$ is a number of vertices. Coordinates of every point of ${P_t}$ edge in relation to a vertex ${V_q}$, $q = \overline {1,{2^m}} $ can be determined with the following formula:

      $\begin{split} &{T_i} = T_i^q + \Delta {T_i}, \;\;\; {\rm{ }}i = \overline {1,m} \\ &{\rm{ }}(\underline {{T_i}} - T_i^q) \le \Delta {T_i} \le (\overline {{T_i}} - T_i^q) \end{split}$

      (2)

      where $\Delta {T_i}$ is the increment of i-th interval parameter, $T_i^q$ is its value in vertex ${V_q}$. Based on introduced indices, we define the edge ${P_T}$ via $R_i^q.$ Each edge of ${P_T}$ is reflected on the complex root plane (Fig. 2) on the basis of the equation

      Figure 2.  Parametric polytope edges

      ${D^q}(s) + \Delta {T_i}{A_i}(s) = 0$

      (3)

      where ${D^q}(s) =\displaystyle\sum\nolimits_{i = 1}^m {T_i^q \cdot {A_i}(s)} + B(s)$ is the vertex characteristic polynomial.

      If ICS with characteristic polynomial (3) has unity feedback, then its open-loop transfer function can be presented as

      $W_i^q(\Delta {T_i},s) = \frac{{\Delta {T_i}{A_i}(s)}}{{{D^q}(s)}}.$

      (4)

      Whereupon the root locus theory, when $\Delta {T_i}$ changes within the interval (2) the roots (3) form one-parameter interval root locus, the branches of which are called edge branches ($RS_i^q$), their starts and ends – a root node (${U_q}$). Herewith, the expressions are correct: $\phi (R_i^q) = RS_i^q$, $\phi ({V_q}) = {U_q}$.

      It is obvious that if two interval parameters ${T_i}$ and ${T_j}$ are changed, then, from one vertex ${V_q}$ a rectangular face ${P_T}$ is formed, which can be depicted through ${G_{ij}}$, and its image as $G{S_{ij}}$ (Fig. 3).

      Figure 3.  Parametric polytope faces

      When each ICS interval parameter is changed along the edge of any boundary vertex, the polynomial roots start moving along the edge branch, which departs from a vertex image at a corresponding angle. Let us define this angle as $\varTheta _i^q$. Based on the root locus theory, at increasing of ${T_i}$ the angle $\varTheta _i^q$ can be calculated with the formula $\varTheta _i^q = 180^\circ - \displaystyle\sum\nolimits_{pol = 1}^n {{\varTheta _{pol}} + \displaystyle\sum\nolimits_{ze = 1}^{vz} {{\varTheta _{ze}}} } $, at decreasing of ${T_i}$, we use the formula $\varTheta _i^q = - \displaystyle\sum\nolimits_{pol = 1}^n {{\varTheta _{pol}}} + \sum\nolimits_{ze = 1}^{vz} {{\varTheta _{ze}}} $, where $\{{\varTheta _{pol}}$ и ${\varTheta _{ze}}\}$ is angles defined by the vectors coming from ${U_q}$ corresponding to pol-th pole and to ze-th zero of transfer function (4). It should be noted that the value $\displaystyle\sum\nolimits_{pol = 1}^n {{\varTheta _{pol}}} $ for all ${T_i}$ is equal, therefore to determine the sequence of edge branches departure angles with ${T_i}$ value, $\displaystyle\sum\nolimits_{pol = 1}^n {{\Theta _{pol}}} $ can be neglected. In case of increasing ${T_i}$, we will get

      $\varTheta _i^q = 180^\circ + \sum\limits_{ze = 1}^{vz} {{\varTheta _{ze}}} $

      (5)

      by decreasing ${T_i}$, we get

      $\varTheta _i^q = \sum\limits_{ze = 1}^{vz} {{\varTheta _{ze}}} .$

      (6)

      Depending on the values found for the departure angles $\varTheta _i^q$, there can be constructed the sequence of how ${T_i}$ parameters are changed from boundary root node. The example on roots departure at changing parameters ${T_i}$, $i = 1,2,3$ from the vertex ${V_q}$ is shown in Fig. 4.

