Sensitivity Analysis of Uniform Control Systems

1. Introduction

The paper can be considered as a logical continuation of the research presented in [1] and is devoted to sensitivity analysis of a special class of multi-input and multi-output (MIMO) control systems called uniform systems. Uniform control systems are widespread in various technical applications, such as aerospace engineering, robotics, chemical and power industry, and many others [2,3,4]. The main structural feature of the uniform systems is that their separate channels have identical transfer functions, and the cross-connections are rigid, i.e. are characterized by a square numerical matrix [4].

The issue of sensitivity analysis of uniform systems to small perturbations of parameters is discussed from the perspective of the Characteristic Transfer Functions (CTFs) (or Characteristic Gain Loci) method which was proposed by A.G. Macfarlane and his colleagues in 1979s [5,6,7,8]. The CTFs method allows one to associate with an N-dimensional MIMO system a set of N independent single-input single-output (SISO) characteristic systems and, thereby, to reduce the stability analysis and design of an interconnected MIMO system to stability analysis and design of N SISO systems [3,4,9].

The sensitivity of characteristic gain loci was first considered in the paper by I. Postlethwaite [10]. The introduced in [10] sensitivity indices characterize the sensitivity of the gain loci to small perturbations of the return-ratio matrix of the MIMO system. The task of robustness of MIMO systems, where the plant is described using the CTFs method, is addressed in [11]. In that paper, a sophisticated two-stage design procedure to improve the system’s robustness is proposed based on the notion of commutative controller [8]. At the first stage, a dynamic pre-compensator is designed which transforms the plant transfer matrix into a normal matrix. At the second stage, a commutative controller is designed for the normalized plant. If the design is correct, then the CTFs of the pre-compensated plant and the commutative controller are multiplied together. Unfortunately, it is mostly impossible to find a realizable matrix compensator that transforms the transfer matrix of the plant into a normal matrix.

A systematic treatment of sensitivity of multivariable control systems from the perspective of the CTFs method is given in [1]. The formulas are derived determining the sensitivity functions of the CTFs and canonical basis axes to small variations of parameters of general type MIMO control systems. The relations between the sensitivity functions of the open-loop and closed-loop MIMO systems are established. However, the results obtained in [1], having generic character, do not allow one to take into account peculiar structural features of some special classes of multivariable control systems. First of all, that concerns the uniform systems which structurally always comprise two different matrix blocks – a scalar transfer matrix of identical separate channels and a numerical matrix of rigid cross-connections. That is why it is important to discern the parameter perturbations of these two blocks since the small perturbations of the transfer functions of separate channels do not affect the canonical basis of the uniform system, and, on the contrary, small perturbations of the numerical matrix of cross-connections change, besides the canonical axes, only the equivalent “gains” of the CTFs of the open-loop system.

Detailed analysis of the above-mentioned points is considered in this paper, which is organized as follows. The canonical representation of uniform control systems is presented in Section 2. Section 3 addresses the sensitivity analysis of uniform control systems to small variations of parameters. The situations of the parameter perturbations of the transfer functions of separate channels and those of the cross-connections matrix are treated separately. Section 4 presents an example of sensitivity analysis of control systems of multirotor unmanned aerial vehicles (UAVs) to small degradations of motors’ efficiency.

2. Canonical Representation of Uniform Control Systems

The matrix block diagram of a linear uniform MIMO system is shown in Figure 1 [4].

In Figure 1, $\phi (s),$$f(s),$$\epsilon (s)$ stand for the Laplace transforms of the N-dimensional input, output, and error vector signals $\phi (t), f(t), \epsilon (t)$, respectively (we shall regard them as elements of some N-dimensional complex space ${ℂ}^N$); $w(s)$is a scalar transfer function of separate channels, which is a proper rational function in complex variable $s$, and $R$ is a numerical matrix of order $N\times N$ describing cross-connections.

It is easy to see that the transfer matrix W(s) of the open-loop uniform system in Figure 1:

$$W(s) = w(s)R$$

(1)

coincides, up to the complex scalar multiplier $w(s)$, with the numerical matrix of cross-connections $R$. This leads to interesting structural and dynamic properties of uniform MIMO systems and allows separating them into an individual class of multivariable control systems [4].

The output $f(s)$ and error $\epsilon (s)$ vectors, where

$$\epsilon (s) = \phi (s) - f(s),$$

are related to the input vector $\phi (s)$ by the following operator equations:

$$f(s) = T(s)\phi (s) , \epsilon (s) = S(s)\phi (s)$$

(2)

where

$$T(s) = {\left[I + w(s)R\right]}^{-1}w(s)R , S(s) = {\left[I + w(s)R\right]}^{-1}$$

(3)

are the transfer function matrices of the closed-loop uniform system with respect to output and error signals, and $I$ is the unit matrix. The transfer matrices $S(s)$ and $T(s)$ (3) are usually called sensitivity function matrix and complementary sensitivity function matrix [2,3,4].

Based on the method of CTFs, the transfer matrix of the open-loop MIMO system $W(s)$ (1) can be represented, using dyadic notation and similarity transformation, in the following canonical forms [4]:

$$W(s) = {\displaystyle \sum _{i = 1}^N{c}_i>q_i(s)<c_i^{+}= }Cdiag\{q_i(s)\}{C}^{-1},$$

(4)

where the complex scalar functions

$$q_i(s)={\lambda }_{i}w(s)$$

(5)

$$i = 1,2,\mathrm{...},N$$

are the CTFs of the open-loop system (further, for simplicity, all ${\lambda }_i$ are assumed distinct); $c_i$ are linearly-independent normalized eigenvectors of $R$ which constitute the canonical basis of the open-loop uniform system; $c_1^{+}$ are vectors dual to $c_i$ (vectors of the dual basis), and the constant modal matrix $C$ is composed of column-vectors $c_i$.

Substitution the canonical representations of $W(s)$ (4) into the complementary sensitivity function matrix $T(s)$ and the sensitivity function matrix $S(s)$ (3) yields [4]

$$T(s) = {\displaystyle \sum _{i = 1}^N{c}_i>\frac{{\lambda }_{i}w(s)}{1 + {\lambda }_{i}w(s)}}<c_i^{+} =Cdiag\left\{{\displaystyle \frac{{\lambda }_{i}w(s)}{1 + {\lambda }_{i}w(s)}}\right\}{C}^{-1}$$

(6)

$$S(s) = {\displaystyle \sum _{i = 1}^N{c}_i>\frac{1}{1 + {\lambda }_{i}w(s)}}<c_i^{+} =Cdiag\left\{{\displaystyle \frac{1}{1 + {\lambda }_{i}w(s)}}\right\}{C}^{-1}$$

(7)

Inspection of (4)-(7) shows that the canonical basis of the closed-loop uniform system coincides with the canonical basis of the open-loop system [4,5,6]. Moreover, the CTFs of the closed-loop MIMO system with respect to the output and error are related to $q_i(s)$ by the very same relationships as common transfer functions of open-loop and closed-loop SISO control systems. Geometrically, all this is illustrated in Figure 2 and Figure 3.

The stability of the uniform system in Figure 1 is determined by roots of the characteristic equation [2,3,4,5]

$$\det [I + w(s)R] = {\displaystyle \prod _{i = 1}^N[1 + {\lambda }_{i}w(s)] = 0}$$

(8)

It is clear from (8) that the uniform system in Figure 1 is stable if all N SISO characteristic systems are stable. Hence, the stability analysis of an N -dimensional uniform system is reduced to stability analysis of N independent characteristic systems which can be performed using standard methods of classical feedback control, for example, using the Nyquist criterion [9].

