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Stable Backstepping Control of Marine Vehicles with Actuator Rate Limits and Saturation⁎
11th IFAC Conference on Control Applications in 11th IFAC Conference on Control Applications in Marine Systems, Robotics, and Vehicles 11th IFAC Conference on Control Applications in Marine Systems, Robotics, and Vehicles Available online at www.sciencedirect.com Opatija, Croatia, September 10-12, 2018 Marine Systems, Robotics, and Vehicles 11th IFAC Conference on Control in Opatija, Croatia, September 10-12,Applications 2018 Opatija,Systems, Croatia, September 10-12, 2018 Marine Robotics, and Vehicles Opatija, Croatia, September 10-12, 2018
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IFAC PapersOnLine 51-29 (2018) 262–267
Stable Backstepping Control of Marine Stable Stable Backstepping Backstepping Control Control of of Marine Marine Vehicles with Actuator Rate Limits and Stable Backstepping Control of Marine Vehicles with Actuator Rate Limits Vehicles with Saturation Actuator Rate Limits and and Vehicles with Saturation Actuator Rate Limits and Saturation KarlSaturation D. von Ellenrieder ∗∗
Karl D. von Ellenrieder ∗∗ Karl D. von Ellenrieder ∗ ∗ D. von Libera Ellenrieder a di Scienze eeKarl Technologie, Universit` a ∗ Facult` Facult` a di Scienze Technologie, Libera Universit` a di di Bolzano, Bolzano, 39100 39100 ∗ ∗ BZ Italy (e-mail: [email protected]). Facult` a di Scienze e Technologie, Libera Universit` a di Bolzano, 39100 BZ Italy (e-mail: [email protected]). ∗ Facult` a di Scienze e Technologie, Libera Universit` a di Bolzano, 39100 BZ Italy (e-mail: [email protected]). BZ Italy (e-mail: [email protected]). Abstract: Abstract: A A six six degree degree of of freedom freedom nonlinear nonlinear control control law law for for the the trajectory trajectory tracking tracking of of marine marine vehicles that operate in the presence of unknown time-varying disturbances, input saturation Abstract: A six degree of freedom nonlinear control law for the trajectory tracking of marine vehicles that operate in the presence of unknown time-varying disturbances, input saturation vehicles thatA rate operate in is the presencenonlinear of unknown time-varying input saturation Abstract: six degree of developed freedom control law observer for thedisturbances, trajectory tracking of marine and actuator actuator limits using disturbance and nonlinear backstepping. and rate limits is developed using aaa disturbance observer and backstepping. and actuator limits using disturbance observer and nonlinear nonlinear backstepping. vehicles that rate operate in is thedeveloped presenceestimates of unknown time-varying disturbances, input saturation The disturbance observer provides of the unknown time-varying disturbances and The disturbance observer provides estimates of the unknown time-varying disturbances and and actuator rate limits is developed using a disturbance observer andsaturation. nonlinear backstepping. a continuously differentiable function is employed to model input The uniform The disturbance observer provides estimates of the unknown time-varying disturbances and a continuously differentiable function is employed to model input saturation. The uniform The disturbance observer provides estimates of the tounknown time-varying disturbances and ultimate boundedness of all all signals in isthe the closed-loop controlinput system is proved. Trajectorya continuously differentiable function employed model saturation. The uniform ultimate boundedness of signals in closed-loop control system is Trajectoryultimate boundedness of all signals closed-loop control system is proved. proved. Trajectoryatracking continuously differentiable functionin underwater isthe employed to model input saturation. The uniform simulations of an autonomous vehicle demonstrate the performance of the tracking simulations of an autonomous underwater vehicle demonstrate the performance of tracking simulations of of an all autonomous vehicle demonstrate performance of the the ultimate boundedness signals in underwater the closed-loop control system the is proved. Trajectoryproposed controller. proposed controller. tracking proposedsimulations controller. of an autonomous underwater vehicle demonstrate the performance of the © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. proposed Keywords:controller. Actuator Saturation, Saturation, Actuator Actuator Rate Rate Limits, Limits, Backstepping, Backstepping, Marine Marine Vehicle Vehicle Control Control Keywords: Actuator Keywords: Actuator Saturation, Actuator Rate Limits, Backstepping, Marine Vehicle Control Keywords: Actuator Saturation, Actuator Rate Limits, Backstepping,functions Marine Vehicle Control 1. INTRODUCTION Hyperbolic Hyperbolic tangent tangent functions are are used used to to model model both both actuactu1. 