# MIMO Passive Control Systems Are Not Necessarily Robust

Authors: Jesús U. Liceaga-Castro; Irma I. Siller-Alcalá; Eduardo Liceaga-Castro; Luis A. Amézquita-Brooks
Journal: Journal of Control Science and Engineering
 (2015)

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## Abstract

Via several cases of study it is shown that a passive multivariable linear control system, contrary to its single input single output counterpart, may not be robust. Moreover, it is shown that lack of robustness can be exposed via the multivariable structure function.


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## Body

## 1. Introduction

The design of controllers for multivariable systems (MIMO) satisfying robustness conditions may be a long and tiresome procedure [1]. Many well-known methodologies had been proposed in order to solve this problem such as LQG, H∞, and QFT. More recently, dissipativity which is related to the conservation, dissipation, and transport of energy is an open loop property that has been used to develop a framework for the design and analysis of control systems. In particular, passive analysis focuses on systems which dissipate energy. This open loop characteristic presents great advantages in the design of robust control systems.For the case of a linear time invariant SISO system described by a rational transfer functionG(s)=N(s)/D(s) is passive if it satisfies the following [2]:(i)
G(s) is stable.(ii)
RG(jω)≥0∀ω∈-∞,∞.These conditions can be translated in terms of the classical control theory as follows: a rational SISO transfer functionG(s)=N(s)/D(s) is passive if(i)
G(s) is stable and minimum phase,(ii)
G(s) has a relative degree ≤ 1,(iii)
G(s) behaves like a lead phase filter,(iv)
the Nyquist plot ofG(jω) lies in the right half plane of the complex plane ∀ω∈-∞,∞.That is, passive SISO systems are robust because they have gain and phase margins ofMg→∞dB’s and Mp≥90°, respectively. Moreover, thanks to this characteristic the parallel, series, or feedback interconnection of two passive systems is also robust. In this context, in general, the strategy for passive control systems is divided in two steps: the passivation of the process and the design of a passive controller—normally a PID controller—based on the amended or passivized process. Also, in order to transform a nonpassive SISO system into a passive one, three methods are normally applied: pre- or postcompensation, feedback, and feed-forward [3, 4].One may cast the conjecture on the existence of these frequency domain properties for linear MIMO systems. If this conjecture proves to be true, the robust design and assessment of multivariable control could be obtained in a similar manner as in SISO systems.A square linear time invariant MIMO system can be represented byy(s)=G(s)u(s), where G(s)∈Cmxm is a rational transfer function matrix; u(t)∈Rm is the input; and y(t)∈Rm the output. Assuming all the poles of G(s) are in R[s]<0, then [2](1)Gsis  passive⟺λminGjω+G∗jω≥0∀ω∈-∞,∞,where G∗(jω) is the conjugated transpose of G(jω).Unfortunately, as it is shown in this paper, the above conjecture is wrong. Even when the open loop MIMO control system satisfies the passivity condition, this does not necessarily result in a robust control system.In the following sections, three cases of passivation of2×2 linear MIMO systems are presented. The results show that in spite of passivating the systems the resulting control systems are fragile, that is, nonrobust. In addition, the cause for the lack of robustness is exposed using the multivariable structure function (MSF) [5] and the well-known Singular Values Analysis (SVA). For the sake of transparency the cases are based on 2×2 linear MIMO systems. However, the results obtained from these cases apply also to the general case of M×M MIMO systems. That is, the MSF applies to the general case of M×M linear MIMO systems [6].