3. Sensitivity of Characteristic Transfer Functions and Canonical Basis Axes of Uniform Systems

In this section, we address the influence of small variations of the system parameters on the dynamics and structural features of uniform systems. In general, this problem is solved by special methods of sensitivity theory [12,13,14,15,16]. Below, the problem of uniform systems sensitivity is considered from the positions of the CTFs method. The main attention is paid to the sensitivity functions of SISO characteristic systems and canonical basis axes in the case of small variations of the uniform system’s parameters.

Let us assume that the open-loop transfer matrix $W(s)$ of the uniform system in Figure 1 depends continuously on $m$ varying parameters ${\alpha }_r (r = 1, 2, \mathrm{...} , m)$, forming the vector $\alpha$. In what follows, to emphasize the dependence of $W(s)$ on $\alpha$ we shall write $W(s,\alpha )$. Denote by ${\alpha }_o$ the vector of nominal values of $\alpha$, and by $\Delta \alpha$ - the vector of variations, i.e. assume $\alpha = {\alpha }_o + \Delta \alpha$. Let us know the CTFs and canonical basis of the uniform system with the nominal values ${\alpha }_o$. A question arises, how the CTFs and canonical basis are changed in case of small variations of parameters $\alpha$, i.e. for ${(\Delta {\alpha }_r)}^2 \approx 0$? We solve this task as a first approximation, based on the perturbation theory of linear operators in finite-dimension Hilbert spaces [17,18].

Due to specific structural features of uniform control systems, we shall discuss two different cases, concerning the sensitivity of the CTFs and canonical basis axes to the parameters variations.

The first case concerns the sensitivity of the uniform system to the changes of m parameters ${\alpha }_r (r = 1, 2, \mathrm{...} , m)$ of the transfer functions $w(s)$. In this case, the scalar transfer matrix $w(s,\alpha )I$ should be replaced by a diagonal matrix $diag\{w(s,{\alpha }_i)\}$, where each m-dimensional vector ${\alpha }_i$ has the form ${\alpha }_i={\alpha }_o+\Delta {\alpha }_i$. Obviously, for unperturbed system (if all $\Delta {\alpha }_i$ are zero) we have $diag\{w(s,{\alpha }_i)\}=w(s,{\alpha }_0)I$. The second case concerns the situation when the varying parameters are elements of numerical matrix of cross-connections, i.e. $R=R(\alpha )$, where $\alpha$ is an m-dimensional vector. In both cases, we will assume that the transfer functions $w(s,{\alpha }_i)$ or the matrix $R(\alpha )$ depend continuously on parameters ${\alpha }_{ir}$ or ${\alpha }_r$.

Let, because of parameters perturbations, the transfer matrix $W(s,{\alpha }_o)$ can be expressed as

$$W(s,\alpha ) = W(s,{\alpha }_o) + \Delta W(s,\alpha ),$$

(9)

where $\Delta W(s,\alpha )$ is the variation of $W(s,{\alpha }_o)$ caused by the perturbations $\Delta \alpha$.

3.1. Open-Loop Uniform System

Consider first the sensitivity of the CTFs and canonical basis axes of the open-loop system.

3.1.1. Sensitivity of CTFs to Small Perturbations of the Transfer Functions $w(s,{\alpha }_i)$

In this case, for each i we have

$$w(s,{\alpha }_i)=w(s,{\alpha }_0)+\Delta w(s,{\alpha }_i)$$

(10)

i = 1,2,…,N

and the transfer matrix $W(s,\alpha )$ (9) has the following form:

$$W(s,\alpha ) =W(s,{\alpha }_o) + \Delta W(s,\alpha ) = diag\{w(s,{\alpha }_i)\}R =w(s,{\alpha }_o)R+diag\{\Delta w_i(s,{\alpha }_i)\}R ,$$

(11)

where

$$W(s,{\alpha }_o) =w(s,{\alpha }_o)R, \Delta W(s,\alpha ) =diag\{\Delta w(s,{\alpha }_i)\}R$$

(12)

and ${\alpha }_i$ is an $m$-dimensional vector with components ${\alpha }_{ir} (r=1,2,\mathrm{...},m)$.

As we assume continuous dependence of $w(s,{\alpha }_i)$ on ${\alpha }_i$, the variation $\Delta W(s,\alpha )$ in (12) can be represented for small $\Delta {\alpha }_{ir}$ with the help of the Taylor series

$$\Delta W(s,\alpha ) = diag\left\{{\displaystyle \sum _{r = 1}^m{U}_{ir}^W(s)\Delta {\alpha }_{ir } + \mathrm{....}}\right\}R,$$

(13)

where

$$\begin{array}{l} U_{ir}^W(s) = {\left.{\displaystyle \frac{\delta w(s,{\alpha }_i)}{\delta {\alpha }_{ir}}}\right|}_{{\alpha }_i = {\alpha }_o} \\ i = 1,2,\mathrm{...},N; r = 1, 2, \mathrm{...} , m\end{array}$$

(14)

are the first-order sensitivity functions of the transfer function $w(s,{\alpha }_i)$ with respect to variations $\Delta {\alpha }_{ir}$, evaluated for the nominal vector ${\alpha }_i = {\alpha }_o$. The dots in (13) and in what follows mean that the terms of the higher order of smallness, as compared with $\Delta {\alpha }_{ir}$, are omitted.

As the transfer functions $w(s,{\alpha }_i)$ are assumed differentiable with respect to ${\alpha }_{ir}$, the following expansions for the CTFs $q_i(s,\alpha )$ are valid:

$$q_i(s,\alpha ) = q_i(s,{\alpha }_o) + {\displaystyle \sum _{r = 1}^m{S}_{i\hspace{0.17em}r}^q}(s)\Delta {\alpha }_{ir} + \mathrm{...}$$

(15)

$$i = 1,2,\mathrm{...},N,$$

where

$$S_{ir}^q(s) = {\left.{\displaystyle \frac{\delta q_i(s,{\alpha }_i)}{\delta {\alpha }_{ir}}}\right|}_{{\alpha }_i = {\alpha }_o}= {\lambda }_i{\left.{\displaystyle \frac{\delta w(s,{\alpha }_i)}{\delta {\alpha }_{ir}}}\right|}_{{\alpha }_i = {\alpha }_o}$$

(16)

$$r = 1, 2, \mathrm{...} ,m$$

are the unknown first-order sensitivity functions of the CTFs $q_i(s,{\alpha }_i)$.

Since (15) are the “eigenvalues” of the perturbed matrix $W(s,\alpha )$ (11), and allowing for (10), we can write down

$$\left[w(s,{\alpha }_o)R+diag\left\{{\displaystyle \sum _{r = 1}^m{U}_{ir}^W(s)\Delta {\alpha }_{ir } + \mathrm{....}}\right\}R\right]c_i = diag\left\{q_i(s,{\alpha }_o) +{\displaystyle \sum _{r = 1}^m{S}_{i\hspace{0.17em}r}^q}(s)\Delta {\alpha }_{ir} + \mathrm{...}\right\}{c}_i$$

(17)

$$i = 1, 2, \mathrm{...} , N.$$

Allowing for $R{c}_i={\lambda }_i{c}_i$ and $q_i(s,{\alpha }_o)={\lambda }_{i}w(s,{\alpha }_o)$ we have immediately

$$\begin{array}{l}{U}_{ir}^W(s)=S_{ir}^q(s) = {\lambda }_i{\left.{\displaystyle \frac{\delta w(s,{\alpha }_i)}{\delta {\alpha }_{ir}}}\right|}_{{\alpha }_i = {\alpha }_o}\\ r = 1, 2, \mathrm{...} ,m,\end{array}$$

(18)

that is the sensitivity function of the ith characteristic system of the open-loop uniform system with respect to non-identical variations of parameters of transfer functions $w(s,{\alpha }_i)$ are equal to the corresponding sensitivities of $w(s,{\alpha }_i)$ multiplied by the eigenvalue ${\lambda }_i$.