      Figure 4.  Root motion direction at changing parameters from vertex of PT

      Due to the fact that the node ${U_q}$ will be boundary node $G{U_q}$, it is needed that the root motion vectors with minimum $\varTheta _1^{Vq}$ and maximum $\varTheta _m^{Vq}$ departure angles will form the boundary angle, lying in a range [0°, 180°]. Introduce this statement on the basis of edge branches departure angles that have been calculated from a positive semiaxis

      $\left| {\varTheta _m^{Vq} - \varTheta _1^{Vq}} \right| < 180^\circ. $

      (7)

      In so doing, a condition (7) allows defining the vertex ${P_T}$, the image of which belongs to an allocation area border ${S_r}$ of a complex root.

    • Suppose the prototypes $R{S_i}$ и $R{S_j}$ are the edges of one face. Consider the angles $R{S_i}$ and $R{S_j}$ departing from the root nodes of one boundary edge branch as $GR{S_k}$. If the sequence of these departure angles at the ends of the edge branch is of the same value, then, $RS_i^q$ and $RS_j^q$ will not intersect (Fig. 5). If for all faces of ${P_T}$ with the common vertex, the same condition is fulfilled, then, the borders of the allocation area of a complex root are determined by non-intersected edge images of ${P_T}$.

      Figure 5.  A case of non-intersecting edge branches along T1 and T2

      If at the ends of the boundary edge branch the sequence of departure angles $RS_i^q$ and $RS_j^q$ is not kept (Fig. 6), then, $RS_i^q$ and $RS_j^q$ are not intersected.

      Figure 6.  A case of intersecting edge branches along T1 and T2

      In this case the border of the allocation area of a complex root will consist of edge branches parts, which will be determined by the intersection points (Fig. 7).

      Figure 7.  Image of a parametric polytope when edge branches intersect

      Let us define the conditions of edge branches intersection. Write down the equation reflecting face plane ${G_{ij}}$:

      ${T_i}{A_i}(s) + {T_j}{A_j}(s) + \sum\limits_k {T_k^q{A_k}(s) + B(s)} = 0.$

      (8)

      If in (8) we pose $s = {s_r} = \alpha +{\rm j}\beta $, $r \in \overline {1,n} $ and derive real and imaginary components, we will get the system of two linear equations, which connects ${s_r}$ with two variables ${T_i}$ and ${T_j}$

      $\left\{\begin{aligned} & {T_i}{\rm{Re}} {A_i}(\alpha ,\beta ) + {T_j}{\rm{Re}} {A_j}(\alpha ,\beta ) + \\ & \quad \quad {\rm{Re}} \left[ {\sum\limits_k {T_k^q{A_k}(s) + B(s)} } \right] = 0 \\ & {T_i}{\rm{Im}} {A_i}(\alpha ,\beta ) + {T_j}{\rm{Im}} {A_j}(\alpha ,\beta ) + \\ & \quad \quad {\rm{Im}} \left[ {\sum\limits_k {T_k^q{A_k}(s) + B(s)} } \right] = 0. \end{aligned}\right.$

      (9)

      Solving the system (9), two cases can be obtained:

      1) The system has the single solution ${T_i} = T_i^*$, ${T_j} = T_j^*$. Then, ${\phi ^{ - 1}}({s_r}) = {P^*}$, ${P^*} = \left( {T_i^*,\,T_j^*} \right)$ and, consequently, the point ${P^*} \in G_{ij}^q$.

      2) The equations are dependent and differ with a constant multiplier. In this case on the plane ${G_{ij}}$, there is a straight line h (${\phi ^{ - 1}}({s_r}) = h$), defined by any equation from the system (9).

      Let us determine the composition of border area ${S_r}$ of a complex root allocation, if ${\varphi ^{ - 1}}({S_r}) = {G_{ij}}$. It is clear that the coordinates ${P^*}$ is the single possible solution (9), then, $RS_i^q$ is the single branch coming through ${s_r}$. In this case the borders ${S_r}$ consist of non-intersected edge images ${G_{ij}}$. If a prototype of root ${s_r}$ is the straight line h, which is in the edge ${G_{ij}}$ marks the interval $\overline {{P_1}{P_2}} $ (points ${P_1}$ and ${P_2}$ belong to the edges ${G_{ij}}$), then, through ${s_r}$ (we call it the intersection node ${U^*}$) many root locus branches go along ${T_i}$ and ${T_j}$, which lie between two intersected edge branches in ${s_r}$. In this case the borders ${S_r}$ will consist of intersected edge images ${G_{ij}}$.

