This paper takes a novel approach to the design of planing craft with active control systems (ACS) by codesigning the longitudinal center of gravity (lcg) and ACS, and compares its performance with a vessel where the lcg and ACS are designed sequentially (traditional approach). The vessels investigated are prismatic in shape. The ACS are modeled as forces on the vessel. The ACS controller is a linear quadratic regulator (LQR) designed using a reduced-order model of the vessel. In the design, only the calm-water drag is optimized. The simulated codesigned vessel had 10% lower calm water and mean seaway drag than the sequentially designed vessel. However, the codesigned vessel's seakeeping was poorer—vertical acceleration doses 25% higher. Results indicate that the traditional sequential design approach does not fully exploit the synergy between a planing craft and its ACS; as a first step, the stability constraints should be relaxed in the design exploration, and the ACS should be considered early in the design stage.

Introduction

Planing craft are used anywhere that speed and simplicity of design are desired, since they represent the simplest crafts in which “high-speeds” are achievable. Because of this, their use is widespread in the boating industry—both in civilian and military applications. Planing craft distinguish themselves from traditional displacement vessels once they get up to design speeds, where a significant part of the planing craft's lift force arise from dynamic lift and not from water displacement—sharing many similarities to an airfoil. The exact speed for which a planing vessel is considered to be planing per se is not clear-cut [1]; but a planing boat operating at a speed coefficient of Cv > 1.5 is generally considered to be in the planing regime [2].

The performance of a vessel is usually measured with three criteria: transport efficiency, seakeeping, and maneuverability. In this research, only transport efficiency and seakeeping are investigated. An improvement in transport efficiency allows the vessel to travel farther and/or faster with less fuel, providing benefits ranging from making boats more environmentally friendly to reducing operational costs and/or acquisition costs. Improvements to vessel seakeeping can make planing boats safer by reducing the risk of operator injury, increase the range of sea conditions the vessel can safely operate, and potentially improve the transport efficiency when operating in a seaway.

Mature research exists exploring the transport efficiency and seakeeping of planing vessels by vessel geometry design (eg., Refs. [24]), most of it conducted in the 1960s and 1970s when the U.S. Navy had an aggressive high-speed vessel research program. From early research, it was known that high-speed vessels could suffer from instability problems such as porpoising and chine-walking (dynamic pitch-heave oscillation and roll oscillation accordingly); for an overview of planing craft dynamic stability, the reader is referred to Ref. [5]. Traditionally, the approach to correcting the controllability problems in planing craft is to modify the vessel hull geometry, change the running trim angle, and/or restrict the operating speeds. While this approach might seem to produce satisfactory vessels, the reality is that some traditional planing craft are designed such that they operate at suboptimal running attitudes (optimal in terms of lift-to-drag ratio)—since the optimal is sometimes unstable [6]. Moreover, there are cases where vessel geometry design optimization alone is not sufficient; in a survey of active-duty personnel operating modern special warfare planing vessels, 65% of the participants reported sustaining at least one injury during service [7]—with the harsh craft's motions being the primary suspect.

Looking for ways to improve upon the performance of planing boats, people have explored implementing active control systems (ACS). ACS are seamlessly embedded to many technologies around us in order to automate or facilitate a process. Uses for ACS in vessels include maintaining a ship inside a sailing path, holding a fixed position in open seas, improving seakeeping, and providing stability during operation (currently, stability preservation is done mainly on hydrofoil vessels). Past research has shown that incorporating ACS to planing craft can be used to address instability problems [8] and improve seakeeping [9]. Additional research on the use of ACS on planing craft can be found in Refs. [915]. However, existing published research has only explored the sequential design of a planing craft and ACS—i.e., the vessel geometry is first selected, and then the ACS is incorporated. And because the ACS and planing craft geometry are coupled systems (both the geometry and ACS affect the vessel dynamics), sequential design might not be optimal [16].

Successes in codesigning ACS and an artifact's geometry are evident in the military aerospace industry. Modern fighter aircraft, such as the F-16, are capable of surprising maneuverability and performance capabilities due the incorporation of ACS into their design. The F-16 aircraft is capable of high maneuverability because the aircraft geometry is inherently unstable (known as “relaxed static stability” in aircraft design); it is capable of controlled flight only because of its ACS (fly-by-wire). On the other hand, planing boats have remained in a path where the implementation of control systems for craft behavior augmentation is almost exclusively considered as a late-design-stage implementation, if not postproduction. To the best of the authors' knowledge, no published research explores the codesign of ACS and a planing craft.