3.1.2. Sensitivity of the Open-Loop Uniform System to Small Perturbations of the Cross-Connection Matrix R

Let us proceed to the case of parameters variations of the numerical matrix of cross-connections R. Matrix R in this case can be written in the form:

$$R=R(\alpha ) = R({\alpha }_o)+\Delta R(\alpha ).$$

(19)

Since the continuous dependence of $R(\alpha )$ on ${\alpha }_r$ is assumed, the variation $\Delta R(\alpha )$in (19) can be represented as

$$\Delta R(\alpha ) = {\displaystyle \sum _{r = 1}^m{U}_r^R\Delta {\alpha }_{r } + \mathrm{....}},$$

(20)

where

$$\begin{array}{l}{U}_r^R= {\left.{\displaystyle \frac{\delta R(\alpha )}{\delta {\alpha }_r}}\right|}_{\alpha = {\alpha }_o} \\ r = 1, 2, \mathrm{...} , m\end{array}$$

(21)

are the sensitivity matrices of $R(\alpha )$ with respect to variations $\Delta {\alpha }_r$.

On the other hand, we have for canonical basis axes (eigenvectors of R) and factors ${\lambda }_i$ (eigenvalues of R) the following expressions:

$$c_i(\alpha ) = c_i({\alpha }_o) + {\displaystyle \sum _{r = 1}^m{\mathrm{\Upsilon }}_{ir}}\Delta {\alpha }_r + \mathrm{....}$$

(22)

where

$${\mathrm{\Upsilon }}_{ir} = {\left.{\displaystyle \frac{\delta c_i(\alpha )}{\delta {\alpha }_r}}\right|}_{\alpha = {\alpha }_o}$$

(23)

are the unknown sensitivity vectors of the ith canonical basis axis $c_i(\alpha )$ with respect to the variation $\Delta {\alpha }_r$ of the rth parameter ${\alpha }_r$, and

$${\lambda }_i(\alpha ) = {\lambda }_i({\alpha }_o) + {\displaystyle \sum _{r = 1}^m{\beta }_{ir\hspace{0.17em}}}\Delta {\alpha }_r + \mathrm{...},$$

(24)

where

$${\beta }_{ir} = {\left.{\displaystyle \frac{\delta {\lambda }_i(\alpha )}{\delta {\alpha }_r}}\right|}_{\alpha = {\alpha }_o}$$

(25)

are the sensitivity functions of the ith eigenvalue ${\lambda }_i(\alpha )$ with respect to the variation $\Delta {\alpha }_r$ of ${\alpha }_r$.

Since (22) and (24) are the eigenvectors and eigenvalues of the perturbed matrix $W(s,\alpha )$ (11) and allowing for (13) we can write down

$$w(s) \left[R({\alpha }_o) + {\displaystyle \sum _{r = 1}^m{U}_r^R\Delta {\alpha }_{r } + \mathrm{....}}\right]\hspace{0.17em}\left[c_i({\alpha }_o) + {\displaystyle \sum _{r = 1}^m{\mathrm{\Upsilon }}_{ir}}\Delta {\alpha }_r + \mathrm{....}\right] =$$

(26)

$$\begin{array}{l}= w(s) \left[{\lambda }_i({\alpha }_o) + {\displaystyle \sum _{r = 1}^m{\beta }_{i\hspace{0.17em}r}}\Delta {\alpha }_r + \mathrm{...}\right]\hspace{0.17em}\left[c_i({\alpha }_o) + {\displaystyle \sum _{r = 1}^m{\mathrm{\Upsilon }}_{ir}}\Delta {\alpha }_r + \mathrm{....}\right]\\ i = 1, 2, \mathrm{...} , N\end{array}$$

Performing the multiplication and comparing the first-order terms with respect to the corresponding perturbations $\Delta {\alpha }_r$ in both sides of (26), we get the following system of $mN$ equations:

$$R({\alpha }_o){\mathrm{\Upsilon }}_{ir}+ U_r^R{c}_i({\alpha }_o)= {\lambda }_i({\alpha }_o){\mathrm{\Upsilon }}_{ir} + {\beta }_{ir}{c}_i({\alpha }_o)$$

(27)

$$i = 1, 2, \mathrm{...} , N, r = 1, 2, \mathrm{...} , m$$

Multiply, in the form of scalar product, both sides of (27) by the vector $c_i^{+}({\alpha }_o)$ dual to the eigenvector $c_i({\alpha }_o)$ of the unperturbed uniform system. Recalling that the first vector in the scalar product should be taken as complex conjugate, we obtain

$$\begin{array}{l}<R({\alpha }_o){\mathrm{\Upsilon }}_{ir},c_i^{+}({\alpha }_o)> +<U_r^R{c}_i({\alpha }_o),c_i^{+}({\alpha }_o)> ={\lambda }_i({\alpha }_o)<{\mathrm{\Upsilon }}_{ir},c_i^{+}({\alpha }_o)>\hspace{0.17em} + \hspace{0.17em}{\beta }_{ir}\\ i = 1, 2, \mathrm{...} , N; r = 1, 2, \mathrm{...} , m.\end{array}$$

(28)

Note that the matrix $R({\alpha }_o)$ in (19) has the canonical representation

$$R({\alpha }_o) = C({\alpha }_o)diag\{{\lambda }_i({\alpha }_o)\}\hspace{0.17em}{C}^{-1}({\alpha }_o).$$

(29)

Conjugation of this matrix yields

$$R^{\ast }({\alpha }_o) = R^T({\alpha }_o) = {(C^{-1}({\alpha }_o))}^{*}diag\{{\tilde{\lambda }}_i({\alpha }_o)\}\hspace{0.17em}{C}^{*}({\alpha }_o).$$

(30)

From here, it is clear that the eigenvalue ${\tilde{\lambda }}_i({\alpha }_o)$ of $R^{\ast }({\alpha }_o)$ corresponds to the dual eigenvector $c_i^{+}({\alpha }_o)$. Therefore

$$<R({\alpha }_o){\mathrm{\Upsilon }}_{ir},c_i^{+}({\alpha }_o)> =\hspace{0.17em} <{\mathrm{\Upsilon }}_{ir},R^{*}({\alpha }_o)c_i^{+}({\alpha }_o)> = {\lambda }_i({\alpha }_o)<{\mathrm{\Upsilon }}_{ir},c_i^{+}({\alpha }_o)>$$

(31)

Substituting (31) into (28) and conjugating both sides in (28), yields the final expression for the sensitivity function ${\beta }_{ir}$:

$${\beta }_{ir} = <c_i^{+}({\alpha }_o),\hspace{0.17em}{U}_r^R{c}_i({\alpha }_o)>$$

(32)

$$i = 1, 2, \mathrm{...} , N, r = 1, 2, \mathrm{...} , m.$$

It is important to note that for any fixed $r$ the right sides in (32) represent (for $i = 1, 2, \mathrm{...} , N$) the diagonal elements of the matrix $U_r^R$ (21), where the latter is evaluated in the basis composed of the vectors $c_i({\alpha }_o)$. Hence, the sensitivity functions ${\beta }_{ir}$ of eigenvalues ${\lambda }_i(\alpha )$ with respect to small perturbations of the $r$th parameter ${\alpha }_r$ are equal to the diagonal elements of the sensitivity matrix $U_r^R$ (21) evaluated in the canonical basis of the unperturbed uniform system.