      It is obvious that the required condition for the intersection node ${U^*} \in {S_r}$ is the straight line h, at least in one from the planes ${P_T}$, which have a common vertex. In order to find the equation linear relationship (9), testifying about the straight line h in the parameters′ space ${T_i}$ and ${T_j}$ and its reflection in ${U^*}(\alpha ,\,{\rm j}\beta )$, it is needed to verify the equation.

      $\begin{split} &\frac{{{\rm{Re}} {A_i}\left( {\alpha ,\beta } \right)}}{{{\rm{Im}} {A_i}\left( {\alpha ,\beta } \right)}} = \frac{{{\rm{Re}} {A_j}\left( {\alpha ,\beta } \right)}}{{{\rm{Im}} {A_j}\left( {\alpha ,\beta } \right)}} =\\ &\quad \frac{{{\rm{Re}} \left[ {\displaystyle\sum\limits_k \begin{gathered} T_k^q{A_k}(\alpha ,\beta ) + B(\alpha ,\beta ) \\ \end{gathered} } \right]}}{{{\rm{Im}} \left[ {\displaystyle\sum\limits_k \begin{gathered} T_k^q{A_k}(\alpha ,\beta ) + B(\alpha ,\beta ) \\ \end{gathered} } \right]}}\end{split}$

      (10)

      from (10) we obtain the following equation system:

      $\left\{\begin{aligned} & {\rm{Re}} {A_i}\left( {\alpha ,\beta } \right){\rm{Im}} {A_j}\left( {\alpha ,\beta } \right) - {\rm{Re}} {A_j}\left( {\alpha ,\beta } \right){\rm{Im}} {A_i}\left( {\alpha ,\beta } \right) = 0 \\ & {\rm{Re}} {A_j}\left( {\alpha ,\beta } \right){\rm{Im}} \left[ {\sum\limits_k {T_k^q{A_k}(\alpha ,\beta ) + B(\alpha ,\beta )} } \right] - \\ & \quad \quad {\rm{Im}} {A_j}\left( {\alpha ,\beta } \right){\rm{Re}} \left[ {\sum\limits_k {T_k^q{A_k}(\alpha ,\beta ) + B(\alpha ,\beta )} } \right] = 0. \end{aligned}\right.$

      (11)

      If the system (11) does not have a solution when $\beta \ne 0$ for all interval parameters combinations, then, in ${S_r}$ there is no ${U^*}$ and the borders of ${S_r}$ consist of non-intersected edge branches.

      Suppose ${A_i}\left( s \right) = \displaystyle\sum\nolimits_{w = 0}^z {{a_{wi}}} {s^w}$, ${A_j}\left( s \right) = \displaystyle\sum\nolimits_{c = 0}^l {{a_{cj}}} {s^w}$. It has been defined if the degree z and l of polynomials ${A_i}\left( s \right)$ and ${A_j}\left( s \right)$ at interval parameters ${T_i}$ and ${T_j}$ are not higher than the second order, then, the analysis geared at the possibilities for edge branches intersection $RS_i^q$ and $RS_j^q$ does not require to solve the system (11), rather to check the condition fulfillment, which has been pointed out within the following statements.

      Statement 1. If ${A_i}\left( s \right)$ and ${A_j}\left( s \right)$ are the first order, then, there is no edge images intersection for face ${G_{ij}}$.

      Proof. The edge images intersections for face ${G_{ij}}$ are possible, if (10) are dependent. Based on Moivre formula, we write down the first equation (10) in trigonometric form

      $\frac{{\displaystyle\sum\limits_{w = 0}^z {{a_{wi}}} {{\left| s \right|}^w}\cos (w\varphi )}}{{\displaystyle\sum\limits_{w = 0}^z {{a_{wi}}} {{\left| s \right|}^w}\sin (w\varphi )}} = \frac{{\displaystyle\sum\limits_{c = 0}^l {{a_{cj}}} {{\left| s \right|}^c}\cos (c\varphi )}}{{\displaystyle\sum\limits_{c = 0}^l {{a_{cj}}} {{\left| s \right|}^c}\sin (c\varphi )}}.$

      The equation from this equality is

      $\begin{aligned} & \sum\limits_{w = 0}^z {{a_{wi}}} {\left| s \right|^w}\cos (w\varphi )\sum\limits_{c = 0}^l {{a_{cj}}} {\left| s \right|^c}\sin (c\varphi ) = \\ &\quad \quad \sum\limits_{w = 0}^z {{a_{wi}}} {\left| s \right|^w}\sin (w\varphi )\sum\limits_{c = 0}^l {{a_{cj}}} {\left| s \right|^c}\cos (c\varphi ). \end{aligned}$