## 2. Cases of Study

Case 1.
Consider the simple2×2 linear time invariant MIMO system of the distillation column thoroughly analysed in [7](2)Gs=87.8-86.4108.2-109.675s+1.
The eigenvalues ofG(jω)+G∗(jω) are shown in Figure 1.
From Figure1 it is clear that one of the eigenvalues of G(jω)+G∗(jω) is negative and therefore the MIMO system of (2) is nonpassive. However, it is possible not only to passivize but also to decouple the system with the static precompensator Kp given by(3)Kp=0.3994-0.31490.3943-0.3200,resulting in the passivized system Gp(s)=G(s)Kp as shown in Figure 2 by the eigenvalues of Gp(jω)+Gp∗(jω):(4)Gps=100175s+1.
As mentioned above, the second step in the passive control strategy is to design a passive controller, normally a PID, based on the amended or passivized system. In this case, it is possible to design a simple decentralized PI controllerK(s) given by(5)Ks=s+0.01333002s+0.026670.01333s,
resulting in the passive open loop control systemGo(s)=Gp(s)K=G(s)KpK(s),(6)Gos=1002swith a stable closed loop matrix transfer function given by(7)Gcls=1s+1002s+2.
Nonetheless, if, for instance, a small parameter perturbation or uncertainty of +4% is introduced in the gain of the individual transfer functiong12(s) of the system in (2) the open loop control system becomes(8)Gos=-0.3628002swhich is nonpassive and closed loop unstable.Figure 1
Eigenvalues ofG(jω)+G∗(jω) (Case 1).Figure 2
Eigenvalues ofGp(jω)+Gp∗(jω) (Case 1).Case 2.
Let the well-known quadruple tank MIMO system described by the matrix transfer function [8](9)G2s=δ13.53363s+11-δ23.5332457s2+102s+11-δ14.6785096s2+147s+1δ24.67891s+1.
If the parametersδ1,δ2∈0,1, which depend on the positions of the system valves, are set to δ1=0.6 and δ2=0.41, this yields a system for which the eigenvalues of G2(jω)+G2∗(jω) show that the quadruple tank is nonpassive (Figure 3).
Following the strategy of Case1 the system can be decoupled by the precompensator:(10)Kp2s=1-170.3s-1.8719774s2+281.9s+1.918-131.3s-2.0845208s2+216.2s+2.1191,
resulting in the decoupled systemGp2(s)=G2(s)Kp2(s). The eigenvalues of Gp2(jω)+Gp2∗(jω) (Figure 4) show that the precompensated system Gp2(s) is also passive.
Therefore, it is possible to design the decentralized PI controllerK2(s) to the passivized system Gp2(s) to obtain the open loop system Go2(s)=Gp2(s)K2(s)=G2(s)Kp2(s)K2(s) with(11)K2s=0.35100s+1000.525100s+1s.
The eigenvalues ofGo2(jω)+Go2∗(jω) (Figure 5) confirm that also the open loop system Go2(s) is passive. Thus, the closed loop system is stable and passive.
However, despite the passivity of the system, it can be destabilized by a small perturbation. That is, a perturbation or uncertainty of +5% in the gain of the individual transfer functiong12(s) of the quadruple tank destabilizes the closed loop system introducing a pair of unstable poles at 0.00074,0.000073.Figure 3
Eigenvalues ofG2(jω)+G2∗(jω) (Case 2).Figure 4
Eigenvalues ofGp2(jω)+Gp2∗(jω) (Case 2).Figure 5
Eigenvalues ofGo2(jω)+Go2∗(jω) (Case 2).Case 3.
Let2×2 MIMO system described by the matrix transfer function G3(s)(12)G3s=8.6s+8.690s+0.90.05s+10.1215s2+3.893s+7.30.03031s3+0.5304s2+2.258s+1.
Similar to two previous cases, the eigenvalues ofG3(jω)+G3∗(jω) (Figure 6) show that G3(s) is nonpassive.
Continuing with the common multivariable control strategy of decoupling via precompensation, the systemG3(s) can be decoupled by the precompensator Kp3(s) given by(13)Kp3s=1-10.47s-0.1047s+1-0.4114s-8.227s2+32.03s+60.061.
Contrary to the two previous examples the precompensated and decoupled systemGp3(s)=G3(s)Kp3(s) remains nonpassive because the eigenvalues of Gp3(jω)+Gp3∗(jω) are not positive at all frequencies as it is shown in Figure 7.
Nonetheless, it is possible to both passivize and control the decoupled systemG3p(s) by the decentralized controller K3(s) comprising two PID + lead-compensator filters:(14)K3s=k1s00k2s,where(15)k1s=0.079s4+0.001461s3+0.007979s2+0.01311s+0.00474ss2+0.1358s+2.071,k2s=7.9s4+0.1462s3+0.7268s2+0.8493s+0.2607ss3+30.13s2+6.097s+62.12.
The systemGo3(s)=Gp3(s)K3(s)=G3(s)Kp3(s)K3(s) satisfies the passivity condition as the eigenvalues of Go3(jω)+Go3∗(jω) (Figure 8) are positive at all frequencies. Consequently, the closed loop system is passive and stable.
Similar to the two preceding cases, regardless of the passivity condition of the open loop control system, its closed loop configuration can be easily destabilized, for instance, a small perturbation in the gain of the individual transfer function (+5%)g12(s) of the original system. This perturbation introduces a pair of closed loop unstable poles at 0.0071±1.4345j.
The cases presented prove that for multivariable control systems passivity may not necessarily imply robustness. That is, a multivariable control system may remain highly sensitive and fragile to parametric perturbations despite having been passivized.
In the next section via the multivariable structure function (MSF) this situation can be easily clarified.Figure 6
Eigenvalues ofG3(jω)+G3∗(jω) (Case 3).Figure 7
Eigenvalues ofGp3(jω)+Gp3∗(jω) (Case 3).Figure 8
Eigenvalues ofGo3(jω)+Go3∗(jω) (Case 3).