Hence, a potential game changer that could significantly improve all three vessel performance criteria is still largely unexplored: codesigning ACS and planing craft. Therefore, the objective of this paper is to start the conversation in the vessel design community on the codesign methodology of high-speed vessels with ACS.

Following the steps of the early works in planing surfaces, this paper will focus on optimizing a planing craft's calm-water L/D by changing its lcg (which in turn changes its trim angle). As previously mentioned, it is known that planing craft are sometimes designed to operate at suboptimal (with respect to the L/D) trim angles in order to prevent porpoising. But the benefits of stabilizing a planing craft with ACS to operate at these attitudes have not been properly addressed in published literature. Thus, in this work, we compare the performance of a planing craft with ACS where the lcg of the vessel and ACS are designed sequentially (traditional approach), with the performance of a vessel in which the lcg and ACS are codesigned.

We assume the calm-water L/D to be a steady-state value when the vessel is in stable operation. Therefore, the ACS's dynamic forces are assumed to be null and the ACS's only role in the calm-water L/D optimization is to provide stability, and no ACS hardware or controller was investigated in this study.

The present results supersede those presented in Ref. [17].

Design Methodology

To simplify the vessel's hydrodynamics (and thus the modeling), the vessels investigated are prismatic in shape, i.e., the beam and deadrise distributions are constant throughout the length of the vessel as shown in Fig. 1. The vessel shape follows that from the models tested by Fridsma in Ref. [4], and the used dimensions and specifications can be seen in Table 1.

In this investigation, we focus on optimizing the vessel's calm-water L/D by changing the lcg—which in turn changes the vessel's running trim angle. Note that because Fz = Δ is considered as a constant, optimizing L/D is equivalent to minimizing RT (RT is the total drag, Fz is the total vertical force, and Δ is the displacement). As it has been found, for a fixed speed and hull geometry, there is a global optimum trim angle [6]. However, the vessel might also have a lcg location where there is porpoising inception [18]; and depending on the speed and hull geometry, the porpoising inception might be before the vessel reaches its optimal L/D [6]. Consequently, the vessel's “optimal” L/D is dependent on whether we want a vessel that is open-loop stable (w/o ACS) or if we allow the vessel to be open-loop unstable and stabilize it with an ACS. In other words, if we would like to have a vessel that is stable w/o ACS, the optimal trim angle would be the one that has the lowest drag but still inside the stable region—this is illustrated in Fig. 2.

As previously mentioned, the ACS's dynamic forces are assumed to be null, and the ACS's only role in the calm-water L/D optimization is to provide stability. Consequently, to investigate the potential from codesigning a planing craft and its ACS, two contrasting investigation procedures are conducted: (1) in one the vessel's design space is explored with the constraint that it has to be stable without an ACS (i.e., open-loop stability at the origin) and (2) and in the other that it has to be stable with an ACS (i.e., locally controllable).

The L/D and local stability investigation was conducted with two different methods. (i) One was by following the procedure in Ref. [1], which uses Lyapunov's indirect method [19] with the nonlinear models described in Ref. [1]—therefore, this method will be referred as “Faltinsen method.” The model used for Faltinsen method is presented in Appendix  A. (ii) The other procedure, which is akin to one used in Ref. [20], conducts a porpoising inception bisection search of the lcg using the time-domain nonlinear simulation program POWERSEA [21]—thus, it will be referred to as “POWERSEA method.”

The optimal L/D was calculated for a range of Cv and β. More specifically, the optimal lcg location (for both design paths) is found for when the vessel has the specifications from Table 1 and for a range of values for Cv ∈ [2.5, 4.5] and β ∈ [5, 30].

Faltinsen Method.

For the Faltinsen method, the two design paths can be seen as the optimization problems shown in Table 2, where x=[lcgτzwl]T, λi are the eigenvalues of the linearized model from Ref. [1], and |WC| is the determinant of the controllability matrix of the linearized model [19]. As previously mentioned, the nonlinear model used for defining RT, Fz, Mcg and the linearization of the system are shown in Appendix  A, and the methods used to test for stability and controllability are stated in Appendix  B. The fmincon optimizer from matlab was used to solve these optimization problems. Apart from the approach taken here, there are multiple ways that one could setup these optimization problems, each with their own advantages and disadvantages. Because the approach taken here leaves a lot of work to the optimizer to gain some efficiency, it is important to give the optimizer as much information as possible so that it does not explore invalid domains. In this case, it is possible to set lower and upper bounds such as 0 < τ < 10 and 0.15b < lcg < 3b, and also the constraint that the vessel should be in the water, e.g., LK > 0.

