- Email: [email protected]
Optimal AUV Control Surface Sizing Using Convex Optimization Methods⋆
Copyright (0 IFAC Manoeuvring and Control of Marine Craft, Brijuni, Croatia, 1997
OPTIMAL AUV CONTROL SURFACE SIZING USING CONVEX OPTIMIZATION METHODS * P. Encarnac;ao' A. Pascoal *," A. Healey"
• [SR / Department of Electrical Engineering [nstituto Superior Tecnico Av. Rovisco Pais, 1096 Lisboa Codex, Portugal •• Department of Mechanical Engineering Naval Postgraduate School Monterey , California 93943, USA
Abstract: This paper addresses the problem of integrated design of AUV plant parameters and feedback controllers to meet mission performance requirements with minimum energy expenditure. Given an AUV that is required to operate over a finite set of representative trim conditions in the vertical plane in the course of a given mission scenario, the objective is to determine the optimal size of the bow and stern planes so that the average propulsion power required to execute that mission is minimized. The minimization process is subject to open loop stability requirements, and to the existence of stabilizing feedback controllers that can meet time and frequency closed loop requirements about each trim condition. The paper introduces a methodology to solve this combined plant/controller optimization problem in the framework of convex optimization theory, and describes its application to the selection of the optimal size of bow and stern planes for a prototype autonomous vehicle. Keywords: Autonomous Underwater Vehicles. Integrated Plant and Controller Design. Convex Optimization.
system design is integrated with the design of the vehicle itself. This is in striking contrast with classical approaches where the plant structure is essentially fixed a priori, and the control system designer is left with the burden of finding an adequate controller to meet certain performance specifications, if at all possible.
1. INTRODUCTION
This paper addresses the problem of integrated design of AUV plant parameters and feedback controllers to optimize vehicle performance over a set of operating conditions arising in the course of a given mission scenario. The motivation for this work stems from the fact that significant energy savings and increased dynamic performance can in principle be obtained if the process of control
As a contribution towards the solution of this general problem, this paper formulates and solves the following simplified problem: given an AUV - with a fixed baseline configuration - that is required to operate over a finite set of representative trim conditions in the vertical plane, determine the optimal size of the bow and stern planes so that a weighted average of the power required at trim is minimized, subject to the conditions that open loop requirements be met , and that
*
The work of P. Encarna"a.o and A. Pascoal was supported by the PRAXIS Programme of JNICT under contract PRAXIS / 3/3.1/ TPR/23/94 and by a Graduate Student Fellowship. The work of A . Pascoa1 was also supported by a NATO Fellowship during his sabbatical leave at the NPS . A . Healey received partial support from the Office of Naval Research (ONR) .
47
stabilizing feedback controllers can be designed to meet time and frequency closed loop requirements about each trim condition. Open loop requirements include the possibility of achieving trim at each operating point and meeting a desired degree of open loop stability; closed loop requirements include maneuverability specifications in response to depth commands , hard limits on surface deflection , one-loop at a time phase and gain margin specifications, and maximum actuator bandwidths. Traditionally, the sizing of underwater vehicle control surfaces has been determined from openloop stability and maneuverability criteria only (Henry, 1995) . This stems from the fact that conventional underwater vehicles are manned , and in this case open-loop criteria impact directly on the quality of vehicle handling. The situation is completely different in the case of AUVs , since they must operate in closed-loop most of the time. This requirement is specially relevant for openloop unstable AUVs. Therefore, closed loop maneuverability criteria must definitely play an integral role in the selection of AUV control surface sizes.
Fig. 1. Prototype AUV which provide an explicit parameterization of all the controllers that stabilize a given plant, and show that many time and frequency specifications are convex functions of the controller parameters. The problem of control system design is then posed and solved in the framework of convex optimization theory by using efficient numerical algorithms that are available with the software package QDES, developed at Stanford University (Boyd and Barratt , 1991 ; Barratt et al., 1987) . Based on the two maps defined above, a powerful graphical procedure is obtained that allows choosing the optimal AUV surface sizes so as to minimize the energy expenditure during trim maneuvers, while allowing to meet open and closed loop dynamic requirements by proper design of dynamic feedback controllers. An application is made to the selection of the optimal size of bow and stern planes for a prototype autonomous vehicle.
At present, no analytic solution exists for even the sim pier combined plant / controller optimization described. The complexity of the problem at hand can be best appreciated by stressing that it aims to find the limits of performance of A UVs as a function of control surface sizes. In particular, this implies that no restrictions should be placed on the structure of the possible controllers adopted, since the choice of a particular control structure (e.g. , PID) would restrict the class of stabilizing feedback stabilizing laws over which to search for the best combination of surface sizes and vehicle controllers.
The organization of the paper is as follows: Section 2 describes a general dynamic model for a prototype AUV in the vertical plane. Section 3 introduces the concept of trim trajectories, and computes the average propulsion power that is required to perform a given mission scenario as a function of control surface sizes. Section 4 computes the limits of the performance that can be achieved with any stabilizing controller about trim trajectories . Finally, Section 5 describes the application of the tools developed in Sections 3 and 4 to the resizing of the control surfaces for the prototype AUV .