      On the base of which the other equation can be made

      $\sum\limits_{w = 0,c = 0}^{z,l} {{a_{wi}}{a_{cj}}} {\left| s \right|^{w + c}}\sin \left( {\left( {{\rm{c}} - w} \right)\varphi } \right) = 0,\,w \ne c.$

      (12)

      Suppose $z = 1$; $l = 1$, then ${a_{0i}}{a_{1j}}{\left| s \right|^1}\sin(\varphi ) - {a_{1i}}{a_{0j}}\times$${\left| s \right|^1}\sin(\varphi ) = 0$. Thus, $\sin(\varphi ) \ne 0,$ then, in solving this equation, we will obtain ${a_{0i}}{a_{1j}} = {a_{1i}}{a_{0j}}$. The result says that when ${T_i}$ and ${T_j}$ are changed, the edge branches $RS_i^q$ and $RS_j^q$ depart from the vertex image at the same angle and coincide with each other. □

      Statement 2. If ${A_i}\left( s \right)$ and ${A_j}\left( s \right)$ are the second order, then, there is no edge images intersections for face ${G_{ij}}$ in case when the inequations are fulfilled for all pairs of the interval parameters as ${T_i}$ and ${T_j}$

      $\begin{split} & ({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}})({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}}) \ge {({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})^2} \\ & {a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}} \le 0 \\ & 4({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}}) \ge 0. \end{split} $

      (13)

      Proof. Suppose $z = 2$; $l = 2$. Then, based on (12) let us write down

      $\begin{aligned} & {a_{0i}}{a_{1j}}{\left| s \right|^1}\sin (\varphi ) - {a_{1i}}{a_{0j}}{\left| s \right|^1}\sin (\varphi ) + {a_{0i}}{a_{2j}}{\left| s \right|^2}\sin (2\varphi ) - \\ & \quad{a_{2i}}{a_{0j}}{\left| s \right|^2}\sin (2\varphi )\!+\!{a_{1i}}{a_{2j}}{\left| s \right|^3}\sin (\varphi )\!-\!{a_{2i}}{a_{1j}}{\left| s \right|^3}\sin (\varphi )\!=\!0. \end{aligned}$

      After the equation rearrangement, we obtain $ \sin (\varphi )\times$$({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}} + {a_{1i}}{a_{2j}}{\left| s \right|^2} - {a_{2i}}{a_{1j}}{\left| s \right|^2})+ 2\sin (\varphi )\cos (\varphi )\times$$({a_{0i}}{a_{2j}}{\left| s \right|^1} - {a_{2i}}{a_{0j}}{\left| s \right|^1}) = 0 $.

      The solution $ \left| s \right| $ (see the equation at the bottom) for this equation will be real and positive.

      If the following conditions are fulfilled.

      1) ${a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}} > 0$.

      2) $4{\cos ^2}(\varphi ){({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})^2} - 4({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}}) \times$$ ({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}}) > 0 $, consequently, ${\cos ^2}(\varphi ) > $ $ \dfrac{{({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}})}}{{{{({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})}^2}}}$. Thus, ${\cos ^2}(\varphi ) < 1$, then, $\dfrac{{({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}})}}{{{{({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})}^2}}} < 1$. Hence, the second condition $({a_{0i}}{a_{1j}}\!-\!{a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}}\!-\!{a_{2i}}{a_{1j}})< $$ {({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})^2}$.

      3) $2\cos (\varphi )({a_{2i}}{a_{0j}} - {a_{0i}}{a_{2j}}) -$

      $ \sqrt {4{{\cos }^2}(\varphi ){{({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})}^2} - 4({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}})} \times$

      $\sqrt { - {a_{2i}}{a_{1j}})} > 0,$ hence, after the rearrangement it follows that $2{\cos ^2}(\varphi )({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}}) > ({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}})\times$$({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}})$.

      Then, ${\cos ^2}(\varphi ) > \dfrac{{({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}})}}{{2{{({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})}^2}}}$, and, hence, $\dfrac{{({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}})}}{{2{{({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})}^2}}} < 1$. In so doing, the third condition is $({a_{0i}}{a_{1j}}\!-\!{a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}}\!-\!{a_{2i}}{a_{1j}}) <$$2{({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})^2}$.