POWERSEA Method.

For the POWERSEA method, the lcg in which porpoising starts was found by using a bisection method with the program POWERSEA. The vessel is simulated without ACS in calm water for 100 s, and if the vessel's pitch at the end of the simulation is not oscillating with more than half of a degree of amplitude, the simulated lcg is considered stable and it is reduced by half for the next simulation. However, if the vessel is found to be porpoising, the next simulated lcg is the average of the last unstable lcg and stable lcg. This algorithm is continued until lcg/b converges within three significant figures.

Calm-Water Controllability.

Moreover, to compare the calm-water controllability between designs, the “controllability index” c from Ref. [22] is used as a metric for a measure of controllability between designs. This index is based on the controllability gramian of the vessel's linearized model; and to keep continuity in the controllability gramian values when the system becomes open-loop unstable at the origin, the procedure in Ref. [23] is used.

Seakeeping Example.

To have a sense on how the calm-water L/D translates to when the vessel is operating in a seaway, the vessels found for the two design paths at Cv = 4.5 and β = 20 deg are simulated for 2500 s in POWERSEA with sea states varying from zero to three. The seaways were simulated with the ISSC spectrum with 100 wave components, following the significant waveheights and periods specified for the Atlantic Ocean in Ref. [1].

For the simulations, the vessel's ACS is modeled as two point forces on the vessel, one at the stern and the other at the bow. The ACS controller is a LQR designed using the same linearized model used to investigate the stability of the vessel; the model is linearized about the calm-water operating trim and heave estimated by Savitsky method [6]. A shortcoming of the controller used is that it requires all states of the vessel to be measured during operation: heave, pitch, heave velocity, and pitch rate.

From the seakeeping simulations, the average L/D and the acceleration metric Dz [24] are computed. It is important to note that to use Dz as a metric for comparison, the time series being compared should have the same time length. The advantage of using Dz as a metric is that its application is simple and it is more sensitive to isolated shocks than the root-mean-square.

Results and Discussion

Faltinsen Method.

When the vessel is required to be open-loop stable, the vessel's optimal efficiency contour is as seen in Fig. 3(a). As expected, the efficiency reduces with increasing deadrise and speed; this is consistent to what vessel designers are familiar with: if you want to go somewhere faster, you will need more energy. Therefore, if a designer's only concern was to design a vessel with the highest efficiency inside this domain, then that design point would correspond to [Cv,β]=[2.5,5deg].

Nonetheless, when the vessel is allowed to operate at any running attitude as long as it is controllable, the optimal efficiency contour changes as seen in Fig. 3(b). In this contour, the points that are open-loop unstable are marked with “×.” In contrast with the previous contour, the efficiency increases when Cv > 3.5. Consequently, rather than having only one optimal point as before, there are now two optimal points: one is the same (since it is also open-loop stable), and the other is [Cv,β]=[4.5,5deg].

The relative change between the efficiencies when open-loop stability constraint is enforced and relaxed can be seen in Fig. 4(a), where RT and RT are the codesign and sequential design vessels' drag, accordingly. It is clear that the efficiency gains from allowing the vessel to operate in an open-loop unstable region increases as the speed increases—reaching an impressive 22% L/D increase for the [Cv,β]=[4.5,5deg] optimal design point. As a rough estimate, these kinds of efficiency gains translate to reducing fuel related costs (volume, weight, and expenses) by 18%.

Nevertheless, there is a difficulty presented if the vessel is required to operate in the new optimal operating point of [Cv,β]=[4.5,5deg]. As one can see from Fig. 4(c), the operating trim angle increases as the designs enter the unstable region with the increasing speed. For the required trim angle at [Cv,β]=[4.5,5deg], the lcg should be located at 74% of the beam's length (from the transom)—if the lcg is forward of this, there would have to be an external positive trim moment to maintain the vessel at this operating point. However, this challenge merely points to the fact that conventional hull designs would need innovation—opening the door for new and unconventional vessel designs.