This paper presents a methodology for the numerical solution of the problem described that builds on a number of key results in the area of convex optimization . The design process starts with the modeling of the vehicle, leading to a mathematical description of its dynamic behaviour that is explicitly parameterized in terms of the control surface sizes. This is followed by a second phase, which produces a map from the space of control surfaces sizes to the set of positive real numbers, defined as the average propulsion power that is spent in steering the vehicle through a sequence of trim maneuvers, as part of an adopted mision scenario. Finally, the minimum settling time in response to depth commands that is achieved with any stabilizing controller (while meeting an extra set of closed loop specifications) is also computed as a function of the surface sizes, for a number of selected trim points. This important step builds on some interesting results in control theory
2. AUV DYNAMIC MODEL IN THE DIVE PLANE The underwater vehicle considered in this study is a modification of the MARIUS AUV depicted in Figure 1; see (Silvestre and Pascoal, 1997) and the references therein . The vehicle has a flatfishlike shape and is 4.5m long, 1.1m wide, and O.6m high . Propulsion is assured by two propellers located behind the main body of the vehicle. Two rudders for vehicle steering are mounted directly aft of the thrusters. For diving maneuvers, the
48
vehicle is equipped with two pairs of all moving control surfaces (bow and stern planes). The AUV dynamic model can be found in (Fryxell, 1994; Silvestre and Pascoal, 1997; Encarna<;ao, 1996).
0..205 .
0.2
O.l~
2.1 Equations of Motion.
-, -
Following standard practice, the equations of motion of the vehicle in the vertical plane can be developed using a global coordinate frame {U} and a body-fixed coordinate frame {B} that moves with the AUV. The following notation is required: (x, Z)T - position of the origin of {B} measured in {U}; (u, w) - velocity of the origin of {B} relative to {U}, expressed in {B} (i.e., body-fixed linear velocity) ; q - angular velocity of {B} relative to {U} , expressed in {B} (Le., body-fixed angular velocity). The symbol ~R(O) denotes the rotation matrix from {B} to {U}, parameterized by the pitch angle O. The extended vector
MRB..i:1.v + CRB"(
.
0.19
O.las
0.1e
O·'7S0~-:0.'':----:'0.:2 --""0.3:--"'0.':---:':0.5:--0-:':.6-0:':: .7--:0:':: .8--:0~ .• - - l Sh Iml J
Fig. 2. Surface sizes required for vehicle trimming at Vt = 1mjs and , = 20°_ lower bound. responding set of equilibrium points, that is, the set of input and state variables for which the net sum of forces and moments acting on the vehicle is zero. This is done formally as follows . An equilibrium (also denoted as trim) point of the vertical plane equations is a vector (ci6,., 00, 8bo ' 8'0' Tof that satisfies
CRB" (
= O.
(2)
In the equation, cio" = (uo, Wo , qo)T . It is assumed that the input variables are restricted by physical constraints that are known in advance (e.g. the stall angles for the control surfaces and the maximum thrust available as a function of total speed) .
(1)
{ O=q
z = -usinO+wcosO
From (2), it follows that the only possible equilibrium points of an AUV in the vertical plane are those that correspond to straight line trajectories . These are easily parameterized in terms of total speed Vt and flight path angle " where , = 0 - Q . Given desired values of Vt and I it is possible, using equations (2), to determine if a corresponding trim condition is achievable, and in the affirmative to compute the required control surface deflections and thrust. Notice that since Ob , 8s , T, Q , and 0 must be computed from the first three equations in (2) and the additional relation I = 0 - Q , one of those variables must be preset . Due to practical considerations, it was decided to set the bow plane deflection to zero at trimming (Encarna<;ao, 1996) .
where MRB" is the mass matrix, CRB" is the matrix of Coriolis and centrifugal terms, and Tv is the vector of external forces and moments. The following notation will also be used in the text : Vt = (u 2 + W 2 )1/2 is the absolute value of the velocity vector and Q = arcsin(wj(u2 + W 2 )1 / 2) is the angle of attack. The problem addressed in this paper required that the vehicle model be parameterized in terms of the sizes of the bow and stern plane. This was done by assuming that: i) the chord c and length d of the control surfaces are such that their aspect ratio AR = %is constant , ii) the control surfaces have a constant profile , and iii) the control surfaces are placed the furthest away from the cent er of mass of the vehicle as physically possible, so as to maximize the moment that they impart to the vehicle. The estimates for the surface lift and drag terms were based on theoretical predictions using thin airfoil theory and on experimental airfoil data. See (Encarna<;ao, 1996) for complete details.
Given a set of values of Vt and, over which the AUV is required to operate, it is now possible to determine the corresponding restrictions on the surface sizes so that trimming can be achieved without incurring surface stall or exceeding the maximum thrust available. A fixed margin of maneuver for the control surfaces abou.t trimming should also be observed, in order to maintain control authority. In the design example considered here, the worst-case trim condition corresponds to Vt Imj s and I 20° since it is the most
3. VEHICLE TRIMMING . AVERAGE PROPULSION POWER Given the nonlinear model of the vehicle in the vertical plane, it is important to compute the cor-
=
49
=
both very small and large surface sizes. Of course, different results are obtained for different mission scenarios. In particular, for missions where the time to transit to an inspection site is negligible, trimming is essentially achieved with all control surfaces set to zero. In this case, the smaller the surfaces the better. However, maneuverability requirements will place further restrictions on the minimum surface sizes, as will be seen later. It can be argued that graphics like that of Figure 3 are a powerful tool to aid in computing the optimal control surface sizes for a given vehicle operating in a specific mission scenario. Notice, however, that no dynamical considerations were taken into account in the figure. This problem is addressed in the next section.
..
, 0.2
r:================:'=OO==~~===:j
Q1~
1M
0.1
0.2
0.3
0.8
0.9
Fig. 3. Average propulsion power as a function of control surface sizes: Level curves. demanding in terms of surface deflection (see Section 5). The region of allowable surface sizes is bounded below by the tradeoff curve depicted in Figure 2. In the absence of bow planes, the minimum size of the stern planes required for vehicle trimming is approximately 0.2m 2 .