      4) $2\cos (\varphi )({a_{2i}}{a_{0j}} - {a_{0i}}{a_{2j}}) +$

      $ \sqrt{4co{s^2}(\varphi ){({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})^2} - 4({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}}) ({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}})} $$>0.$

      Then, the fourth condition is $4({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}} - $${a_{2i}}{a_{1j}}) < 0.$

      Statement 2 leads to two conclusions.

      Conclusion 1. If $z = 1$, $l = 2$, then, there is no edge images intersections in face ${G_{ij}}$ in case when the inequations are fulfilled for all pairs of the interval parameters as ${T_i}$ and ${T_j}$

      $\begin{split} & {a_{1i}}{a_{2j}}({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}}) \ge {({a_{0i}}{a_{2j}})^2}\\ &4{a_{1i}}{a_{2j}}({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}}) \ge 0. \end{split} $

      (14)

      Conclusion 2. If $z = 2$, $l = 1$, then, there is no edge images intersections in face ${G_{ij}}$ in case when the inequalities are fulfilled for all pairs of the interval parameters as ${T_i}$ and ${T_j}$

      $\begin{split} & {a_{2i}}{a_{1j}}({a_{1i}}{a_{0j}} - {a_{0i}}{a_{1j}}) \ge {({a_{2i}}{a_{0j}})^2}\\ &4{a_{2i}}{a_{1j}}({a_{1i}}{a_{0j}} - {a_{0i}}{a_{1j}}) \ge 0. \end{split} $

      (15)

      Consequently, the methodology on the possibility analysis geared at edge images intersection for face ${G_{ij}}$ consists of the following stages.

      1) Write down ICP as (1).

      2) If the degree of all polynomials at interval parameters is not higher than the second order, then, it is necessary to check if the conditions (13)–(15) are properly fulfilled.

      3) If the conditions (13)–(15) are not fulfilled, then, there is edge images intersections for face ${G_{ij}}$.

      4) If among polynomials at interval parameters, there are polynomials of the third order and higher, then, it is necessary to choose an optional vertex ${V_q}$, $q \in \overline {1,{2^m}} $ and to solve the equation system for all faces concurrent in it (11).

    • If the edge branch point, which is the nearest one to an imaginary axis, is one of the edge ends as shown in Fig. 8 (a), then, this edge branch can be classified as the first type. If the nearest to an imaginary axis is one of the inner roots of the edge branch, it is referred to the second type (Fig. 8 (b)). The types of boundary edge branches are important to know when defining the root quality indices. For example, if the branch is of the first type, then, in order to define the minimal degree of stability and the maximum degree of oscillability, there is no need to build this edge branch, but it is enough to define the roots at the edge ends.

      Figure 8.  Edge branches of parametric polytope

      $\left| s \right| = \frac{{2\cos (\varphi )({a_{2i}}{a_{0j}} - {a_{0i}}{a_{2j}}) \pm \sqrt {\begin{array}{*{20}{l}} {4{{\cos }^2}(\varphi ){{({a_{0i}}{a_{2j}} - {a_{2i}}{a_{0j}})}^2} - }{4({a_{0i}}{a_{1j}} - {a_{1i}}{a_{0j}})({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}})} \end{array}} }}{{2({a_{1i}}{a_{2j}} - {a_{2i}}{a_{1j}})}}.$

      Condition 1. If polynomials ${A_i}(s)$ at interval parameters ${T_i}$ are the polynomials of the first degree or of even and odd degree s, as well as a product of two polynomials, then, the edge branch $RS_i^q$ is the branch of the first type. For other polynomials ${A_i}(s)$, the following condition is valid.

      Condition 2. If the condition is fulfilled as

      $\frac{{\partial \arg ((\overline {{{\rm{T}}_i}} - \underline {{{\rm{T}}_i}} ){{\rm{A}}_i}({\rm{j}}\beta ))}}{{\partial \beta }} \le \left| {\frac{{\sin (2\arg ((\overline {{{\rm{T}}_i}} - \underline {{{\rm{T}}_i}} ){{\rm{A}}_i}({\rm{j}}\beta )))}}{{2\beta }}} \right|$

      (16)

      then, the edge branch $RS_i^q$ can be the first-type one.