1. INTRODUCTION INTRODUCTION Hyperbolic tangent functions are used to model both actuator saturation and rate limits. The authors remark ator saturation and rate limits. The authors remark that that ator saturation andfunctions rate limits. authors remark that INTRODUCTION tangent areThe used model both actutheir can to the control of Control design design for for1.marine marine vessels is is often often done done under under the the Hyperbolic their approach approach can be be extended extended to thetotracking tracking control of Control vessels saturation andsuch rate limits. The authors remark that nonlinear systems, as robotic manipulators. Control design foractuator marine vessels is often done under can the ator their approach can be extended to the tracking control of assumption that saturation and dynamics nonlinear systems, such as robotic manipulators. assumption that actuator saturation and dynamics can their approach can be extended to the tracking control of Control design foractuator marineinvessels is often done under can the nonlinear systems, such as robotic manipulators. be neglected. However, a real-world implementation assumption that saturation and dynamics non handling be neglected. However, in a real-world implementation The The use use of ofsystems, non adaptive adaptive control formanipulators. handling input input magmagsuch ascontrol roboticfor assumption that actuatorin saturation andimplementation dynamics can nonlinear the neglected. commanded control inputs calculated by aa tracking tracking be However, a real-world nitude and rate saturation in nonlinear MIMO systems the commanded control inputs calculated by The use of non adaptive control for handling input magnitude and rate saturation in nonlinear MIMO systems the commanded control inputs calculated by a tracking be neglected. However, in a real-world implementation controller are are constrained constrained by by the the maximum maximum forces forces and and The useand of non adaptive control for use handling input magis less common. Du et al. (2016) dynamic surface controller nitude rate saturation in nonlinear MIMO systems is less common. Du et (2016) use surface the commanded control such inputs calculated by a tracking moments that propellers, thrusters and controller are actuators, constrained by as maximum forces is less and common. Ducoupled et al. al. with (2016) use dynamic dynamic surface rate saturation in nonlinear MIMO systems moments that actuators, such asthe propellers, thrusters and nitude control techniques, a nonlinear disturbance techniques, coupled with a nonlinear disturbance controller areproduce constrained byand maximum forces rudders can can (Fossen Berge, 1997). This and can control moments that actuators, such asthe propellers, thrusters is less common. et al. with (2016) use dynamic surface observer, to aacoupled controller for dynamic positioning rudders produce (Fossen and Berge, 1997). This can control techniques, nonlinear disturbance observer, to design designDu controller for athe the dynamic positioning moments thatproduce actuators, such as possibly propellers, thrusters and cause degraded performance and instability of the rudders can (Fossen and Berge, 1997). This can control techniques, coupled with a nonlinear disturbance of surface vessels with actuator saturation and exogenous cause degraded performance and possibly instability of the observer, to design a controller for the dynamic positioning cause degraded performance possibly of can the of surface vessels with actuator saturation and exogenous rudders can produce (Fossenand Berge, instability 1997). the This controlled system, especially inand situations where forces observer, design aal. controller for the dynamic disturbances. Su (2017) propose aa nonlinear proporcontrolled system, especially in situations where the forces of surfaceto vessels with actuator saturation andpositioning exogenous disturbances. Su et et al. (2017) propose nonlinear proporcause degraded performance and possibly instability of the generated by environmental disturbances approach controlled system, especially in situations where the forces disturbances. Su (PD) et al. controller (2017) propose a dynamic nonlinear proporsurface vessels with actuator saturation and exogenous generated by environmental disturbances approach the of tional derivative for the positiontional derivative (PD) controller for the dynamic positioncontrolled system, especially indisturbances situations where the forces output capabilities of thrusters (Sarda et al., 2016) or when generated by