Complex-Coefficient Frequency Domain Stability Analysis Method for a Class of Cross-Coupled Antisymmetrical Systems and Its Extension in MSR Systems

This paper develops a complex-coefficient frequency domain stability analysis method for a class of cross-coupled two-dimensional antisymmetrical systems, which can greatly simplify the stability analysis of the multiple-input multiple-output (MIMO) system. Through variable reconstruction, themultiple-inputmultiple-output (MIMO) system is converted into a single-input single-output (SISO) systemwith complex coefficients.The pole locations law of the closed-loop system after the variable reconstruction has been revealed, and the controllability as well as observability of the controlled plants before and after the variable reconstruction has been studied too, and then the classical Nyquist stability criterion is extended to the complex-coefficient frequency domain. Combined with the rigidmagnetically suspended rotor (MSR) systemwith heavy gyroscopic effects, corresponding stability criterion has been further developed. Compared with the existing methods, the developed criterion for the rigid MSR system not only accurately predicts the absolute stability of the different whirling modes, but also directly demonstrates their relative stability, which greatly simplifies the analysis, design, and debugging of the control system.

Commonly, the cross-coupled systems can be classified into two kinds: the cross-coupled symmetrical and antisymmetrical systems [13,14].As for the controlled plant with antisymmetrical cross-coupling characteristics, antisymmetrical cross-decoupling controller is often needed to realize the decoupling control of the system.This paper focuses on the stability analysis of the cross-coupled two-dimensional antisymmetrical system.For the sake of simplicity and clarity, the whole control system including the cross-coupled antisymmetrical controlled plant and its antisymmetrical crossdecoupling controller is called the cross-coupled antisymmetrical system in this paper.
Cross-coupled antisymmetrical system is a class of special multi-input multi-output (MIMO) systems.As for the stability analysis of the MIMO system, the most traditional method resorts to resolve the characteristic roots [15].However, as for the high-order cross-coupled antisymmetrical system, it is very tedious and even impossible to obtain its analytical expressions.Although this issue may be alleviated by employing the Routh-Hurwitz criterion [16][17][18], this method cannot present the whole information about the system stability.Additionally, rebuilding a proper Lyapunov function is not easy since there is no rule to follow [19], especially for a highorder MIMO system.Therefore, when employed to judge the stability of the cross-coupled antisymmetrical system, these traditional methods are relatively complex.Moreover, the resolved relative stability information is abstract since they lack specific physical meanings.As a result, these methods have some difficulty in combining the stability analysis with the controller design.
Accordingly, it is quite interesting and significative to judge directly and simply the stability of the cross-coupled antisymmetrical system, which is also a basic theoretical issue to be urgently addressed in engineering applications.An ideal stability criterion must be combined tightly with the design process of the controller, which can effectively guide the system design and further reveal the influence of the variation of the system parameters on the system stability [20].Obviously, the above methods can not satisfy this requirement, and it will be more difficult when considering simultaneously the variations of the multiple system parameters.
The Nyquist stability criterion of the single-variable system has many advantages [21][22][23][24].It not only can judge the closed-loop stability, but also can directly reveal the relationships between the relative stability of the system and its gain.Although this method has been extended to the multivariable case [25], such as characteristic locus method [26][27][28][29][30][31] and the inverse Nyquist array (INA) method [32,33], these extended methods are quite complex since they depend on some special computer-aided-design (CAD) softwares.Especially when diagonal dominance does not hold, the stability of the MIMO system cannot be judged by the INA method.
Among all the cross-coupled antisymmetrical systems, the MSR system is a class of typical cases in which stability analysis is quite important since the MSR system is an openloop unstable system with high rotor speed [34][35][36].Especially for the MSR system with significant gyroscopic effects, its whirling modes stability analysis is more challenging.However, the existing stability analysis methods [37][38][39][40] can only judge the stability of the whole whirling modes and cannot present the stability information of the different whirling modes, which is the key concern of the controller design and debugging.To solve this issue, the whirling modes stability criterion [41] based on complex coefficient frequency characteristics has been proposed, which can judge the stability of the different whirling modes.This method is of great value to the development of stability analysis of control theory.However, this method is only suited to the special MSR system, and it does not suit a class of cross-coupled antisymmetrical systems.How to extend the method to a class of cross-coupled systems is a great interest in engineering.
To simplify the absolute stability judgment of the crosscoupled antisymmetrical system and reveal directly its relative stability, a complex-coefficient frequency domain stability analysis method for a class of cross-coupled antisymmetrical systems is presented in this paper.