Proceed now to calculation of the sensitivity vectors ${\mathrm{\Upsilon }}_{ir}$(23) of the canonical basis axes of the open-loop system. Multiplication of both sides of (27) by the dual vectors $c_k^{+}({\alpha }_o)$ $(k = 1, 2, \mathrm{...} , N;$ $k \ne i)$ yields after some simple transformations:

$$<c_k^{+}({\alpha }_o),\hspace{0.17em}{\mathrm{\Upsilon }}_{ir}> = \hspace{0.17em}{\displaystyle \frac{<c_k^{+}({\alpha }_o),U_r^R{c}_i({\alpha }_o)>}{\left[{\lambda }_i({\alpha }_o) - {\lambda }_k({\alpha }_o)\right]}}$$

(33)

$$k = 1, 2, \mathrm{...} , N; k \ne i, r = 1, 2, \mathrm{...} , m,$$

where $<c_k^{+}({\alpha }_o),U_r^R{c}_i({\alpha }_o)> = n_{ki}^r$ are the elements of the $i$th column (except for $n_{ii}^r$) of the matrix $U_r^R$ (21), represented in the canonical basis of the unperturbed system. Notice that the scalar products on the left side of (33) are the coordinates of the vector ${\mathrm{\Upsilon }}_{ir}$ along the $k$th axis $c_k({\alpha }_o)$ of the initial canonical basis. It is easy to show, using the normalizing condition for the varied eigenvector $c_i({\alpha }_o)$ (22) [17], that the coordinate of ${\mathrm{\Upsilon }}_{ir}$ along the $i$th axis $c_i({\alpha }_o)$ is equal to zero. Therefore, we can write

$$\begin{array}{l}{\mathrm{\Upsilon }}_{ir}= \hspace{0.17em}{\displaystyle \sum _{\begin{array}{l}k = 1\\ k \ne i\end{array}}^N\hspace{0.17em}\frac{n_{ki}^r}{\left[{\lambda }_i({\alpha }_o) - {\lambda }_k({\alpha }_o)\right]}}\hspace{0.17em}{c}_k({\alpha }_o),\\ i = 1, 2, \mathrm{...} , N; r = 1, 2, \mathrm{...} , m.\end{array}$$

(34)

Thus, expressions (32) – (34) determine the sensitivity functions ${\beta }_{ir}$ of the “factors” ${\lambda }_i(\alpha )$ of the CTFs $q_i(s,\alpha )={\lambda }_i(\alpha )w(s)$ of the open-loop uniform system and the sensitivity vectors ${\mathrm{\Upsilon }}_{ir}$ of the canonical axes $c_i(\alpha )$ with respect to small perturbations of the parameters ${\alpha }_r$ of the matrix $R(\alpha )$. These expressions show that for evaluating ${\beta }_{ir}$ and ${\mathrm{\Upsilon }}_{ir}$ it is enough to find the representation

$$\begin{array}{l}{N}_r = C^{-1}({\alpha }_o)U_r^{R}C({\alpha }_o)\\ r = 1, 2, \mathrm{...} , m\end{array}$$

(35)

of the matrices $U_r^R$ (21) in the canonical basis of the unperturbed uniform system. The diagonal elements $n_{ii}^r$ of the matrix $N_r$ (35) are equal to the sensitivity functions ${\beta }_{ir}$ (25), and the non-diagonal elements $n_{ik}^r$, after dividing by the differences $\Delta =$ ${\lambda }_i({\alpha }_o) - {\lambda }_k({\alpha }_o)$, give the coordinates of the sensitivity vectors ${\mathrm{\Upsilon }}_{ir}$ (23) in the mentioned basis.

3.2. Closed-Loop Uniform System

In this section we discuss the sensitivity functions of the closed-loop uniform systems and their relation to the corresponding sensitivity functions of open-loop systems. For brevity, only the case of sensitivity transfer matrix $S(s)$ will be considered. The case of complementary transfer matrix $T(s)$ can be treated analogously.

3.2.1. Sensitivity of the CTFs of Closed-Loop Uniform Systems with Respect to Parameters Variations of the Transfer Functions $w(s,{\alpha }_i)$

In this case, the canonical representation of the sensitivity transfer matrix $S(s,\alpha )$ via the similarity transformation has the form:

$$S(s,\alpha ) = Cdiag\left\{S_i(s,{\alpha }_i)\right\}{C}^{-1}$$

(36)

where

$$S_i(s,\alpha )={\displaystyle \frac{1}{1 + {\lambda }_{i}w(s,{\alpha }_i) }}$$

(37)

$$i = 1, 2, \mathrm{...} , N$$

are the CTFs of $S(s,\alpha )$.

Since the modal matrix C of uniform systems is constant, we immediately get from (36) the following expressions for sensitivity of CTFs (37) with respect to parameter variations ${\alpha }_{ir}$ of transfer functions $w(s,{\alpha }_i)$:

$$S_{ir}^S(s) \hspace{0.17em}= \hspace{0.17em}{\left.{\displaystyle \frac{\delta S_i(s,{\alpha }_i)}{\delta {\alpha }_{ir}}}\right|}_{\alpha = {\alpha }_o}=-{\displaystyle \frac{1}{{\left[1 + {\lambda }_{i}w(s,{\alpha }_0)\right]}^2 }} {\lambda }_i{\left.{\displaystyle \frac{\delta w(s,{\alpha }_i)}{\delta {\alpha }_{ir}}}\right|}_{\alpha = {\alpha }_o}$$

(38)

or, allowing for (16),

$$\begin{array}{l}{S}_{ir}^S(s) \hspace{0.17em}=-{\displaystyle \frac{1}{{\left[1 + {\lambda }_{i}w(s,{\alpha }_0)\right]}^2 }}{S}_{ir}^q(s) \\ i=1,2,\mathrm{...},N; r = 1, 2, \mathrm{...} , m.\end{array}$$

(39)

where

$$S_{ir}^q(s) ={\lambda }_i{\left.{\displaystyle \frac{\delta w(s,{\alpha }_i)}{\delta {\alpha }_{ir}}}\right|}_{\alpha = {\alpha }_o}$$

Thus, as can be seen from (39), the sensitivity functions of CTFs of the open-loop and closed-loop uniform systems in case of parameters variations of the transfer functions $w(s,{\alpha }_i)$ are related by the same expressions as in the common SISO case [9].

3.2.2. Sensitivity of the Closed-Loop Uniform Systems to Variation of Elements of the Cross-Connections Matrix R.

Consider now the sensitivity of the transfer matrix $S(s)$ (3). Assume that because of parameter perturbations $\Delta \alpha$ the initial transfer matrix $S(s,{\alpha }_o)$ becomes

$$S(s,\alpha ) = S(s,{\alpha }_o) + \Delta S(s,\alpha ),$$

(40)

where $\Delta S(s,\alpha )$ is the variation of $S(s,{\alpha }_o)$ caused by the perturbations $\Delta \alpha$.