      Consequently, when these conditions are used, we can define the type for all edge branches arriving into the boundary route.

    • Considered research resulted in the methodology, based on constructing a vertex-edge route, applicable to determine the root indices of ICS robust quality. The methodology includes the following stages:

      1) Deriving an ICP (1).

      2) Defining the coordinates of polytope ${P_T}$ vertices.

      3) Calculating a polynomial complex root ${U_q}$ for the arbitrary ${V_q}$, $q \in \overline {1,{2^m}} $.

      4) Finding m angles $\Theta _i^q$, $i \in \overline {1,{{m}}} $ for ${U_q}$ based on (5) and (6).

      5) Verification of inhering ${U_q}$ to the border ${S_r}$ based on (7). If at least one condition (7) is not fulfilled, it should be chosen the other vertex ${P_T}$ and repeat the attempt, points from 3) to 5) above.

      6) For the value found $G{U_q}$ the consequence of departure angles based on the interval parameters ${T_i}$ for edge branches should be composed.

      7) Based on the consequence $\Theta _i^q$, $i \in \overline {1,{{m}}} $ the direct edge route can be built, which will depart from $G{U_q}$ and include $2m$ of edges.

      8) Defining faces ${G_{ij}}$, edge images $RS_i^q$, which can intersect.

      9) If two consequent edges $R_i^q$ and $R_j^q$ of the edge route are the edges of face ${G_{ij}}$ and their images can intersect, then, two opposite edges of this face should be added to the direct edge route.

      10) If while constructing the route we get repeated edges, they should be united.

      11) Defining the type of edge branches entering the constructed edge route.

      12) If the edge branch $RS_i^q$ is referred to the first type, then, it is deleted from the edge route. If the edge branches of the first type are connected consequently, then, in the route only vertices connecting them are left.

      13) Introducing a boundary vertex-edge route to the root plane and defining the root indices of ICS robust quality (a degree of robust oscillability and stability) according to allocation areas of ICP roots.

    • Let us consider a system responsible for automated position stabilization in a charging station to be merged with a tether for autonomous unmanned underwater vehicles. The structural scheme is described in Fig. 9.

      Figure 9.  System structural diagram

      In Fig. 9, ${k_{eng}}$ is the voltage transfer coefficient of an engine; ${k_{amp}}$ is transfer coefficient of an amplifying device; $C{\rm{ = }}\displaystyle\frac{C_{\rm{1}}}{l}$ is tether hardness coefficient, ${C_{\rm{1}}} = {10^5}\;{\rm N}$ is specific tether hardness coefficient; ${\rm{\chi = }}\displaystyle\frac{{\rm{\chi }}_1}{l}$ is the relative loss coefficient for tether elasticity, ${\chi _1} = {10^4}\;{\rm N} \cdot {\rm s}$ is specific coefficient of tether elasticity loss; $r = 0.1\;{\rm m}$ is hoist drum radius; ${k_1} = 1-$transfer coefficient of PI-controller, ${k_2} = 0.01$ is time constant for PI-controller, $m = [50;\,500]\;{\rm kg}$ is charging station mass and underwater vehicle mass; $l = [50;\,100]\;{\rm m}$ is tether length; $k = {k_{amp}} \cdot {k_{eng}} \cdot {k_v} =$[5; 15] is transfer coefficient of electric drive.

      As a result of structural transformations we obtain the interval characteristic polynomial:

      $D(s) = [{d_4}]{s^4} + [{d_3}] \cdot {s^3} + [{d_2}] \cdot {s^2} + [{d_1}] \cdot s + [{d_0}]$

      (17)

      where $\left[ {{d_0}} \right]=\left[ m \right]{{{C}}_{y\partial }}\left[ k \right]{k_1}{r^2}$; $[{d_1}]=({C_{y\partial }}(J + [m]{r^2}(1 + {k_2}$$[k])) + $${{\rm{\chi }}_{y\partial }}[m]{r^2}{k_1}k)$; $[{d_2}] = ({T_m}{C_{y\partial }}({r^2}[m] + J) + {{\rm{\chi }}_{y\partial }}(J + $$[m]{r^2}(1 + {k_2}k)))$; $[{d_3}] = (J[l][m] + {T_m}{\chi _{y\partial }}({r^2}[m] + J)$; $[{d_4}] = $$J[l][m]{T_m} $.