4. LIMITS OF DYNAMIC PERFORMANCE Given a model of a physical plant and a set of design goals that include performance specifications, robustness requirements, and control law constraints, the problem of control system design is to find a suitable feedback controller that meets the above requirements, or determine that none exists. As in any engineering design problem, this involves examining tradeoffs. A simple example is the case of a mechanical system where larger actuator signals - forces or moments - are required to obtain shorter time responses to input command signals. Unfortunately, other tradeoffs exist which defy simple analysis. For example, the pattern of open loop zeros and poles of a linear plant imposes fundamental restrictions on the closed loop performance that can be achieved with any stabilizing controller (Freudenberg and Looze, 1988) . In particular, unstable, nonminimum phase plants must tradeoff low sensitivity to output disturbances at low frequencies with high sensitivity in the complementary range of frequencies. This result is independent of the particular stabilizing controller used. The reader is referred to (Freudenberg and Looze, 1988; Boyd and Barratt, 1991) for an introduction to this circle of ideas, and its application to determining basic limits of performance of linear control systems.
Consider now a mission scenario where the vehicle is required to go through a finite set of trim trajectories, and that each trajectory is allocated a given time percentage of the complete mission duration. Then, it is important to compute the average power that is spent in trimming the vehicle at those cruising conditions, as a function of the control surface sizes. From the discussion above, it is clear how to compute the thrust force T required for each achievable trim condition. The corresponding power that must be input to the thruster motors can then be evaluated using a nonlinear model of the vehicle propulsion system (Fryxell, 1994; Silvestre and Pascoal, 1997). The cverage propulsion power is then computed as E [PSCT] =
L PiPSCT
(Vti'
"Yi),
where PSCT is the power required for trimming the vehicle along a trajectory with velocity Vti and flight path angle "Yi, and Pi is the percentage of the total mission time that is spent at that trim condition. Figure 3 represents the level curves of E [Pscr ] for an hypothetical three phase mission where E [PseT] = 0.4 X PSCT (I, -20°) + 0.2 x P SCT (2.5,0°) + 0.4 x P scr (I, 20°). It is interesting to point out that the convex aspect of the surface in Figure 3 results from the combination of two effects: i) when the surfaces are small, they must deflect considerably to achieve trimming at low speed and high flight path angles; this will in turn increase the total drag, and ii) when the control surfaces are large, the deflection angles that are required for vehicle trimming at "Y #- 0° are small; however, since the total drag contains a term that is proportional to total surface size, the total drag increases again. This justifies why in this particular case the average propulsion power increases for
Motivated by the work mentioned above, this paper addresses explicitly the problem of determining the basic limits of performance achievable with an AUV about trim trajectories, as a function of the surface sizes. To make the problem tractable, it is assumed that the AUV motion about trim trajectories can be described by its linearized equations of motion. Given a linear plant, a control engineer will often be able to find a suitable stabilizing controller, when it exists, using any of the available control design methods. However, even if he fails to design a suitable controller, he cannot be sure that there is no controller that will meet the specifications required. To tackle
50
C.2 - the maximum control surface deflection should not exceed 30° , in order to avoid surface stalling. C.3 - gain and phase margins of 8dB and 35° respectively should be observed on the bow and stern plane control channels (one-loop at a time analysis), to provide some form of robustness in presence of parametric uncertainties; C.4 - the bandwiths of the bow and stern plane actuation channels should not exceed 5radj s in order to not violate the normal maximum bandwith of the DC motors that drive the control surfaces.
Fig. 4. Settling time (in seconds) as a function of control surface sizes; Vt = 2.5mj s - level curves
The results of the constrained optimization procedure are summarized in Figure 4, which shows the evolution of the minimum settling time achievable as a function of control surface sizes for the case when the AUV is moving at 2.5mj s .. The overall trend can be summarized by saying that the larger the control surfaces are , the smaller the settling time is. However, as seen in the next section, other requirements will impose upper bounds on the surface sizes.
this issue, Stephen Boyd and his coworkers have proposed in (Boyd and Barratt, 1991 ; Barratt et al., 1987) a numerical optimization design method that can effectively solve the controller design problem for linear plants, there is, it allows finding a suitable controller that meets the design specifications if they are indeed achievable, or it determines that they are not. Their pioneering work led to a software package called QDES (Barratt et al., 1987) that implements that numerical optimization design method in the discrete time domain. The package developed builds on convex optimization techniques, and relies on an explicit parameterization of all the linear controllers that stabilize a given plant (Boyd and Barratt, 1991) .