Let us find the expression for $S(s,\alpha )$ assuming that the corresponding variation $\Delta W(s,\alpha )$ of the open-loop uniform system is known. In that case, the matrix $S(s,\alpha )$ in (3) can be represented in the form

$$S(s,\alpha ) = {[I + w(s)R({\alpha }_o) + w(s)\Delta R(\alpha )]}^{-1}.$$

(41)

It is easy to see from (3) and (41) that the variation $\Delta W(s,\alpha )$ is equal to the difference of the inverse transfer matrices $S^{-1}(s,\alpha )$ and $S^{-1}(s,{\alpha }_o)$, i.e.

$$S^{-1}(s,\alpha ) - S^{-1}(s,{\alpha }_o) = w(s)\Delta R(\alpha )$$

Multiplying this relation from the left by $S(s,{\alpha }_o)$ and from the right by $S(s,\alpha )$ we get

$$S(s,{\alpha }_o) - S(s,\alpha ) = -\Delta S(s,\alpha ) = w(s)S(s,{\alpha }_o)\Delta R(\alpha )S(s,\alpha )$$

(42)

from which, taking into account (40), after simple algebraic transformations we obtain the following expression for the variation $\Delta S(s,\alpha )$ of the transfer matrix $S(s,{\alpha }_o)$ caused by the variation $\Delta W(s,\alpha )$:

$$\Delta S(s,\alpha ) = -w(s){[I + w(s)S(s,{\alpha }_o)\Delta R(\alpha )]}^{-1}S(s,{\alpha }_o)\Delta R(\alpha )S(s,{\alpha }_o).$$

(43)

In principle, this expression is valid for any, not necessarily small, variations $\Delta W(s,\alpha )$. Recalling that the task above was solved as a first approximation, that is preserving only the terms linear with respect to $\Delta {\alpha }_r$, let us determine this approximation for $\Delta S(s,\alpha )$. To this end, expand the inverse matrix on the right side of (41) into the infinite Neumann series [18]:

$${[I + S(s,{\alpha }_o)w(s)\Delta R(\alpha )]}^{-1} = {\displaystyle \sum _{i = 0}^{\infty }{[S(s,{\alpha }_o)w(s)\Delta R(\alpha )]}^i}$$

(44)

This series converges for $\left|\right|S(s,{\alpha }_o)w(s)\Delta R(\alpha )\left|\right| < \hspace{0.17em}1$ , which always takes place in practice for small $\Delta {\alpha }_r$. Substituting (44) in (43) and neglecting the terms of higher order of smallness gives a first approximation for the variation $\Delta S(s,\alpha )$

$$\Delta S(s,\alpha ) = -w(s)S(s,{\alpha }_o)\Delta R(\alpha )S(s,{\alpha }_o).$$

(45)

Due to the continuous dependence of the transfer matrix $S(s,\alpha )$ on parameters ${\alpha }_r$ the variation $\Delta S(s,\alpha )$ can be written in the form

$$\Delta S(s,\alpha ) = \hspace{0.17em}{\displaystyle \sum _{r = 1}^m{U}_r^S(s)\Delta {\alpha }_r \hspace{0.17em}+ \hspace{0.17em}\mathrm{....}},$$

(46)

where

$$\begin{array}{l}{U}_r^S(s) \hspace{0.17em}= \hspace{0.17em}{\left.{\displaystyle \frac{\delta S(s,\alpha )}{\delta {\alpha }_r}}\right|}_{\alpha = {\alpha }_o}, \\ r = 1, 2, \mathrm{...} , m\end{array}$$

(47)

are the first-order sensitivity matrices of the transfer matrix $S(s,\alpha )$ with respect to the variations of the $r$th parameter ${\alpha }_r$. Then, based on (45), (46), and (21), we find the following formulae establishing relationships between the sensitivity matrices (21) and (47) of the open-loop and closed-loop MIMO system transfer matrices

$$\begin{array}{l}{U}_r^S(s) \hspace{0.17em}= \hspace{0.17em}-w(s)S(s,{\alpha }_o)U_r^{R}S(s,{\alpha }_o) , \\ r = 1, 2, \mathrm{...} , m\end{array}$$

(48)

The CTFs $S_i(s)$ of the closed-loop MIMO system have the following form:

$$\begin{array}{l}{S}_i(s,\alpha ) = {\displaystyle \frac{1}{1 + q_i(s,\alpha )}} = {\displaystyle \frac{1}{1 + {\lambda }_i(\alpha )w(s)}} \\ i = 1, 2, \mathrm{...} , N.\end{array}$$

Expand these functions into the Taylor series:

$$\begin{array}{l}{S}_i(s,\alpha ) \hspace{0.17em}= \hspace{0.17em}{S}_i(s,{\alpha }_o) + {\displaystyle \sum _{r = 1}^m{S}_{ir}^S}(s)\Delta {\alpha }_r \hspace{0.17em}+ \hspace{0.17em}\mathrm{....}\hspace{0.17em} \\ i = 1, 2, \mathrm{...} , N,\end{array}$$

(49)

where

$$\begin{array}{l}{S}_{ir}^S(s) \hspace{0.17em}= \hspace{0.17em}{\left.{\displaystyle \frac{\delta S_i(s,\alpha )}{\delta {\alpha }_r}}\right|}_{\alpha = {\alpha }_o}=-{\left. {\displaystyle \frac{1}{{\left[1 + {\lambda }_i({\alpha }_0)w(s)\right]}^2}}{\displaystyle \frac{𝜕q_i(s,\alpha )}{𝜕{\alpha }_r}}\right|}_{\alpha = {\alpha }_o}=\\ =-{\left. {\displaystyle \frac{1}{{\left[1 + {\lambda }_i({\alpha }_0)w(s)\right]}^2}}w(s){\displaystyle \frac{𝜕{\lambda }_i(\alpha )}{𝜕{\alpha }_r}}\right|}_{\alpha = {\alpha }_o}= {\displaystyle \frac{1}{{\left[1 + {\lambda }_i({\alpha }_0)w(s)\right]}^2}}w(s){\beta }_{ir}\\ r = 1, 2, \mathrm{...} , m\end{array}$$

(50)

are the first-order sensitivity functions of the CTFs $S_i(s,\alpha )$ with respect to the variations of the $r$th parameter ${\alpha }_r$.

Since the canonical bases of the open-loop and closed-loop MIMO systems coincide, the expansion (22) is also valid for the axes $c_i(s,\alpha )$ of the closed-loop system, where the designations ${\mathrm{\Upsilon }}_{ir}^S(s)$ should be introduced instead of ${\mathrm{\Upsilon }}_{ir}(s)$. Proceeding as in analyzing the open-loop uniform system it is easy to show that for evaluating $S_{ir}^S(s)$ and ${\mathrm{\Upsilon }}_{ir}^S(s)$ it is necessary to find the representation of the matrices $U_r^S(s)$ (47) in the canonical basis of the unperturbed uniform system. That representation has the form

$$\begin{array}{l}{M}_r(s) \hspace{0.17em}= \hspace{0.17em}{C}^{-1}({\alpha }_o)U_r^S(s)C({\alpha }_o) \\ r = 1, 2, \mathrm{...} , m.\end{array}$$

(51)

The diagonal elements $m_{ii}^r(s)$ of the matrix $M_r(s)$ are equal to the sensitivity functions $S_{ir}^S(s)$ (50), and non-diagonal elements $m_{ki}^r(s)$, divided by the differences $\Delta = [S_i(s,{\alpha }_o) \hspace{0.17em}-\hspace{0.17em} S_k(s,{\alpha }_o)]$ $(i,k =$$1, 2, \mathrm{...} ,N,$ $k \ne i)$, yield the sensitivity vectors ${\mathrm{\Upsilon }}_{ir}^S(s)$ of the $i$th canonical basis axis of the closed-loop uniform system.