While it requires innovative vessel designs to operate at these new optimal operating points, the new designs could actually be more controllable. To compare the “ease of control” in those cases, we compute the relative change in the controllability indexes; the results are shown in Fig. 4(b), where c and c are the controllability metric of the codesign and sequential design vessel, respectively. According to this metric, when the vessel is required to be open-loop stable at the origin, the system is very close to being uncontrollable (the value ≈ 0); this makes the relative change values to be very large—perhaps misleadingly large. Consequently, to have a sense of the difference in controllability between designs, the nonlinear model from Ref. [1] was simulated when the initial state was perturbed from the origin and controlled by a LQR controller. The LQR controller was tuned so that the vessel responses were as similar as possible; this was attempted by an optimization method with the LQR tuning parameters as the design variables. An example result can be seen in Fig. 5, where the larger controllability index corresponds to the vessel design that required open-loop origin local stability (solid line). From these time-domain results, it is noticeable that even with increased control effort for the vessel with open-loop stability, it has worse performance than the vessel that is open-loop unstable. Nonetheless, note that these results are for calm water.

POWERSEA Method.

This method was used primarily as additional confirmation to the results suggested by using Faltinsen method. From the optimal lift-to-drag contours shown in Figs. 6(a) and 6(b), we can see that the efficiency contours share some similarities to the ones found with Faltinsen method. Moreover, comparing Figs. 3(b) and 6(b), we can observe that Faltinsen method is more conservative on estimating the domain where the vessel's lift-to-drag optimum is open-loop unstable.

Both the POWERSEA method results and the Faltinsen method results suggest that the L/D improvement changes with β, with the highest L/D gains at the lower β (see Figs. 4(a) and 6(c)). The improvements in L/D agree with those found by Faltinsen method, reaching an impressive 25% L/D increase for the [Cv,β]=[4.5,5deg] optimal design point.

Seakeeping Example.

The seakeeping results can be seen plotted in Fig. 7. From the L/D results, it can be observed that the L/D gains are maintained in a seaway. However, the seakeeping performance was considerably worsened at sea states higher than one—increasing the vertical acceleration doses by approximately 25%.

Conclusion

When the vessel and ACS are codesigned, any designs that would be otherwise discarded because they have open-loop instabilities are now viable candidates because the ACS can stabilize the system. Vessels can now be designed to operate on much more efficient running attitudes. Moreover, the calm-water controllability investigation shows that the unstable vessels resulting from the codesign approach are actually more controllable (in calm water) than the vessels from the traditional approach, with this result increasing in strength with higher speeds—a result which indicates a “win–win” situation between calm-water controllability and calm-water L/D optimization.

The results suggest that codesigned vessels provide designs that are of greater performance, for the metric considered, than traditional vessels. The improvement in transport efficiency but worsening in seakeeping at the higher sea states suggest that there might be a tradeoff between the two; therefore, future research will investigate techniques to include seakeeping as a design metric in the codesign method. Moreover, the maneuverability of the vessel is an additional design metric that would be affected by the ACS; as a result, if the maneuverability of the vessel is important, this metric should be also incorporated into the codesign study.

Nonetheless, these first results investigating the codesign of a planing craft and its ACS indicate that the traditional design approach does not fully exploit the synergy between a planing craft and its ACS; and as a first step, the stability constraints should be relaxed in the design exploration. But more importantly, a designer should incorporate, if possible, the ACS early in the design stage.

Acknowledgment

This work was supported by the Naval Engineering Education Consortium (NEEC) and by the ONR (N00014-11-1-0831 and N00014-11-1-0832, Program Director: Ms. Kelly Cooper). The authors are appreciative of the significant help provided by Mr. Richard Akers at Main Marine Composites in assisting with POWERSEA simulations; and Professor Robert F. Beck for his comments on the effects on maneuverability during the presentation of this work at OMAE 2015s Professor Robert F. Beck Honoring Symposium on Marine Hydrodynamics.