      The interval parameters $[m]$, $[l]$, $[k]$ are linearly included into ICP coefficients (17) (set the affine coefficients uncertainty) and are formed interval parametric polytope ${P_T}$. The polytope ${P_T}$ possesses 8 vertices: ${V_1}(\underline m ,\underline l ,\underline k )$, ${V_2}(\underline {{m_1}} ,\,\overline l ,\,\underline k )$, ${V_3}(\underline m ,\,\overline l ,\,\overline k )$, ${V_4}(\overline m ,\,\underline l ,\,\underline k )$, ${V_5}(\overline m ,\,\overline l ,\,\underline k )$, ${V_6}(\overline m ,\,\overline l ,\,\overline k )$, ${V_7}(\overline m ,\,\underline l ,\,\overline k )$,${V_8}(\underline m ,\,\underline l ,\,\overline k )$. Then, ICP (1) with the affine coefficients uncertainty looks as

      $[{T_1}] \cdot {A_1}(s) + \frac{1}{{[{T_2}]}} \cdot {A_2}(s) + [{T_3}] \cdot {A_3}(s) + B(s) = 0$

      (18)

      where $[{T_1}] = [l]$; $[{T_2}] = [m]$; $[{T_3}] = [k]$; ${A_1}(s) = J{s^3}({T_m}s + $ $ 1)$; ${A_2}(s) = Js(({T_m}s + 1)({\chi _1}s + {C_1}))$; ${A_3}(s) = {r^2}(({\chi _1}s + $$ {C_1})({k_2}s + {k_1}))$; $B(s) = {r^2}s(({\chi _1}s + {C_1})({T_m}s + 1))$. It is necessary to determine the vertices and edges of the polytope ${P_T}$, which will help to define the root indices of the robust quality in a system able to stabilize a position of a charging station to be merged.

      According to the algorithm, the polynomial roots in the first vertex have been defined in (18) [–10; –31.3; –6.3 – j4.88; –6.3 + j4.88], and the roots of polynomial ${A_1}(s)$ as well: [0; 0; 0; –10], ${A_2}(s)$: [0; 0; –10], ${A_3}(s)$: [–100; –10]. Further, for the image of the first vertex ${U_1} =- 6.3 +$$ {\rm j}4.88$ based on (5) and (6) the departure angles of the edge branches have been calculated with the following interval parameters: $\varTheta _{{T_1}}^{V1} = {146.34^ \circ }$, $\varTheta _{{T_2}}^{V1} = {93.94^ \circ }$, $\varTheta _{{T_3}}^{V1} = $82°. As long as the condition (7) is fulfilled, then, ${U_1}$ is a boundary vertex and belongs to the edge route. By virtue of the fact that $\varTheta _{{T_3}}^{V1} < \varTheta _{{T_2}}^{V1} < \varTheta _{{T_1}}^{V1}$, then, in the edge route the interval parameters depart from the vertex ${V_1}$ in the following consequence: ${T_3} \to {T_2} \to {T_1} \to {T_3} \to$$ {T_2} \to {T_1}$. This consequence accords with the direct edge route as shown in Fig. 10.

      Figure 10.  Direct edge route

      Let us verify if boundary edge route has intersected edge branches. In the vertex ${V_1}$ three faces ${G_{32}}$, ${G_{21}}$, ${G_{31}}$ meet, where the indices comply with the indices of the interval parameters. Given that the polynomials ${A_1}(s)$, ${A_2}(s)$ have a higher than the second degree, then, for faces ${G_{32}}$, ${G_{21}}$, ${G_{31}}$ three equation systems can be composed in (11). Having solved these equations, we obtain two roots: ${s_{1,2}} = - 5.55 \pm {\rm j}8.96$, corresponding to the coordinates of a possible intersection for the edge images of faces ${G_{32}}$, ${G_{21}}$, ${G_{31}}$. In so doing, boundary edge route will be viewed as shown in Fig. 11.