environmental approach the disturbances. Su et al. (2017) propose a nonlinear proporing of surface vessels, including both actuator saturation output capabilities of thrusters (Sarda et al., 2016) or when tional derivative (PD) controller for the dynamic positionof surface vessels, including both actuator saturation et in generated by environmental disturbances approach the a vehicle’s vehicle’s mission requires it to to operate wide range range of ing output capabilities of thrusters (Sarda al.,aa2016) or when tional derivative (PD)input controller for theactuator dynamic positionand faults. However, rate limits and the exogenous a mission requires it operate in wide of ing of surface vessels, including both saturation adifferent vehicle’s mission requires it to(Sarda operate wide range of and faults. However, input rate limits and the exogenous output capabilities of thrusters et in al.,a2016) or when orientations for which which its actuator configuration ing of surface vessels, including both actuator disturbances are not different orientations for its actuator configuration and faults. However, input rate limits and the saturation exogenous disturbances are not addressed. addressed. a vehicle’s mission requires it to operate in a wide range of may not be optimal (Bertaska and von Ellenrieder, 2018). different orientations for which its actuator configuration disturbances are not addressed. faults. However, input rate limits and the exogenous may not be optimal (Bertaska and von Ellenrieder, 2018). and different orientations for whichand its von actuator configuration may not be optimal (Bertaska Ellenrieder, 2018). Here, Here, non-adaptive non-adaptive backstepping and aa nonlinear nonlinear disturdisturare notbackstepping addressed. and Mostnot recent work in in(Bertaska which actuator actuator saturation and rate disturbances may be optimal and vonsaturation Ellenrieder, 2018). bance observer are used to develop a control law takes Most recent work which and rate Here, non-adaptive backstepping and a nonlinear bance observer are used to develop a control law that thatdisturtakes response are included during nonlinear multiple input mulMost recent work in which actuator saturation and rate bance observerunknown are used to develop and a control law thatdisturtakes non-adaptive backstepping a nonlinear response are included during nonlinear multiple input mul- Here, into account time-varying disturbances, input into account unknown time-varying disturbances, input Most recent work in during which actuator saturation and rate tiple output (MIMO) control design involves the use of response are included nonlinear multiple input mulobserver are used to develop a The control law that takes saturation, and actuator rate limits. approach is novel tiple output (MIMO) control design involves the use of bance into account unknown time-varying disturbances, input and actuator rate limits. The approach is novel response are included during nonlinear multiple input muladaptive and/or neural network control techniques. For tiple output (MIMO) control design involves the use of saturation, into account unknown time-varying disturbances, input in that it does not rely on adaptive control approaches adaptive and/or neural network control techniques. For saturation, and actuator rate limits. The approach is novel that it does not rely on adaptive control approaches adaptive and/or neuralcontrol control techniques. tiple output (MIMO) design involves useFor of in example, Rezazadegan etnetwork al. (2015) (2015) explore the the trajectory in that it does not rely rate on adaptive control approaches saturation, andtoactuator limits. The approach is while novel for robustness disturbances/unmodeled dynamics, example, Rezazadegan et al. explore the trajectory for robustness to disturbances/unmodeled dynamics, while adaptive and/or neuraletnetwork control techniques. For tracking of an underactuated autonomous underwater veexample, Rezazadegan al. (2015) explore the trajectory for that robustness tonot disturbances/unmodeled dynamics, while it does rely on adaptive control approaches also handling both input magnitude and rate limits. tracking of an underactuated autonomous underwater ve- in handling both input magnitude and rate limits. example, Rezazadegan et al.of(2015) explore the trajectory hicle (AUV) (AUV) inunderactuated six degrees degrees freedom withunderwater input magnitracking of anin autonomous ve- also for to disturbances/unmodeled dynamics, hicle six of freedom with input magnialsorobustness handling both input magnitude and rate limits. while hicle (AUV) six degrees of autonomous freedom withunderwater input magnitracking of aninunderactuated ve- also handling both