Nomenclature

     
  • b =

    beam of planing vessel

  •  
  • c =

    controllability index [22]

  •  
  • Cv =

    speed coefficient =  V/gb

  •  
  • CΔ =

    load coefficient =  Δ/(γb3)

  •  
  • Dz =

    acceleration dose [24]

  •  
  • Fz =

    lift, total vertical force

  •  
  • g =

    gravitational acceleration

  •  
  • L/D =

    lift-to-drag ratio (i.e., Fz/RT)

  •  
  • LC =

    chine wetted length

  •  
  • LK =

    keel wetted length

  •  
  • lcg =

    longitudinal distance of center of gravity from transom (measured parallel to keel)

  •  
  • LOA =

    length overall

  •  
  • Rg =

    radius of gyration

  •  
  • RT =

    drag, total horizontal resistance

  •  
  • V =

    horizontal velocity of vessel

  •  
  • vcg =

    vertical distance of center of gravity from keel (measured normal to keel)

  •  
  • zwl =

    vertical distance of center of gravity to the calm water line

  •  
  • β =

    deadrise angle

  •  
  • γ =

    specific weight of water

  •  
  • Δ =

    displacement

  •  
  • η3 =

    vertical displacement of the center of gravity relative to the calm water zwl

  •  
  • η5 =

    rotation of the body relative to the calm water τ

  •  
  • τ =

    trim angle

Appendix A: Planing Craft Model

In this section, the equations used to find the linearized values of the system—for using Faltinsen method—will be presented in a programmatic fashion, i.e., the equations are presented in a sequence which can be sequentially calculated. No derivations or explanations will be presented; for the interested reader, the derivations can be found in Ref. [1] unless otherwise stated.

The added mass coefficients are “based on a high-frequency free-surface condition and strip theory” [1]. For the damping coefficients B33 and B55, a quasi-steady approach is used using the hydrodynamic lift forces estimated by Savitsky's equations [6]; and for B35 and B53, a rough estimation is done by using the Euler beam equation applied to high-frequency rigid-body oscillations [1]. And the restoring coefficients are the linearizations of Savitsky's equations rewritten following Troesch's procedure [25].

Note that these equations could be defined as a function of any of the used variables. For the purpose of making the linearized model used in this paper, the equations are made functions of η3 and η5.

Reduced-Order Model Equations.
Keel wetted length 
LK=lcg+vcgtan(τ+η5)(zwl+η3)sin(τ+η5)

Note: If LK < 0, the vessel is out of the water and so it should be set to LK = 0.

Distance from keel/water-line intersection to start of wetted chine 
xs=0.5btan(β)(1+zmaxUt)(τrad+η5)

where zmax/Ut is the coefficient of maximum pressure coordinate, which can be interpolated from Table 3.

Chine wetted length 
LC={LKxsifLK>xs0otherwise

Note: If LC = 0, the vessel is running “chines-dry.”

Mean wetted length–beam ratio 
λW=LK+LC2b
Adjusted distance from keel/water-line intersection to start of wetted chine 
x̃s={xsifLC>0LKotherwise
(A1)

Note: Equation (A1) is an adjustment made to Ref. [1].

Longitudinal distance from center of gravity to keel/waterline intersection. 
xG=LKlcg
K constant [1] 
K=cot(β)[πsin(β)Γ(1.5βrad/π)Γ2(1βrad/π)Γ(0.5+βrad/π)1]

where Γ is the gamma function.

Added mass coefficients for dry chine region 
κ=(1+zmax)(τrad+η5)A33(1)=13ρκ2Kx̃s3A35(1)=A53(1)=A33(1)(xG34x̃s)A55(1)=A33(1)(xG232xGxs+35x̃s2)

where ρ is the water density.

Added mass coefficients for wetted chine region; thus if LC = 0 then Ajk(2)=0, else 
C1=2tan2(β)πKA33(2)=ρb3C1π8LCbA35(2)=A53(2)=ρb4C1π16[(LKb)2(x̃sb)2]+xGA33(2)A55(2)=ρb5C1π24[(LKb)3(x̃sb)3]ρb4C1π8xG[(LKb)2(x̃sb)2]+xG2A33(2)
Total added mass coefficients 
Ajk=Ajk(1)+Ajk(2)
Draft from the spray root 
d={0.5btan(β)ifLC>0(1+zmaxUt)(τrad+η5)LKotherwise
Two-dimensional added mass coefficient in heave for a wedge [1] 
a33=ρKd2
Hydrodynamic lifting force coefficient for β = 0 deg 
CL0=(180π)1.10.012(τrad1.1+η5)λW0.5
Hydrodynamic lift coefficient for β > 0 
CLβ=CL00.0065βdegCL00.60
Derivatives of the hydrodynamic lift coefficients 
CL0τ=(180π)1.10.0132(τrad+η5)0.1λW0.5CLβτ=CL0τ(10.0039βdegCL00.4)
Damping coefficients 
B33=ρ2Vb2CLβτB35=V(A33+a33lcg)B53=B33(0.75λWblcg)B55=Va33lcg2
Wetted surface area for dry chine region 
S(1)=x̃sb2cos(β)
Wetted surface area for wetted chine region 
S(2)=bLCcos(β)
Total wetted surface area 
S=S(1)+S(2)
Average bottom velocity, from Ref. [26] 
Vm=V10.0120λWτrad1.10.0065βrad(0.0120λWτrad1.1)0.6λWcos(τ)
Reynold's number based on the average bottom velocity 
Rn=VmλWbν

where ν is the fluid's viscosity.