## 3. Robustness Analysis

The MSF is a key element of the framework of analysis and design for linear MIMO control systems known asIndividual Channel Analysis and Design (ICAD) [9–11]. ICAD allows the application of the well proved SISO classical control theory in the analysis and design of linear MIMO control systems. An explanation of ICAD is out of the scope of this paper; however, a comprehensive description and applications of ICAD can be found in [12–14].Let2×2 MIMO system described by the matrix transfer function (16)Gs=g11sg12sg21sg22s.Then, the MSFγ(s) of G(s) is defined by(17)γs=g12sg21sg11sg22s.In [5], it was proved that 2×2 MIMO system is highly sensitive to parametric uncertainties or perturbations and nonmodeled dynamics if the MSF γ(jω) is close to the point (1,0) at some range of frequencies ω∈ω0,ω1. This condition is known within ICAD as structurally nonrobust. This indicates that the number of minimum and nonminimum phase zeros can be modified by uncertainties or perturbations. Therefore, via the MSF, it is possible to determine the possibilities of a MIMO control system to satisfy the requirements of design and the conditions the controller must satisfy to comply with these requirements ensuring appropriate robustness.Case 1 (robust analysis).
The systemG(s) of (2) has no transmission zeros and its MSF γ(s) is given by(18)γs=g12sg21sg11sg22s=0.9715.
Hence, the system of (2) is highly sensitive at all frequencies due to the closeness of its MSF to point (1,0). This condition can likewise be explained via the condition number κ(jω):(19)κGjω=σmaxGjωσminGjω,where σmax(G(jω)) and σmax(G(jω)) are the maximum and minimum Singular Values (SV) of G(s).
The condition numberκ(jω) for the system of (2) is κ(jω)=141.732. That is, the system is ill-conditioned at all frequencies.Case 2 (robust analysis).
The systemG2(s) of (9) is minimum phase with transmission zeros at -0.0431,-0.0004,-0.011,-0.159. On the other hand, the Nyquist plot of the MSF γ2(s) of G2(s) is shown in Figure 9.
From Figure9, it is clear that γ2(jω) is close to the point (1,0) at ω=0rad/sec. Hence, the system G2(s) is highly sensitive or ill-conditioned at ω=0rad/sec as shown in Figure 10 by its condition number κ2(jω).
It is due to this condition that when the small perturbation of +5% was introduced in the individual transfer functiong12s, G2(s) became non-minimum phase with a zero at +0.000076 and consequently the closed loop control system becomes unstable.Figure 9
Nyquist plot ofγ2(s) (Case 2).Figure 10
Condition numberκ2(jω) (Case 2).Case 3 (robust analysis).
SystemG3(s) of (12) is minimum phase with zeros at -28.58,-0.0.0677±1.437j. The Nyquist plot of the MSF γ3(s), depicted in Figure 11, indicates that G3(s) is highly sensitive at frequencies around ω=1.44rad/sec.
This situation is confirmed by the plot of the condition numberκ3(jω) (Figure 12) which clearly shows that the system is ill-conditioned in a region of frequencies around ω=1.44rad/sec. This explains why the closed loop system was destabilized by the small perturbation of +5% in the gain of the individual transfer function g12(s). That is, due to this perturbation, the system became non-minimum phase with a pair of zeros at 0.0076±1.44j and as a result, like the previous case, closed loop unstable.Figure 11
Nyquist plot ofγ3(s) (Case 3).Figure 12
Condition numberκ3(jω) (Case 3).


## 4. Conclusions

In this paper, it is shown that in the case of linear MIMO control systems passivity does not necessarily imply good robustness properties. This was exemplified by means of 3 cases where despite the passivation of the open loop control systems it was possible to destabilize their closed loops configurations by small parametric perturbations. That is, the control systems are highly sensitive regardless of its passive condition. This situation was exposed via the multivariable structure function and the condition number. In particular, it was shown that the lack of structural robustness, ill-conditioning, cannot be amended via passivation. The three cases of study represent the three possible conditions in which a system is not structurally robust or ill-posed: at all frequencies, at one particular frequency, and along a region of frequencies.It can be concluded that in order to exploit all the benefits of passivity in MIMO control systems it is necessary to consider the lack of structural robustness or ill-conditioning of the process before attempting any passivation strategy.Finally, it is possible to extend the application of the multivariable structure function to the passive analysis of the general case ofM×M MIMO systems.



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