Complex-Coefficient Frequency Domain
Stability Analysis Method  of the cross-coupled antisymmetrical controlled plant can be written as where A  , A  ∈  × , B ∈  ×1 , and C ∈  1× .According to Figure 1, we assume that the state vector of the controlled plant P() is [ ]  , where ,  ∈  ×1 .The variable reconstruction is performed as follows: Then, after the variable reconstruction, the original MIMO system with real coefficients is converted into a SISO control system with complex coefficients, which is shown in Figure 2.
Based on the information of the original controlled plant, the system realization of the SISO controlled plant   () can be written as And the open-loop transfer function of the SISO system is given by   () =  ()   () (  () +   ()) . (3)

Pole Locations Law before and after Variable Reconstruction
Theorem 1. Suppose that the linear differential equations have the form where  is the differential operator, A ∈  2×2 , and dim() = dim().If all the solutions [ ]  for the above satisfy the following differential equations: where Ã ∈  × , then the eigenvalues of the A and Ã satisfy the following relationships: (2) If  +  ∉   ( Ã) and  −  ∉   ( Ã), then  ±  ∉   (A).
Define the following function space: From the proposition, it can be obtained that   ⊂   .According to the property of the general solution for the ordinary differential equations, one can obtain that dim(  ) =  = dim(  ).Accordingly,   =   .

Theorem 2. The systems before and after variable reconstruction satisfy the pole locations law presented in Theorem 1.
Proof.Note that system matrix determines the pole locations; therefore, it should be obtained at first.Since  1 () =  2 (), their system realization is given by And the state variables of  1 () and  2 () are x 1 , x 2 , respectively.
Similarly the system realization of  1 () and  2 () is given by And the state variables of  1 () and  2 () are, respectively, x 1 and x 2 .Also, the system realization of  1 () and  2 () can be described as with the state variables of x 1 and x 2 , respectively.
Then the system matrix of the whole system before variable reconstruction is given by Define the new state variables χ = according to (12) and Theorem 1, the corresponding system matrix Ã of χ can be derived as As for the SISO system with complex coefficients, the system realization of   () +   () can be derived from the system realizations of   () and   () as follows: Similarly, the complex-coefficient system matrix of the SISO system after variable reconstruction can be derived as From ( 13) and ( 15), it can be drawn that A  = Â.Accordingly, the original system and the new system are in line with the same pole locations laws with that of Theorem 1.This concludes the proof.

Controllability and Observability before and after Variable Reconstruction
Proof.When  = 1, the conclusion holds obviously.Suppose that the conclusion also holds when  = ; then when Similarly, This concludes the proof.
Theorem 4. If the SISO system (2) is controllable and observable, that is, rank ) = , where Ã = A  +A  , then the MIMO system (1) is controllable and observable too.
Similarly, it can be proven that if the SISO system (2) is observable, then the MIMO system (1) is observable too.Proof of the proposition is completed.

Complex-Coefficient Frequency Domain Stability Criterion.
According to Theorems 1-4, the following corollary can be achieved.