5. CONTROL SURFACE DESIGN EXAMPLE The rest of the paper describes a design exercise in which the tools developed in the previous sections were applied to the resizing of the control surfaces for the MARIUS AUV. As an example, the following three phase mission was considered (see also Section 3):
The rest of this section describes how the QDesign methodology was used to determine the limits of performance of the AUV as a function of the control surface sizes. The first step in the methodology required linearizing the AUV model about the set of trim points that define the three phase mission example introduced before. For simplicity of presentation, it was assumed that the maneuverability requirements are only important when the vehicle is inspecting the seabed in straight, level flight . In this case, it is important that the vehicle be able to change depth quickly in order to avoid unforeseen obstacles. For a finite set of control surface configurations, the vehicle model was linearized about the trim point determined by cio. = (uo,O,O)T,Oo = 0, and (8 b ,8s ,T)T = (O,O , To)T, with Uo = 2.5mjs, and discretized at 10Hz using a zero order hold. The following constrained optimization problem was then posed and solved (for each surface configuration) using the QDES software package: Find a controller that minimizes the settling time of the A UV response to a 1m dive command, subject to the following constraints:
i) Diving to 600m depth at speed Vt = 1mj s and flight path angle, = -20° . Duration: 30m . ii) Vehicle on level flight, at speed Vt = 2.5mj s and flight path angle, = 0°. Duration: 15m. iii) Resurfacing at speed Vt = 1mj s and flight path angle, = +20°. Duration: 30m . The constraints on the bow and stern surface sizes so that the vehicle will be able to sustain a flight path angle I = 20° at Vt = 1mj shave been analyzed in Section 3 and are summarized in Figure 2. The curve of average propulsion power as a function of surface sizes is depicted in Figure 3. While on level flight, it is required that the vehicle avoid unforeseen obstacles or terrain features . Suppose this requirement converts into the specification that the vehicle have a settling time ts S 3s in response to a 1m depth command. The corresponding constraint on surface sizes can be easily extracted from Figure 4, which reflects also the additional constraints C.1-C.4 described in Section 4. Suppose that an additional constraint on open loop stability is desired for the case when the vehicle is on straight, level flight, at Vt = 2.5mj s and I = 0°. The corresponding constraints on the control surface sizes can be easily evaluated by
C.1 - the closed-loop system should exhibit zero steady-state error in response to constant depth commands (integral action required).
51
allows searching for better tradeoffs among the design objectives.
0.' 0.8
6. CONCLUSIONS
0.7L~_~:..:-.-.=lU--_CC .
The paper introduced a methodology for integrated design of AUV plant parameters and feedback controllers to meet mission performance requirements with minimum energy expenditure. The methodology addresses both static and dynamic mission requirements, and departs considerably from classical procedures since it does not restrict a priori the types of feedback controllers considered . Furthermore, it provides a powerful graphical procedure for surface sizing that allows assessing the tradeoffs among competing performance requirements. An application was made to the resizing of the bow and stern planes of an existing prototype AUV. The design example described illustrated two basic concepts: i} optimal surface sizes are mission dependent, and ii} for some mission scenarios, it is clearly advantageous to have both bow and stern planes instead of stern planes only. Future work will address the challenging problem of optimal AUV surface sizing for operation under wave disturbances.
0.6
.. ~?
."
o.• ~_______~ . ~ ~-~1
u
~
u
u
u Sh
u
V
~
~
1m 2 )
Fig. 5. Open Loop Stability Index: Level Curves -
Average power level cwve,..
ScttJing time level curve for 1=3s •.••..
C)-
0.•
0.8 \
0.7 0.6
!
."
0.5
Stability index level QU'VC for i=.(). 15 Trimming con...ltaint
,
,
, ~3
...-0.1.5. ,_
,
'37
. . .. +-O._l~ .
0.4
0.3 0.2
- - - - - -
~\-,
-
-t60- - - - - - - - - - - - - - - - - - - -
7. REFERENCES
0. ,~==3'~60E==========~ 0.1
0.2
0.3
0."
O.S
S"
0.6
0.7
0.'
Barratt, C., N. Khraishi and S. Norman (1987). UNIX QDES Manual Entry. Boyd, S. and C. Barratt (1991). Linear Controller Design: Limits of Performance. Prentice Hall, Inc. Englewood Cliffs, New Jersey. Encarna<;iio, P. (1996). Convex Optimization Techniques with Applications to the Integrated Design of Underwater Robotic Vehicles and Controllers. Master's thesis. Dept. Electrical Engineering, 1ST. Lisboa, Portugal. Fossen, T . (1994). Guidance and Control of Ocean Vehicles. John Wiley & Sons. Chichester, England. Freudenberg, J . and D. Looze (1988). Frequency Domain Properties of Scalar and Multivariable Feedback Systems. Lecture Notes in Control and Information Sciences. SpringerVerlag. Fryxell, D. (1994). Modeling, Identification, Guidance and Control of an Autonomous Underwater Vehicle. Master's thesis. Dept. of Electrical Engineering, 1ST. Lisboa, Portugal. Henry, C. (1995) . On Optimum Turning Configurations with fixed-fin stabilization. IEEE Journal of Oceanic Engineering 20(4), 268275. Silvestre, C. and A. Pascoal (1997). Control of an AUV in the vertical and horizontal planes: system design and tests at sea. To appear in the Transactions of the Institute of Measurement and Control.