input magnitude and rate limits. tude saturation using adaptive backstepping (exogenous 2. PROBLEM FORMULATION tude saturation using adaptive backstepping (exogenous 2. tude saturation using backstepping hicle (AUV) in degrees with disturbances do six not actadaptive on of thefreedom system). Cui input et(exogenous al. magni(2017) 2. PROBLEM PROBLEM FORMULATION FORMULATION disturbances do not act on the system). Cui et al. (2017) tude saturation using adaptive backstepping (exogenous propose a combination of neural network, reinforcement disturbances do not act on the system). Cui et al. (2017) 2. PROBLEM FORMULATION propose a combination of neural network, reinforcement 2.1 Governing equations equations and and assumptions assumptions disturbances do notand act adaptive on neural the system). Cuithe et trajectory al. (2017) 2.1 Governing learning techniques techniques control for propose a combination of network, reinforcement learning and adaptive control for the trajectory 2.1 Governing equations and assumptions learning techniques and adaptive control for the trajectory propose aofcombination of neural network, reinforcement tracking a fully-actuated AUV in the horizontal plane 2.1 Governing equations of andmotion assumptions tracking of a fully-actuated AUV in the horizontal plane Consider the equations tracking of a fully-actuated AUV inboth the actuator horizontal plane Consider the equations of motion for for aa marine marine vehicle vehicle learning techniques and adaptive control the trajectory with disturbances, and saturawith exogenous exogenous disturbances, and both for actuator satura- Consider maneuvering in (mostly) calm, flat water, which be the equations of motion for a vehicle which can can be tracking ofdeadzone. a fully-actuated AUV inboth theuse horizontal plane maneuvering in (mostly) calm, flat water,marine tion and Zou et al. (2016) backstepping with exogenous disturbances, and actuator saturamaneuvering in (mostly)ofcalm, flat for water, which can be tion and deadzone. Zou et al. (2016) use backstepping Consider the equations motion a marine vehicle represented as represented as with exogenous disturbances, and both actuator saturacoupled with robust and adaptive control techniques to tion and deadzone. Zou et al. (2016) use backstepping coupled with robust and adaptive control techniques to maneuvering in (mostly) calm, flat water, which can be ˙ η = J (η)v (1) represented as coupled with robusttracking and et adaptive control to η˙ = J (η)v (1) tion deadzone. Zou al. (2016) use backstepping designand an attitude controller for techniques spacecraft in design an attitude tracking controller for spacecraft in represented as ˙ η = J (η)v (1) coupled with robust and adaptive control to three of with exogenous disturbances. design an attitude tracking controller for techniques spacecraft in and and three degrees degrees of freedom freedom with exogenous disturbances. η˙ = J (η)v (1) design an attitude tracking controller for spacecraft in and M (2) three degrees of freedom with exogenous disturbances. This work was sponsored, in part, by the US National Science Mv v˙˙ + + C(v)v C(v)v + + D(v)v D(v)v + + g(η) g(η) = =τ τ (u) (u) + +d d (2) and ˙ M v + C(v)v + D(v)v + g(η) = τ (u) + d (2) This degrees work was of sponsored, part, exogenous by the US National Science three freedomin with disturbances. in body-fixed coordinates (Fossen, 2011). The terms apFoundation (Award #1526016). This work was sponsored, in part, by the US National Science in body-fixed coordinates (Fossen, 2011). The terms apFoundation (Award #1526016). ˙ M v + C(v)v + D(v)v + g(η) = τ (u) + d (2) pearing in equations are in 1. n in body-fixed coordinates 2011). The terms The author also sponsored, with the Dept. Ocean Mechanical Engineering, Foundation (Award #1526016). This workis in part, by& US National Science pearing in these these equations (Fossen, are defined defined in Table Table 1. Here Hereapn The author is was also with the Dept. Ocean &the Mechanical Engineering, in body-fixed coordinates (Fossen, 2011). The terms apFlorida Atlantic University, DaniaOcean Beach,&FL 33004 USA is the number of degrees of freedom (DOF) of the motion pearing in these equations are defined in Table 1. Here n The author is also with the Dept. Mechanical Engineering, Foundation (Award #1526016). Florida Atlantic University, Dania Beach, FL 33004 USA is the number of degrees of freedom (DOF) of the motion Florida Atlantic University, DaniaOcean Beach,&FL 33004 USA is the number of equations degrees of are freedom (DOF) of the pearing in these defined in Table 1. motion Here n The author is also with the Dept. Mechanical Engineering, Florida Atlantic University, Dania Beach, FL 33004 USA is the number ofLtd. degrees of freedom 2405-8963 © 2018 2018, IFAC (International Federation of Automatic Control) by Elsevier All rights reserved. (DOF) of the motion Copyright © IFAC 262 Hosting Copyright 2018 IFAC 262 Control. Peer review© responsibility of International Federation of Automatic Copyright © under 2018 IFAC 262 10.1016/j.ifacol.2018.09.513 Copyright © 2018 IFAC 262