Expressions (48) and (51) enable us to determine the relationship between sensitivity functions of the open-loop and closed-loop uniform system, i.e., to determine the effect caused on the system sensitivity by the introduction of feedback loop. Substituting (48) into (51) and using the canonical representation of the transfer matrix $S(s,{\alpha }_o)$(7) we get after simple transformations:

$$\begin{array}{l}{M}_r(s) \hspace{0.17em}= -w(s)diag\left\{{\displaystyle \frac{1}{1 + {\lambda }_i({\alpha }_0)w(s)}}\right\}{N}_r diag\left\{{\displaystyle \frac{1}{1 + {\lambda }_i({\alpha }_0)w(s)}}\right\}\\ r = 1, 2, \mathrm{...} , m.\end{array}$$

(52)

From here, finding the diagonal elements on the right side, we get the same expression (50):

$$\begin{array}{l}{S}_{ir}^S(s) \hspace{0.17em}=-{\displaystyle \frac{1}{{[1 + {\lambda }_i({\alpha }_0)w(s)]}^2}}w(s){\beta }_{ir} = -{\displaystyle \frac{1}{{[1 + {\lambda }_i({\alpha }_0)w(s)]}^2}}\hspace{0.17em}{S}_{ir}^q(s)\\ i = 1, 2, \mathrm{...} ,N; r = 1, 2, \mathrm{...} , m.\end{array}$$

(53)

Expressions (53) relate the sensitivity functions of the closed-loop and open-loop SISO characteristic systems and are completely analogous to those well-known in the classical control theory [9]. If we let $s = j\omega$, then from (53) it is clear that for those frequencies $\omega$ for which $|q_i(j\omega ,{\alpha }_o)| \hspace{0.17em}>> \hspace{0.17em}1$ (usually it is the low-frequency region), the introduction of feedback decreases the sensitivity of the CTFs to parameters variations. In the high-frequency region, where $|q_i(j\omega ,{\alpha }_o)| \hspace{0.17em}\approx \hspace{0.17em}0$, the sensitivity of the closed- and open-loop CTFs is approximately the same. Finally, in the region of the resonance frequencies, where $|S_i(j\omega ,{\alpha }_o)| \hspace{0.17em}>\hspace{0.17em} 1$, the feedback deteriorates the sensitivity of the CTFs.

The non-diagonal elements $m_{ki}^r(s)$ of the matrix $M_r(s)$ (52) have the form

$$\begin{array}{l}{m}_{iki}^r(s) \hspace{0.17em}= -\hspace{0.17em}w(s){\displaystyle \frac{n_{ik}^r}{[1 + q_i(s,{\alpha }_o)]\hspace{0.17em}[1 + q_k(s,{\alpha }_o)]}} \\ \hspace{0.17em}i,k\hspace{0.17em} = 1, 2, \mathrm{...} , N; k \ne i.\end{array}$$

(54)

Dividing both sides of (54) by the differences $[S_i(s,{\alpha }_o) \hspace{0.17em}-\hspace{0.17em} S_k(s,{\alpha }_o)]$, we obtain the following equalities:

$$\begin{array}{l}{\displaystyle \frac{m_{ki}^r(s)}{S_i(s,{\alpha }_o) \hspace{0.17em}-\hspace{0.17em} S_k(s,{\alpha }_o)}} \hspace{0.17em}= \hspace{0.17em}{\displaystyle \frac{n_{ki}^r}{{\lambda }_i({\alpha }_o) \hspace{0.17em}-\hspace{0.17em} {\lambda }_k({\alpha }_o)}}\hspace{0.17em},\\ i,k \hspace{0.17em}= 1, 2, \mathrm{...} , N; k \ne i\end{array}$$

(55)

But according to the stated above, the left-hand terms in (55) are equal (for some $i$) to the coordinates of the sensitivity vectors ${\mathrm{\Upsilon }}_{ir}^S(s)$ of the $i$th canonical axis of the closed-loop uniform system, and the right-hand terms are equal to the coordinates of the sensitivity vectors ${\mathrm{\Upsilon }}_{ir}(s)$ of the corresponding (in fact, the same) axis of the open-loop system. Besides, all these coordinates are determined with respect to the same canonical basis of the unperturbed uniform system.

Consequently, the vectors ${\mathrm{\Upsilon }}_{ir}^S(s)$ and ${\mathrm{\Upsilon }}_{ir}(s)$ are equal, i.e. the introduction of feedback does not affect the sensitivity of canonical basis axes of the uniform system to small variations of parameters.

It should be noted that the results of the last sections have more general character and can be used in cases when both the parameters of the transfer function w(s) and of the matrix R are perturbed simultaneously.

4. Example

Let us consider the sensitivity of control systems of multirotor UAVs. Two typical examples of multirotor UAVs – a quadcopter and an octocopter - are shown in Figure 4, where the number of rotors (motors) is denoted NR [19,20]. Irrespective of the number $NR$, the flight altitude $z$ and the vector of rotation angles (roll $\varphi$, pitch $\theta$, and yaw $\psi$) are usually chosen as four control variables in the underactuated control systems of multirotor UAVs [19,20].

Below, we shall assume that the angles and angular velocities of the UAV are so small that nonlinear terms in the dynamics equations of rotational motions can be neglected, cosines of all rotation angles are approximately equal to one, and sines are zero. On these conditions, the dynamics equations of multirotor UAVs take on the following linear form [19,20]:

$${\displaystyle \frac{d^{2}z(t)}{d{t}^2}}={\displaystyle \frac{1}{m}}{T}_{\mathrm{\Sigma }}(t)-g,$$

(56)

$$J{\displaystyle \frac{d\omega (t)}{dt}}=\tau (t),$$

(57)

where m is the mass of the UAV; g – the gravitational constant; J – diagonal tensor of inertia with the components $I_x,$ $I_y,$ $I_z$ on the principal diagonal; $\omega (t)$ - vector of angular velocities in the body-fixed frame; vector $\tau (t)={[{\tau }_x,{\tau }_y,{\tau }_z]}^T$ combines the principal non-conservative forces and moments applied to the UAV airframe by the aerodynamics of the $NR$ rotors (assuming no external disturbances);

$$T_{\mathrm{\Sigma }}={\displaystyle {\sum }_{i=1}^{NR}{T}_i}$$

(58)

- the total thrust at hover, where $T_i$ is the thrust generated by the $i$th rotor.

Denoting by $\overline{T}$ the $NR$-dimensional vector of thrusts $T_i$ ($\overline{T}={[T_1,T_2,\mathrm{...},T_{NR}]}^T$), the mapping of $\overline{T}$ to the vector ${\left[T_{\mathrm{\Sigma }},\tau \right]}^T$ can be written in a matrix form

$$\left[\begin{array}{c}{T}_{\mathrm{\Sigma }}\\ \tau \end{array}\right]=D_M{\mathrm{\Lambda }}_M\overline{T}, {\mathrm{\Lambda }}_M=diag\left\{{\lambda }_i^M\right\},$$

(59)

where the $4\times NR$ full-rank numerical matrix $D_M$ (often called a control allocation matrix [19]) depends on the UAV’s geometry, number of rotors, etc. [19,20], and ${\lambda }_i^M$ $(0<{\lambda }_i^M\le 1)$ are the motors’ efficiency degradation parameters. For properly functioning motors the matrix ${\mathrm{\Lambda }}_M$ is equal to the identity matrix $I$ (or $I_{NR\times NR}$, to indicate the order $NR$ of the matrix $I$). Actually, $D_M$ is a matrix of kinematic cross-connections between separate channels of the control system.