0.9
IIII~ I
Fig. 6. Control Surface Restrictions plotting the largest real part of the eigenvalues of the state space matrix of the vehicle's linearized model. This is done in Figure 5. Notice from the figure that without the stern plane the vehicle is open-loop unstable. It is also interesting to point out that in general, to maintain a given degree of open-loop stability, the bow and stern planes sizes are restricted to lie in a strip region. It is now possible to put together all the design constraints in Figure 6, and arrive at the optimal size of the bow and stern planes Sb = 0.3m 2 and S. = 0.52m 2 , for a required stability index of -0.15. This design example captures two key concepts: i} optimal surface sizing is mission dependent, and ii} in some cases, it is advantageous to use both bow and stern planes instead of simply stern planes. The latter case can be simply argued by considering the situation where there are no bow planes. In this case, it is impossible to meet the 3s settling time constraint while meeting the -0.15 stability index figure. This value of the stability index can still be achieved by allowing the settling time to increase to approximately 3.5s (see Figure 4). However, the average propulsion power will be increased in the process. The addition of the bow planes allows for an extra degree of freedom that
52
OPTIMAL AUV CONTROL SURFACE SIZING USING CONVEX OPTIMIZATION METHODS * P. Encarnac;ao' A. Pascoal *," A. Healey"
• [SR / Department of Electrical Engineering [nstituto Superior Tecnico Av. Rovisco Pais, 1096 Lisboa Codex, Portugal •• Department of Mechanical Engineering Naval Postgraduate School Monterey , California 93943, USA
Abstract: This paper addresses the problem of integrated design of AUV plant parameters and feedback controllers to meet mission performance requirements with minimum energy expenditure. Given an AUV that is required to operate over a finite set of representative trim conditions in the vertical plane in the course of a given mission scenario, the objective is to determine the optimal size of the bow and stern planes so that the average propulsion power required to execute that mission is minimized. The minimization process is subject to open loop stability requirements, and to the existence of stabilizing feedback controllers that can meet time and frequency closed loop requirements about each trim condition. The paper introduces a methodology to solve this combined plant/controller optimization problem in the framework of convex optimization theory, and describes its application to the selection of the optimal size of bow and stern planes for a prototype autonomous vehicle. Keywords: Autonomous Underwater Vehicles. Integrated Plant and Controller Design. Convex Optimization.
system design is integrated with the design of the vehicle itself. This is in striking contrast with classical approaches where the plant structure is essentially fixed a priori, and the control system designer is left with the burden of finding an adequate controller to meet certain performance specifications, if at all possible.
1. INTRODUCTION
This paper addresses the problem of integrated design of AUV plant parameters and feedback controllers to optimize vehicle performance over a set of operating conditions arising in the course of a given mission scenario. The motivation for this work stems from the fact that significant energy savings and increased dynamic performance can in principle be obtained if the process of control
As a contribution towards the solution of this general problem, this paper formulates and solves the following simplified problem: given an AUV - with a fixed baseline configuration - that is required to operate over a finite set of representative trim conditions in the vertical plane, determine the optimal size of the bow and stern planes so that a weighted average of the power required at trim is minimized, subject to the conditions that open loop requirements be met , and that
*
The work of P. Encarna"a.o and A. Pascoal was supported by the PRAXIS Programme of JNICT under contract PRAXIS / 3/3.1/ TPR/23/94 and by a Graduate Student Fellowship. The work of A . Pascoa1 was also supported by a NATO Fellowship during his sabbatical leave at the NPS . A . Healey received partial support from the Office of Naval Research (ONR) .
47
stabilizing feedback controllers can be designed to meet time and frequency closed loop requirements about each trim condition. Open loop requirements include the possibility of achieving trim at each operating point and meeting a desired degree of open loop stability; closed loop requirements include maneuverability specifications in response to depth commands , hard limits on surface deflection , one-loop at a time phase and gain margin specifications, and maximum actuator bandwidths. Traditionally, the sizing of underwater vehicle control surfaces has been determined from openloop stability and maneuverability criteria only (Henry, 1995) . This stems from the fact that conventional underwater vehicles are manned , and in this case open-loop criteria impact directly on the quality of vehicle handling. The situation is completely different in the case of AUVs , since they must operate in closed-loop most of the time. This requirement is specially relevant for openloop unstable AUVs. Therefore, closed loop maneuverability criteria must definitely play an integral role in the selection of AUV control surface sizes.
Fig. 1. Prototype AUV which provide an explicit parameterization of all the controllers that stabilize a given plant, and show that many time and frequency specifications are convex functions of the controller parameters. The problem of control system design is then posed and solved in the framework of convex optimization theory by using efficient numerical algorithms that are available with the software package QDES, developed at Stanford University (Boyd and Barratt , 1991 ; Barratt et al., 1987) . Based on the two maps defined above, a powerful graphical procedure is obtained that allows choosing the optimal AUV surface sizes so as to minimize the energy expenditure during trim maneuvers, while allowing to meet open and closed loop dynamic requirements by proper design of dynamic feedback controllers. An application is made to the selection of the optimal size of bow and stern planes for a prototype autonomous vehicle.
At present, no analytic solution exists for even the sim pier combined plant / controller optimization described. The complexity of the problem at hand can be best appreciated by stressing that it aims to find the limits of performance of A UVs as a function of control surface sizes. In particular, this implies that no restrictions should be placed on the structure of the possible controllers adopted, since the choice of a particular control structure (e.g. , PID) would restrict the class of stabilizing feedback stabilizing laws over which to search for the best combination of surface sizes and vehicle controllers.
The organization of the paper is as follows: Section 2 describes a general dynamic model for a prototype AUV in the vertical plane. Section 3 introduces the concept of trim trajectories, and computes the average propulsion power that is required to perform a given mission scenario as a function of control surface sizes. Section 4 computes the limits of the performance that can be achieved with any stabilizing controller about trim trajectories . Finally, Section 5 describes the application of the tools developed in Sections 3 and 4 to the resizing of the control surfaces for the prototype AUV .