Corollary 5.
As for the complex-coefficient system (15), the sufficient and necessary condition for the system stability is that all the characteristic roots lie in the left half plane (LHP).And the sufficient and necessary condition for the system critical stability is that there is at least a pure imaginary root and there is no right-half plane (RHP) pole.
From the analysis above, it can be drawn that the stability of the cross-coupled antisymmetrical system can be demonstrated by that of the corresponding complex-coefficient system.Note that the argument principle applies to the complex variable function which is defined on the complex plane, and the complex-coefficient open-loop transfer function   also belongs to this class of complex variable functions.Since the Nyquist stability criterion is based on the argument principle and the system controllability and observability are the precondition of Nyquist stability criterion, the corresponding SISO system can be employed to analyze the stability of the original control system by using complex-coefficient Nyquist stability criterion.Accordingly, one can achieve the following stability analysis method based on complex-coefficient frequency characteristics.Denote enc  (  (), −1 + 0) as the number of anticlockwise encirclements of the −1 + 0 point by the   () locus as a representative point  traces on the path  in the clockwise direction, where  is the semicircle contour that encloses the entire right-half  plane.Then one can obtain the following stability criterion for twodimensional cross-coupled antisymmetrical systems.Criterion 1.As for the cross-coupled two-dimensional antisymmetrical system (12), if its corresponding SISO controlled plant ( 2) is both controllable and observable, and its number of the RHP poles is   , then the sufficient and necessary condition for the stability of system ( 12) is that enc  (  (), −1 + 0) =   .
Compared with the traditional Nyquist stability criterion for the multi variable systems, the proposed one greatly simplified the stability analysis since the complex multivariable system has become a single-variable system.
In fact, as for a special controlled plant, this criterion can be further developed according to its practical physical meaning, for example, the MSR system with heavy gyroscopic effects; it can be proven that the closed-loop righthalf-poles and left-half-poles of the complex-coefficient SISO system denote the forward and backward whirling modes of the MSR respectively, which is the theoretical base for the whirling modes stability criterion [41].

Simulation and Experimental Results
Note that the large magnetically suspended control moment gyroscope (MSCMG) has a rigid MSR with significant crosscoupling effects which is a typical coupled-coupled antisymmetrical system; thus, it is taken as the example.To demonstrate the effectiveness and superiority of the proposed stability criterion, comparative simulation between the proposed method and the traditional one (root locus with respect to rotor speed) and experiments have been performed.

Experiment Setup.
Figure 3 shows the experimental setup, where the MSR has two 2-degree-of-freedom (DOF) radial MBs and two single-DOF axial MBs, which are driven by five H-bridge unipolar switching power amplifiers independently [42], realizing 5-DOF active control.
The well-known decentralized PID plus filtered cross feedback control approach [34] is adopted.The main parameters of the MSCMG and the controller coefficients used in simulation and experiments have been listed in Tables 1 and 2.
The experiments are performed in a TMS320C32 digital signal processor (DSP) board.The total digital control system delay can be tested, which is about 150 s.To facilitate drawing the root locus diagrams [43] and compare fairly, the delay unit is simplified as its second-order Taylor's series approximation when drawing the root locus and the complexcoefficient frequency characteristics plots; that is, where  = 0.00017.In fact, this approximate has little influence on the stability analysis for the MSR system since the interested critical stability frequencies of the whirling modes are relatively low.That is, the phase errors at the critical stability frequencies are tolerant.The closed-loop rotation dynamic model with the antisymmetrical cross-decoupling controller can be given by [41,44] where   and   are the momenta of inertia of the rotor about the radial and axial directions, respectively; Ω is the rotor speed;   and   are the distances from the geometry center of the MSR to the radial and the radial sensor;  and  are the rotor angular displacements about the and axes;   ,   ,   ,   , and   denote the basic controller, the antisymmetrical cross-decoupling controller, the switching power amplifier, the antialias filter, and the control system delay unit, respectively;  ℎ is the force-displacement factor of the radial magnetic bearings, and   is the proportional gain of the displacement sensor.The open-loop transfer function of the SISO system with complex coefficients can be derived as [44]   () = 2          ()   ()   (  () +   ()) According to the definition used by logarithmic frequency stability criterion [45], we make the following definitions:  + ,  − denote the number of the positive and negative phase cross-over points of   () in the set of   = { |  ≥ 0, log |  ()| ≥ 0}, when  changes from 0 to +∞;  + ,  − denote the number of the positive and negative phase cross-over points of  cmp () in   = { |  ≥ 0, log |  ()| ≥ 0}, when  decreases from +∞ to 0;  + ,  − are the number of the positive and negative phase crossover points of  cmp () in   , when  increases from 0 to +∞; and  + ,  − denote the number of the positive and negative phase cross-over points of   () in   = { |  ≤ 0, log |  ()| ≥ 0} when  changes from −∞ to 0. According to the proposed Criterion 1 and the whirling modes stability criterion for a MSR with bending modes [41], one can further obtain the following whirling modes stability criterion for the rigid MSR without bending modes.Criterion 2. As for the rigid MSR system (19), the sufficient and necessary conditions for the nutation and precession modes stability are   =  + +  + −  − −  − and   =  + +  + −  − −  − , respectively.