      Figure 11.  Edge route

      As a final stage let us define the type of edge branches in the edge route to be constructed. For the polynomial ${A_1}(s)$ the condition 1 is fulfilled, hence, the branches along ${T_1}$ have the first type. However, this condition does not cover the polynomials ${A_2}(s)$ and ${A_3}(s)$. Therefore, in order to define a branch type along ${T_2}$ and ${T_3}$ we need to verify the condition (16) and we will get $\dfrac{{\partial \arg \!\left(\!\!\left(\dfrac{1}{{\underline {{T_2}} }}\!-\!\dfrac{1}{{\overline {{T_2}}}}\right)\!{{{A}}_2}({\rm{j}}\beta )\!\right)}}{{\partial \beta }}\!>\!\left|\!{\dfrac{{\sin (2\arg \!\left(\!\!\left(\dfrac{1}{{\underline {{T_2}} }}\!-\!\dfrac{1}{{\overline {{T_2}} }}\right)\!{{{A}}_2}({\rm{j}}\beta )\!\right)}}{{2\beta }}}\!\right|$, $\dfrac{{\partial \arg \left(\left(\overline {{T_3}} - \underline {{T_3}}\right){{{A}}_3}({\rm{j}}\beta )\right)}}{{\partial \beta }} > \left| {\dfrac{{\sin (2\arg ((\overline {{T_3}} - \underline {{T_3}} ){{{A}}_3}({\rm{j}}\beta ))}}{{2\beta }}} \right|$, it shows that the edge branches along ${T_2}$ and ${T_3}$ are the second type ones. Consequently, the vertex-edge route has the view as seen in Fig. 12.

      Figure 12.  Boundary vertex-edge route

      With the aim of defining the root quality indices we put the route on the root plane (Fig. 13). As seen in Fig. 13, the degree of the robust system stability responsible for the stabilization of the charging station at merging is $\alpha = 1.62$, the degree of its robust oscillability is μ =8.13; it corresponds to a sector with the angle $\varphi = \pm {82^ \circ }$. These quality indices are defined by a vertex image. ${V_6}(\overline {{T_1}} ;\,\overline {{T_2}} ;\,\overline {{T_3}} )$. It should be noted that sufficiently high oscillability in a merged station position stabilization can be explained by tether elasticity properties in combination with a low coefficient of damping effect.

      Figure 13.  Vertexes and edges in boundary route

      Fig. 14 presents transient processes in two vertices of a boundary route, one of them corresponds to a minimum oscillability degree of the system stability (V6), the second one - to maximal value of oscillability degree (V4).

      Figure 14.  Vertexes and edges in boundary route

      As seen in Fig. 14, the minimum constant time is tmax = 0.58 (in ${V_4}(\overline {{T_1}} ;\,\underline {{T_2}} ;\,\underline {{T_3}} )$), and the maximum constant time is tmax = 1.98 (in ${V_6}(\overline {{T_1}} ;\,\overline {{T_2}} ;\,\overline {{T_3}} )$). The latter index corresponds to the found degree of the robust stability $\alpha $ that proves the correctness in evaluation of the system quality root indices.

    • The research gives the ground to conclude that when using the interval and affine uncertainty of ICP, the more accurate root allocation area is obtained with affine uncertainty. It is located inside the area constructed after reducing ICP coefficients to the interval view. It states that in transmitting from the interval parameters towards the interval coefficients as ICP, the control quality indices can be significantly decreased. Let us reaffirm this conclusion with the comparison of the robust quality analysis accuracy of the system studied above, when ICP has the affine uncertainty of the coefficients. For the second case, in Fig. 15, we introduced the polynomial vertices of ICP coefficients obtained through construction of the vertex route.

      Figure 15.  Vertexes of boundary route

      The figure illustrates that the system, which is relatively stable at affine uncertainty of ICP coefficients, and is responsible for charging station position stabilization in merging process, turned out to be non-stable after the coefficients have been reduced to the interval view.

    • For ICS with affine uncertainty of ICP coefficients the following properties of boundary vertex-edge route of a parametric polynomial have been set:

      1) The route can be composed of non-intersected edge branches $RS_i^q$ in the ordering corresponding to the departure angles sequence $RS_i^q$ from any boundary pole;

      2) The route can include intersecting edge branches $RS_i^q$ and $RS_j^q$, which can be defined via the algebraic conditions in the view of the proven statements;

      3) The first type edge branches can be deleted from the route, having left only their boundary root nodes.

      On the basis of the properties presented above, we developed the algorithm enabling to construct a boundary vertex-edge polytope with the interval system parameters. Its projection to the root plane defines the root robust quality indices of ICS.

      It is shown that in transmitting from the interval system parameters towards the interval ICP coefficients the root allocation area of ICP is significantly enlarged. That results in reducing the robust quality indices of the system.

    • This work was supported by the Ministry of Education and Science of the Russian Federation (No. 2.3649.2017/PCh).

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