This paper presents a methodology for the numerical solution of the problem described that builds on a number of key results in the area of convex optimization . The design process starts with the modeling of the vehicle, leading to a mathematical description of its dynamic behaviour that is explicitly parameterized in terms of the control surface sizes. This is followed by a second phase, which produces a map from the space of control surfaces sizes to the set of positive real numbers, defined as the average propulsion power that is spent in steering the vehicle through a sequence of trim maneuvers, as part of an adopted mision scenario. Finally, the minimum settling time in response to depth commands that is achieved with any stabilizing controller (while meeting an extra set of closed loop specifications) is also computed as a function of the surface sizes, for a number of selected trim points. This important step builds on some interesting results in control theory
2. AUV DYNAMIC MODEL IN THE DIVE PLANE The underwater vehicle considered in this study is a modification of the MARIUS AUV depicted in Figure 1; see (Silvestre and Pascoal, 1997) and the references therein . The vehicle has a flatfishlike shape and is 4.5m long, 1.1m wide, and O.6m high . Propulsion is assured by two propellers located behind the main body of the vehicle. Two rudders for vehicle steering are mounted directly aft of the thrusters. For diving maneuvers, the
48
vehicle is equipped with two pairs of all moving control surfaces (bow and stern planes). The AUV dynamic model can be found in (Fryxell, 1994; Silvestre and Pascoal, 1997; Encarna<;ao, 1996).
0..205 .
0.2
O.l~
2.1 Equations of Motion.
-, -
Following standard practice, the equations of motion of the vehicle in the vertical plane can be developed using a global coordinate frame {U} and a body-fixed coordinate frame {B} that moves with the AUV. The following notation is required: (x, Z)T - position of the origin of {B} measured in {U}; (u, w) - velocity of the origin of {B} relative to {U}, expressed in {B} (i.e., body-fixed linear velocity) ; q - angular velocity of {B} relative to {U} , expressed in {B} (Le., body-fixed angular velocity). The symbol ~R(O) denotes the rotation matrix from {B} to {U}, parameterized by the pitch angle O. The extended vector
MRB..i:1.v + CRB"(
.
0.19
O.las
0.1e
O·'7S0~-:0.'':----:'0.:2 --""0.3:--"'0.':---:':0.5:--0-:':.6-0:':: .7--:0:':: .8--:0~ .• - - l Sh Iml J
Fig. 2. Surface sizes required for vehicle trimming at Vt = 1mjs and , = 20°_ lower bound. responding set of equilibrium points, that is, the set of input and state variables for which the net sum of forces and moments acting on the vehicle is zero. This is done formally as follows . An equilibrium (also denoted as trim) point of the vertical plane equations is a vector (ci6,., 00, 8bo ' 8'0' Tof that satisfies
CRB" (
= O.
(2)
In the equation, cio" = (uo, Wo , qo)T . It is assumed that the input variables are restricted by physical constraints that are known in advance (e.g. the stall angles for the control surfaces and the maximum thrust available as a function of total speed) .
(1)
{ O=q
z = -usinO+wcosO
From (2), it follows that the only possible equilibrium points of an AUV in the vertical plane are those that correspond to straight line trajectories . These are easily parameterized in terms of total speed Vt and flight path angle " where , = 0 - Q . Given desired values of Vt and I it is possible, using equations (2), to determine if a corresponding trim condition is achievable, and in the affirmative to compute the required control surface deflections and thrust. Notice that since Ob , 8s , T, Q , and 0 must be computed from the first three equations in (2) and the additional relation I = 0 - Q , one of those variables must be preset . Due to practical considerations, it was decided to set the bow plane deflection to zero at trimming (Encarna<;ao, 1996) .
where MRB" is the mass matrix, CRB" is the matrix of Coriolis and centrifugal terms, and Tv is the vector of external forces and moments. The following notation will also be used in the text : Vt = (u 2 + W 2 )1/2 is the absolute value of the velocity vector and Q = arcsin(wj(u2 + W 2 )1 / 2) is the angle of attack. The problem addressed in this paper required that the vehicle model be parameterized in terms of the sizes of the bow and stern plane. This was done by assuming that: i) the chord c and length d of the control surfaces are such that their aspect ratio AR = %is constant , ii) the control surfaces have a constant profile , and iii) the control surfaces are placed the furthest away from the cent er of mass of the vehicle as physically possible, so as to maximize the moment that they impart to the vehicle. The estimates for the surface lift and drag terms were based on theoretical predictions using thin airfoil theory and on experimental airfoil data. See (Encarna<;ao, 1996) for complete details.
Given a set of values of Vt and, over which the AUV is required to operate, it is now possible to determine the corresponding restrictions on the surface sizes so that trimming can be achieved without incurring surface stall or exceeding the maximum thrust available. A fixed margin of maneuver for the control surfaces abou.t trimming should also be observed, in order to maintain control authority. In the design example considered here, the worst-case trim condition corresponds to Vt Imj s and I 20° since it is the most
3. VEHICLE TRIMMING . AVERAGE PROPULSION POWER Given the nonlinear model of the vehicle in the vertical plane, it is important to compute the cor-
=
49
=
both very small and large surface sizes. Of course, different results are obtained for different mission scenarios. In particular, for missions where the time to transit to an inspection site is negligible, trimming is essentially achieved with all control surfaces set to zero. In this case, the smaller the surfaces the better. However, maneuverability requirements will place further restrictions on the minimum surface sizes, as will be seen later. It can be argued that graphics like that of Figure 3 are a powerful tool to aid in computing the optimal control surface sizes for a given vehicle operating in a specific mission scenario. Notice, however, that no dynamical considerations were taken into account in the figure. This problem is addressed in the next section.
..
, 0.2
r:================:'=OO==~~===:j
Q1~
1M
0.1
0.2
0.3
0.8
0.9
Fig. 3. Average propulsion power as a function of control surface sizes: Level curves. demanding in terms of surface deflection (see Section 5). The region of allowable surface sizes is bounded below by the tradeoff curve depicted in Figure 2. In the absence of bow planes, the minimum size of the stern planes required for vehicle trimming is approximately 0.2m 2 .