Simulation and Experimental Results
. At first, under the conditions that the cross proportion is 48 Hz, the Bode diagram is drawn, respectively, at the rotation speed of 140 Hz and 145 Hz, which is shown in Figure 4. Substituting the system parameters and rotor speed into (20), it can be calculated that   = 0 and   = 0. From Figure 4 Therefore, the backward whirling mode is stable, while the forward whirling mode is unstable at the speed of 145 Hz.The root locus with respect to the rotor speed is drawn, which is shown in Figure 5.The speed range is from 0 Hz to 150 Hz and it is plotted every 5 Hz.According to Figure 5, it can be derived that the system is stable at the speed of 140 Hz while unstable at the speed of 145 Hz; that is, the margin speed is located between 140 Hz and 145 Hz, which is in accordance with the conclusion drawn its radial displacement responses in -direction (including channels   and   ) and the corresponding frequency spectrum characteristics before and after the whirling mode loses its stability.Besides, the Lissajous figure of terminal  of the MSR, which is generated by ℎ  and ℎ  , is also tracked as shown in Figure 8(c).From Figure 8(a), it can be seen that the instability is caused by rotational mode instead of translational mode since the motion directions of channels   and   are opposite, which can also be testified from Figure 8(c).From Figure 8(b), the critical rotor speed is about 52 Hz, and its corresponding whirling frequency is about 70 Hz, which agrees well with the simulation results.Additionally, from Figure 8(c), it can be obtained that the instability mode is backward whirl instead of forward whirl since the whirling direction is opposite to the rotation.

Conclusion
To analyze directly and simply the stability of the crosscoupled two-dimensional antisymmetrical system, this paper presents a stability analysis method based on complexcoefficient frequency characteristics.Through variable reconstruction, the MIMO system is converted into a SISO system with complex coefficients.The closed-loop pole locations law and the controllability as well as observability before and after the variable reconstruction are studied and revealed, Pole map  which demonstrate that the two systems before and after variable reconstruction possess the same absolute and relative stability.Then the Nyquist stability criterion is extended from a real-coefficient system to a complex-coefficient system according to argument principle.
Combined with the rigid MSR system with significant gyroscopic effects, simulation and experimental results demonstrate that the proposed stability analysis method is feasible, correct, and quite simple to implement.Compared with the existing methods, the presented one can greatly

Figure 3 :
Figure 3: Photograph of the experimental setup.

Figure 7 :
Figure 7: Root locus with respect to rotor speed from 0 to 220 Hz.Note: the arrowhead denotes the increased direction of rotor speed from 0 to 150 Hz.

Figure 8 :
Figure 8: Experimental results.(a) Radial displacement responses of channels   and   .(b) Frequency spectrum of displacement ℎ  .(c) Lissajous figure of terminal  of the MSR.Note: the arrowhead denotes the direction of the rotation and the circle drawn by dotted line means the auxiliary bearing.

Table 1 :
System parameters of the MSCMG.

Table 2 :
Parameters of the controller.