ITTC 1957 friction drag coefficient 
Cf=0.075(log10(Rn)2)2
Friction drag coefficient correction for hull roughness, from Ref. [27] 
103ΔCf=44[(AHR/(λWb))1/310Rn1/3]+0.125

where AHR is the average hull roughness, and a value of AHR = 150 × 10−6 was used.

Friction drag 
Rf=12ρ(Cf+ΔCf)SV2
Hydrodynamic lift 
FLβ=12CLβρV2b2
Total drag force (horizontal force with respect to calm water) 
RT=FLβtan(τ+η5)+Rfcos(τ+η5)
Sum of forces along z (about η3) 
Fz=FLβRfsin(τ+η5)
Longitudinal position of the center of pressure 
lp=λWb(0.7515.21(Cv/λW)2+2.39)
Longitudinal position of the center of drag 
lf=btan(β)(16S(1)+14S(2))S(1)+S(2)
(A2)

Note: Equation (A2) is the geometric center of the wetted area.

Lift's normal force with respect to keel 
FN=FLβcos(τ+η5)
Moment about CG (about η5) 
Mcg=FN(lcglp)+Rf(lfvcg)

where the coordinate system used follows that of Ref. [1], with positive toward aft and up.

Linearized restoring coefficients in heave and pitch 
C3k=F3cgηk=FzηkC5k=F5cgηk=Mcgηk
(A3)

where the derivatives in Eq. (A3) were found numerically by using the complex-step method [28]—using this method, the accuracy of the derivatives can be found up to machine precision.

State-Space Representation.
The vessel's unforced equation of motion can be written as 
Amη¨+Bη˙+Cη+Du=0Am=[Δg00Δrg2g]+[A33A35A53A55]B=[B33B35B53B55]C=[C33C35C53C55]D=[11lcglcgLOA]
(A4)

where η=[η3η5]T,u=[faftffwd]T, and the control inputs faft and ffwd are the ACS forces at the stern and bow, respectively. The right-hand side of Eq. (A4) is zero only when τ and zwl are chosen so that Fz = 0 and Mcg = 0 (vessel is in equilibrium—but not necessarily stable).

By using the state η̃=[η3η5η˙3η˙5]T, we can rewrite Eq. (A4) in state-space form as 
η̃˙=Ãη̃+B̃uÃ=[01Am1CAm1B]B̃=[0000Am1[11lcgLOAlcg]]
(A5)

The local stability and controllability can now be easily investigated with Eq. (A5).

Appendix B: Stability and Controllability

This Appendix is only meant to be a concise and practical presentation of the tools used in this paper to investigate local stability and local controllability, with no proofs. For proofs and discussion on these techniques, the reader is referred to Ref. [19].

For the following discussions, it is assumed that the nonlinear system has an equilibrium point at x* = 0, i.e., f(x*)=f(0)=0. Moreover, we are considering the system 
x˙=f(x,u)Ax+BuA=fx(x,u)|x=0,u=0B=fu(x,u)|x=0,u=0

Therefore, the following statements only apply if f(x, u) is continuously differential within a neighborhood of the origin—which is not true when the vessel is operating precisely between LC = 0 and LC > 0.

Stability: Lyapunov's Indirect Method.

Lyapunov's indirect method states that (i) the origin (x* = 0) is asymptotically stable if Reλi < 0 for all eigenvalues of A, (ii) and the origin is unstable if Reλi > 0 for one or more of the eigenvalues of A [19]. This method is inconclusive if Reλi ≤ 0 for all eigenvalues of A.

Controllability.
Following from Lyapunov's indirect method and linear systems theory, a nonlinear system can be locally controlled by linear feedback control if the controllability matrix Wc, Eq. (B1), is invertible. 
Wc=(BABAn1B)
(B1)

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