4. LIMITS OF DYNAMIC PERFORMANCE Given a model of a physical plant and a set of design goals that include performance specifications, robustness requirements, and control law constraints, the problem of control system design is to find a suitable feedback controller that meets the above requirements, or determine that none exists. As in any engineering design problem, this involves examining tradeoffs. A simple example is the case of a mechanical system where larger actuator signals - forces or moments - are required to obtain shorter time responses to input command signals. Unfortunately, other tradeoffs exist which defy simple analysis. For example, the pattern of open loop zeros and poles of a linear plant imposes fundamental restrictions on the closed loop performance that can be achieved with any stabilizing controller (Freudenberg and Looze, 1988) . In particular, unstable, nonminimum phase plants must tradeoff low sensitivity to output disturbances at low frequencies with high sensitivity in the complementary range of frequencies. This result is independent of the particular stabilizing controller used. The reader is referred to (Freudenberg and Looze, 1988; Boyd and Barratt, 1991) for an introduction to this circle of ideas, and its application to determining basic limits of performance of linear control systems.
Consider now a mission scenario where the vehicle is required to go through a finite set of trim trajectories, and that each trajectory is allocated a given time percentage of the complete mission duration. Then, it is important to compute the average power that is spent in trimming the vehicle at those cruising conditions, as a function of the control surface sizes. From the discussion above, it is clear how to compute the thrust force T required for each achievable trim condition. The corresponding power that must be input to the thruster motors can then be evaluated using a nonlinear model of the vehicle propulsion system (Fryxell, 1994; Silvestre and Pascoal, 1997). The cverage propulsion power is then computed as E [PSCT] =
L PiPSCT
(Vti'
"Yi),
where PSCT is the power required for trimming the vehicle along a trajectory with velocity Vti and flight path angle "Yi, and Pi is the percentage of the total mission time that is spent at that trim condition. Figure 3 represents the level curves of E [Pscr ] for an hypothetical three phase mission where E [PseT] = 0.4 X PSCT (I, -20°) + 0.2 x P SCT (2.5,0°) + 0.4 x P scr (I, 20°). It is interesting to point out that the convex aspect of the surface in Figure 3 results from the combination of two effects: i) when the surfaces are small, they must deflect considerably to achieve trimming at low speed and high flight path angles; this will in turn increase the total drag, and ii) when the control surfaces are large, the deflection angles that are required for vehicle trimming at "Y #- 0° are small; however, since the total drag contains a term that is proportional to total surface size, the total drag increases again. This justifies why in this particular case the average propulsion power increases for
Motivated by the work mentioned above, this paper addresses explicitly the problem of determining the basic limits of performance achievable with an AUV about trim trajectories, as a function of the surface sizes. To make the problem tractable, it is assumed that the AUV motion about trim trajectories can be described by its linearized equations of motion. Given a linear plant, a control engineer will often be able to find a suitable stabilizing controller, when it exists, using any of the available control design methods. However, even if he fails to design a suitable controller, he cannot be sure that there is no controller that will meet the specifications required. To tackle
50
C.2 - the maximum control surface deflection should not exceed 30° , in order to avoid surface stalling. C.3 - gain and phase margins of 8dB and 35° respectively should be observed on the bow and stern plane control channels (one-loop at a time analysis), to provide some form of robustness in presence of parametric uncertainties; C.4 - the bandwiths of the bow and stern plane actuation channels should not exceed 5radj s in order to not violate the normal maximum bandwith of the DC motors that drive the control surfaces.
Fig. 4. Settling time (in seconds) as a function of control surface sizes; Vt = 2.5mj s - level curves
The results of the constrained optimization procedure are summarized in Figure 4, which shows the evolution of the minimum settling time achievable as a function of control surface sizes for the case when the AUV is moving at 2.5mj s .. The overall trend can be summarized by saying that the larger the control surfaces are , the smaller the settling time is. However, as seen in the next section, other requirements will impose upper bounds on the surface sizes.
this issue, Stephen Boyd and his coworkers have proposed in (Boyd and Barratt, 1991 ; Barratt et al., 1987) a numerical optimization design method that can effectively solve the controller design problem for linear plants, there is, it allows finding a suitable controller that meets the design specifications if they are indeed achievable, or it determines that they are not. Their pioneering work led to a software package called QDES (Barratt et al., 1987) that implements that numerical optimization design method in the discrete time domain. The package developed builds on convex optimization techniques, and relies on an explicit parameterization of all the linear controllers that stabilize a given plant (Boyd and Barratt, 1991) .