The matrix block diagram of the UAV’s control system is shown in Figure 5 [1,20].

Commonly, the matrix regulator $K_{\mathrm{Reg}}(s)$ in such systems is taken in the form

$$K_{\mathrm{Reg}}(s)=K_{R}diag\{w_i^R(s)\},$$

(60)

where $K_R$ is a constant matrix, and $w_i^R(s)$ are scalar transfer functions of the regulators in separate channels. In practice, the standard Proportional-Integral-Derivative (PID) regulators [9] are usually used as $w_i^R(s)$ in (60). In what follows, we shall assume for simplicity that all regulators $w_i^R(s)$ are identical, that is $w_i^R(s)=w^R(s)$.

It is appropriate to transform the matrix block diagram in Figure 5 to equivalent four-dimensional form in Figure 6, where the vectors $\zeta (s)$, ${\rho }_{\mathrm{Out}}(s)$ of size $4\times 1$ and the $4\times 4$ diagonal matrix $M_{\mathrm{\Sigma }}$ are given by the following expressions:

$$\zeta (s)=\left[\begin{array}{c}{z}_{\mathrm{Ref}}(s)\\ {\eta }_{\mathrm{Ref}}(s)\end{array}\right], {\rho }_{\mathrm{Out}}=\left[\begin{array}{c}z(s)\\ \eta (s)\end{array}\right], M_{\sum }=\left[\begin{array}{cccc}m& 0& 0& 0\\ 0& I_x& 0& 0\\ 0& 0& I_y& 0\\ 0& 0& 0& I_z\end{array}\right],$$

(61)

where the components of the three-dimensional vector $\eta$ in the four-dimensional vector ${\rho }_{\mathrm{Out}}(s)$ are the roll $\varphi$, pitch $\theta$, and yaw $\psi$ angles, i.e. $\eta ={[\varphi ,\theta ,\psi ]}^T$.

Figure 6. Transformed block diagram of the UAV’s control system.

Figure 6. Transformed block diagram of the UAV’s control system.

Let us denote $R$ the following numerical matrix:

$$R=M_{\mathrm{\Sigma }}^{-1}{D}_M{\mathrm{\Lambda }}_M{K}_R.$$

(62)

The transfer matrix $W(s)$of the open-loop control system in Figure 6 in case of ${\mathrm{\Lambda }}_M\ne I$ has the form:

$$W(s) =w(s) R,$$

(63)

where

$$w(s)={\displaystyle \frac{w^R(s)}{s^2}}.$$

(64)

Consequently, the control systems of multirotor UAVs belong to the class of uniform systems.

The matrix $K_R$ in (62) is usually chosen as

$$K_R=D_M^{+}{M}_{\mathrm{\Sigma }},$$

(65)

where $D_M^{+}$ denotes the Moore-Penrose pseudoinverse of $D_M$ ($D_M^{+}=D_M^{-1}$ for $NR=4$) [19,20].

In case of ideal motors (if ${\mathrm{\Lambda }}_M=I_{NR\times NR}$), from (62) and (65) we have $R=I_{4\times 4}$ for any $NR$, i.e. kinematic cross-connections between four separate channels of the control system in Figure 6 are compensated and the control system splits into four isolated identical SISO systems (Figure 7). For that reason, the regulator $K_{\mathrm{Reg}}(s)$ (60), which incorporates a matrix part $K_R=D_M^{+}M$ (65), is usually called decoupling regulator [2,3,4].

On the other hand, if there are some motors’ efficiency degradations, i.e. if ${\mathrm{\Lambda }}_M\ne I_{NR\times NR}$, then the matrix $R$ (62) is not an identity matrix and the control system of the UAV belongs to the class of uniform systems.

As a typical example, consider the control system of a quadcopter with the so-called I4 scheme of the motors’ allocation and the following parameters: $m=2.5 kg$, $I_x=$ $I_y=0.5 kg\cdot m^2$, $I_z=1.5 kg\cdot m^2$. Matrices $D_M$ and $M_{\mathrm{\Sigma }}$ in this case are equal to

$$D_M= \left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 0.1& 0& -0.1\\ -0.1& 0& 0.1& 0\\ -0.3& 0.3& -0.3& 0.3\end{array}\right] , M_{\sum }=\left[\begin{array}{cccc}2.5& 0& 0& 0\\ 0& 0.5& 0& 0\\ 0& 0& 0.5& 0\\ 0& 0& 0& 1.5\end{array}\right].$$

(66)

and the matrix $K_R$ (65) is

$$K_R=D_M^{+}{M}_{\mathrm{\Sigma }}=D_M^{-1}{M}_{\mathrm{\Sigma }}= \left[\begin{array}{cccc}0.625& 0& -2.5& -1.25\\ 0.625& 2.5& 0& 1.25\\ 0.625& 0& 2.5& -1.25\\ 0.625& -2.5& 0& 1.25\end{array}\right].$$

(67)

The identical proportional-derivative (PD) regulators with the first order filter $w^R(s)$ in the separate channels are taken in the form:

$$w^R(s)=0.00559+{\displaystyle \frac{0.639s}{0.00875s+1}}.$$

(68)

The frequency and step response characteristics of the separate channels in case of ideal motors (when $R=I$) are shown in Figure 8, Figure 9 and Figure 10. The black line in Figure 8 is the graph of the plant, the blue line represents the PD regulator (68), and the transfer function of the resulting open-loop channel is shown by the brown line.

Figure 8. Nyquist plots of the isolated channels (${\mathrm{\Lambda }}_M=I$).

Figure 8. Nyquist plots of the isolated channels (${\mathrm{\Lambda }}_M=I$).

Figure 9. Frequency-response characteristic of the complementary sensitivity function.

Figure 9. Frequency-response characteristic of the complementary sensitivity function.

Figure 10. Step response of the isolated channels in case ${\mathrm{\Lambda }}_M=I$.

Figure 10. Step response of the isolated channels in case ${\mathrm{\Lambda }}_M=I$.

Let us suppose now that the matrix ${\mathrm{\Lambda }}_M$ in (59) has the following form

$${\mathrm{\Lambda }}_M=\left[\begin{array}{cccc}0.85& 0& 0& 0\\ 0& 0.9& 0& 0\\ 0& 0& 0.75& 0\\ 0& 0& 0& 0.8\end{array}\right]$$

(69)

Suppose also that the matrix $K_R$ in $K_{\mathrm{Reg}}(s)$ (60) is an identity matrix $I$. Practically, this means that the cross-connections between separate channels are not compensated and are described by the matrix

$$R_0=M_{\mathrm{\Sigma }}^{-1}{D}_M= \left[\begin{array}{cccc}0.4& 0.4& 0.4& 0.4\\ 0& 0.2& 0& -0.2\\ -0.2& 0& 0.2& 0\\ -0.2& 0.2& -0.2& 0.2\end{array}\right]$$

(70)

The eigenvalues of this matrix are complex-conjugated and equal to

$$\begin{array}{l}{\lambda }_{1,2} = 0.1798 \pm j0.4150, \\ {\lambda }_{3,4} = 0.3202 \pm j\mathrm{0.1504.}\end{array}$$

(71)

The frequency response and step response characteristics of the quadcopter control system with cross-connections matrix $R_0$ (70) are shown in Figure 11, Figure 12 and Figure 13. In Figure 11, the four black lines are the graphs of the plant, the blue line represents the identical PD regulators (68) in separate channels, and the gain loci of the corrected system are shown by brown lines. The vinous lines in Figure 12 represent the majorant and minorant of the generalized frequency response characteristics and the black lines depict the complementary sensitivity functions of the closed-loop characteristic systems.