5. CONTROL SURFACE DESIGN EXAMPLE The rest of the paper describes a design exercise in which the tools developed in the previous sections were applied to the resizing of the control surfaces for the MARIUS AUV. As an example, the following three phase mission was considered (see also Section 3):
The rest of this section describes how the QDesign methodology was used to determine the limits of performance of the AUV as a function of the control surface sizes. The first step in the methodology required linearizing the AUV model about the set of trim points that define the three phase mission example introduced before. For simplicity of presentation, it was assumed that the maneuverability requirements are only important when the vehicle is inspecting the seabed in straight, level flight . In this case, it is important that the vehicle be able to change depth quickly in order to avoid unforeseen obstacles. For a finite set of control surface configurations, the vehicle model was linearized about the trim point determined by cio. = (uo,O,O)T,Oo = 0, and (8 b ,8s ,T)T = (O,O , To)T, with Uo = 2.5mjs, and discretized at 10Hz using a zero order hold. The following constrained optimization problem was then posed and solved (for each surface configuration) using the QDES software package: Find a controller that minimizes the settling time of the A UV response to a 1m dive command, subject to the following constraints:
i) Diving to 600m depth at speed Vt = 1mj s and flight path angle, = -20° . Duration: 30m . ii) Vehicle on level flight, at speed Vt = 2.5mj s and flight path angle, = 0°. Duration: 15m. iii) Resurfacing at speed Vt = 1mj s and flight path angle, = +20°. Duration: 30m . The constraints on the bow and stern surface sizes so that the vehicle will be able to sustain a flight path angle I = 20° at Vt = 1mj shave been analyzed in Section 3 and are summarized in Figure 2. The curve of average propulsion power as a function of surface sizes is depicted in Figure 3. While on level flight, it is required that the vehicle avoid unforeseen obstacles or terrain features . Suppose this requirement converts into the specification that the vehicle have a settling time ts S 3s in response to a 1m depth command. The corresponding constraint on surface sizes can be easily extracted from Figure 4, which reflects also the additional constraints C.1-C.4 described in Section 4. Suppose that an additional constraint on open loop stability is desired for the case when the vehicle is on straight, level flight, at Vt = 2.5mj s and I = 0°. The corresponding constraints on the control surface sizes can be easily evaluated by
C.1 - the closed-loop system should exhibit zero steady-state error in response to constant depth commands (integral action required).
51
allows searching for better tradeoffs among the design objectives.
0.' 0.8
6. CONCLUSIONS
0.7L~_~:..:-.-.=lU--_CC .
The paper introduced a methodology for integrated design of AUV plant parameters and feedback controllers to meet mission performance requirements with minimum energy expenditure. The methodology addresses both static and dynamic mission requirements, and departs considerably from classical procedures since it does not restrict a priori the types of feedback controllers considered . Furthermore, it provides a powerful graphical procedure for surface sizing that allows assessing the tradeoffs among competing performance requirements. An application was made to the resizing of the bow and stern planes of an existing prototype AUV. The design example described illustrated two basic concepts: i} optimal surface sizes are mission dependent, and ii} for some mission scenarios, it is clearly advantageous to have both bow and stern planes instead of stern planes only. Future work will address the challenging problem of optimal AUV surface sizing for operation under wave disturbances.
0.6
.. ~?
."
o.• ~_______~ . ~ ~-~1
u
~
u
u
u Sh
u
V
~
~
1m 2 )
Fig. 5. Open Loop Stability Index: Level Curves -
Average power level cwve,..
ScttJing time level curve for 1=3s •.••..
C)-
0.•
0.8 \
0.7 0.6
!
."
0.5
Stability index level QU'VC for i=.(). 15 Trimming con...ltaint
,
,
, ~3
...-0.1.5. ,_
,
'37
. . .. +-O._l~ .
0.4
0.3 0.2
- - - - - -
~\-,
-
-t60- - - - - - - - - - - - - - - - - - - -
7. REFERENCES
0. ,~==3'~60E==========~ 0.1
0.2
0.3
0."
O.S
S"
0.6
0.7
0.'
Barratt, C., N. Khraishi and S. Norman (1987). UNIX QDES Manual Entry. Boyd, S. and C. Barratt (1991). Linear Controller Design: Limits of Performance. Prentice Hall, Inc. Englewood Cliffs, New Jersey. Encarna<;iio, P. (1996). Convex Optimization Techniques with Applications to the Integrated Design of Underwater Robotic Vehicles and Controllers. Master's thesis. Dept. Electrical Engineering, 1ST. Lisboa, Portugal. Fossen, T . (1994). Guidance and Control of Ocean Vehicles. John Wiley & Sons. Chichester, England. Freudenberg, J . and D. Looze (1988). Frequency Domain Properties of Scalar and Multivariable Feedback Systems. Lecture Notes in Control and Information Sciences. SpringerVerlag. Fryxell, D. (1994). Modeling, Identification, Guidance and Control of an Autonomous Underwater Vehicle. Master's thesis. Dept. of Electrical Engineering, 1ST. Lisboa, Portugal. Henry, C. (1995) . On Optimum Turning Configurations with fixed-fin stabilization. IEEE Journal of Oceanic Engineering 20(4), 268275. Silvestre, C. and A. Pascoal (1997). Control of an AUV in the vertical and horizontal planes: system design and tests at sea. To appear in the Transactions of the Institute of Measurement and Control.
0.9
IIII~ I
Fig. 6. Control Surface Restrictions plotting the largest real part of the eigenvalues of the state space matrix of the vehicle's linearized model. This is done in Figure 5. Notice from the figure that without the stern plane the vehicle is open-loop unstable. It is also interesting to point out that in general, to maintain a given degree of open-loop stability, the bow and stern planes sizes are restricted to lie in a strip region. It is now possible to put together all the design constraints in Figure 6, and arrive at the optimal size of the bow and stern planes Sb = 0.3m 2 and S. = 0.52m 2 , for a required stability index of -0.15. This design example captures two key concepts: i} optimal surface sizing is mission dependent, and ii} in some cases, it is advantageous to use both bow and stern planes instead of simply stern planes. The latter case can be simply argued by considering the situation where there are no bow planes. In this case, it is impossible to meet the 3s settling time constraint while meeting the -0.15 stability index figure. This value of the stability index can still be achieved by allowing the settling time to increase to approximately 3.5s (see Figure 4). However, the average propulsion power will be increased in the process. The addition of the bow planes allows for an extra degree of freedom that
52