Let us analyze now the sensitivity of the closed-loop control system with the cross-connection matrix $R_0$ (70) to small degradations of motors’ parameters ${\lambda }_i^M$. For brevity, below we present the results only for the sensitivities ${\beta }_{i1}$ and ${\mathrm{\Upsilon }}_{i1}$ of eigenvalues ${\lambda }_i$ (71) and eigenvectors $c_i$ of the matrix $R_0$ (70) with respect to small degradations of the efficiency of the first motor (with respect to small negative perturbations of ${\lambda }_1^M$).

The calculations yield the following results:

$${\beta }_{11}=0.3238-j0.2012=0.3812\exp (-j{31.86}^{\circ }), {\beta }_{21}=0.3238-j0.2012=0.3812\exp (j{31.86}^{\circ }),$$

$${\beta }_{31}=0.1762-j0.2012=0.1767\exp (-j{3.98}^{\circ }), {\beta }_{41}=0.1762-j0.2012=0.1767\exp (j{3.98}^{\circ })$$

$${\mathrm{\Upsilon }}_{11}=\left[\begin{array}{c}0\\ -0.07-j0.37\\ 0.02+j0.18\\ -0.14+j0.11\end{array}\right], {\mathrm{\Upsilon }}_{21}=\left[\begin{array}{c}-0.07+j0.37\\ 0\\ -0.14-j0.11\\ 0.02-j0.18\end{array}\right] ,$$

$${\mathrm{\Upsilon }}_{31}=\left[\begin{array}{c}-0.18-j0.33\\ -0.37+j0.06\\ 0\\ 0.098+j0.15\end{array}\right] , {\mathrm{\Upsilon }}_{41}=\left[\begin{array}{c}-0.37-j0.06\\ -0.18+j0.33\\ 0.098-j0.15\\ 0\end{array}\right]$$

The sensitivity functions with respect to loss of efficiency of other motors are determined analogously.

Finally, we discuss the case when the matrix $K_R$ in $R$ (62) is taken as in (65) and there are motors’ efficiency degradations, i.e. ${\mathrm{\Lambda }}_M\ne I$. The cross-connections matrix $R$ in this case can be written in the form of similarity transformation

$$R=Cdiag\left\{{\lambda }_i^M\right\}{C}^{-1}$$

(72)

where

$$C=M_{\mathrm{\Sigma }}^{-1}{D}_M.$$

(73)

As can be seen from (72) and (73), the columns of the matrix $C$ constitute the canonical basis of the cross-connected control system in Figure 5 and Figure 6, and the parameters ${\lambda }_i^M$ are the “gains” ${\lambda }_i$ of the CTFs $q_i(s)$ in (5). Moreover, the form of the matrix $N_r$ (35) in which ${\alpha }_r={\lambda }_r^M$ shows that in this case we have

$$N_r = C^{-1}{U}_r^{R}C= C^{-1}{\displaystyle \frac{\delta }{\delta {\lambda }_r^M}}\left[Cdiag\left\{{\lambda }_i^M\right\}{C}^{-1}\right]C={\displaystyle \frac{\delta }{\delta {\lambda }_r^M}}\left[diag\left\{{\lambda }_i^M\right\}\right]={{\rm E}}_r$$

(74)

$$r = 1, 2, 3, 4,$$

where ${{\rm E}}_r$ is a $4\times 4$ matrix in which the $r$th entry on the principal diagonal is one (${\beta }_{rr}=1$) and all other entries are zero (${\beta }_{kr}=0, k\ne r$). Consequently, if ${\mathrm{\Lambda }}_M\ne I$ and $K_R=D_M^{-1}{M}_{\mathrm{\Sigma }}$, then the small degradation of the $r$th motor’s efficiency does not affect at a first approximation the canonical basis of the uniform system and the “gains” ${\lambda }_i$ of the CTFs for $i\ne r$. Besides, the sensitivity of the $r$th eigenvalue is equal to one, that is small degradations of ${\lambda }_r^M$ lead to the same degradations of eigenvalue ${\lambda }_r$.

Figure 14. Nyquist plots of the isolated channels in case ${\mathrm{\Lambda }}_M\ne I$ and $K_R=D_M^{+}{M}_{\mathrm{\Sigma }}$.

Figure 14. Nyquist plots of the isolated channels in case ${\mathrm{\Lambda }}_M\ne I$ and $K_R=D_M^{+}{M}_{\mathrm{\Sigma }}$.

Figure 15. Generalized frequency-response characteristics of the complementary sensitivity function matrix.

Figure 15. Generalized frequency-response characteristics of the complementary sensitivity function matrix.

Figure 16. Step responses of the control system in case ${\mathrm{\Lambda }}_M\ne I$ and $K_R=D_M^{+}{M}_{\mathrm{\Sigma }}$.

Figure 16. Step responses of the control system in case ${\mathrm{\Lambda }}_M\ne I$ and $K_R=D_M^{+}{M}_{\mathrm{\Sigma }}$.

The frequency response and other characteristics of the quadcopter’s control system with cross-connections matrix $R_0$ (72 ), where ${\mathrm{\Lambda }}_M$ is given by (69), are shown in Figure 14, Figure 15 and Figure 16.

5. Discussion

In the paper, the issue of sensitivity of uniform control systems is discussed from the position of the CTFs method. In a sense, the results of the paper can be regarded as an extension to uniform systems of the results presented in [1] and concerning the sensitivity of the CTFs and canonical basis axes of general type MIMO systems. The point is that, as compared with general type MIMO systems, uniform systems always comprise structurally two different matrix blocks, specifically, a scalar transfer matrix of identical separate channels and a numerical matrix of rigid cross-connections. The parameter perturbations of these two blocks result in distinguishable changes of the CTFs and canonical basis axes of uniform systems. In the paper, these two situations are treated separately, and the formulas are derived determining the sensitivity of the CTFs and canonical basis axes when parameter perturbations are completely concentrated in the transfer matrix of separate channels or in the numerical matrix of cross-connections.

Practical application of the results is demonstrated by the sensitivity analysis of control system of a quadcopter to small degradations of the motors’ efficiency.

The numerical calculations and modeling of dynamics of the quadcopter control system presented in Figure 8 – Figure 16 are obtained by a software package MIMO Control Toolbox which is being developed at the Aerial Robotics Center of the National Polytechnic University of Armenia [20]. The toolbox works in the MATLAB environment and is designed for analysis and design of MIMO control systems in robotic , mechatronics, and in other fields. The central feature of the MIMO Control Toolbox is that the design of any N-dimensional MIMO control system is reduced to the design of a certain fictitious SISO control system. Another key feature of the toolbox is that it includes about 250 new MATLAB classes (objects) describing all the main structural types of MIMO control systems known from scientific and technical literature.

The frequency and time response characteristics of the quadcopter’s control system in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 were obtained with the help of a special interactive graphical user interface (GUI) which is a part of the package MIMO Control Toolbox. That GUI can be viewed as an extension to multivariable case of the well-known GUI Control Systems Designer in MATLAB [21]. The MIMO Control Toolbox can be used both as a computer-aided control systems design tool in various areas of industry and technics, and for teaching the fundamentals of classical and modern feedback control at educational institutions.