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
Definition 2. Let the measure of tracking be ˜˙ + Λ˜ η. (6) s := η˙ − η˙ r = η Assumption 3. The plant is intput to state stable (ISS). Assumption 4. The system of actuators has the following properties:
Table 1. Variables used in (1) and (2). Term
Dimension
Description
M
Rn
Inertia tensor
C(v)
Rn × Rn
Coriolis and centripetal matrix Hydrodynamic damping matrix
g(η)
Rn × R n
Rn
Gravity and buoyancy forces
J(η)
Transformation matrix
v
Rn × Rn
Rn
Velocity/angular rate vector
η
Rn
Position/attitude vector
u
Rn
Vector of actuator states
τ (u)
Rn
Actuator forces/moments
d
Rn
Vector of disturbances
D(v)
×
Rn
(including added mass effects) (including added mass effects)
and r is the number of actuators. In general, a marine craft with actuation in all DOFs, such as an underwater vehicle, requires a n = 6 DOF model for model-based controller and observer design, while ship and semi-submersible control systems can be designed using an n = 3, or 4 DOF model. In 6 DOF η is a composite vector of translations in 3 DOF and Euler angle rotations in 3 DOF. Thus, in 6 DOF, its dimensions are often denoted as R3 × S 3 . Assumption 1. In the following formulation, the coordinate transformation matrix J (η), which is used to convert the representation of vectors between a body-fixed coordinate system and an Earth-fixed North-East-Down coordinate system, is based on the use of Euler angles. The transformation matrix J (η) has singularities at pitch angles of θ = ±π/2. Here, it is assumed that |θ| < π/2. Remark 1. The singularities at θ = ±π/2 are not generally a problem for surface vessels. However, an underwater vehicle may occassionally approach these singularities if performing extreme maneuvers. In such cases, as suggested in Fossen (2011), the kinematic equations could be described by two Euler angle representations with different singularities and the singular points can be avoided by switching between the representations. The terms M , C(v) and D(v) have the useful mathematical properties shown in Table 2. The control objective is to make the system track a desired trajectory η d . (3)
Assumption 2. The trajectory η d and its derivatives η d , ¨ d , and η˙ d are smooth and bounded. η Definition 1. Let ˜ := η − η d (3) η be the Earth-fixed tracking error. Then define the reference trajectories in body-fixed and Earth-fixed coordinates as η (4) η˙ r := η˙ d − Λ˜ and (5) v r := J −1 (η)η˙ r , where Λ > 0 is a diagonal design matrix. Table 2. Mathematical properties of M , C(v) and D(v). M = M T > 0 ⇒ xT M x > 0, ∀x = 0
C(v) = −C T (v) ⇒ xT C(v)x = 0, ∀x D(v) > 0 ⇒
1 T x [D(v) 2
263
+ D T (v)]x > 0, ∀x = 0
263
a) The vehicle is fully-actuated or overactuated (r ≥ n). b) The vector τ (u) ∈ Rn is the result of the combined effects of a control allocation scheme and r actuators. A control allocation scheme would generally accept a control input uc ∈ Rn and then send a control signal to each of the r actuators. Each actuator would have its own rate limit and saturation response. When r > n it is assumed that the combined effects of control allocation and overactuation can be modeled as a set of n control inputs uc , which are rate-limited to a set of n actuator states u. Owing to saturation, the vector of forces and moments output by the actuators is τ (u). c) Integrator backstepping, requires smoothly differentiable functions. Each component of τ (u) is modeled by a smooth function bounded by uiMax := supui τi (ui ) and uiMin := inf ui τi (ui ). The Jacobian of τ (u) is smooth and nonsingular, such that its inverse exists. Each component of τ (u) is only dependent on the corresponding component of the actuator states, i.e. ∂τi /∂uj = 0, ∀i = j such that ∂τ /∂u is diagonal. d) The saturation limits and the rate limit of each component of τ (u) are independent. The time response of the combined set of actuators is governed by the first order differential equation (7) T u˙ = −u + uc where T = T T > 0 ∈ Rn × Rn is a diagonal matrix of time constants (see Fossen and Berge (1997), for example). e) The error between the forces/moments commanded by the controller and the actual forces/moments produced by the system of combined actuators, uc − τ (u), and its time derivative, are bounded.
Remark 2. The saturation function ui , |ui /uiM | < 1, sat(ui /uiM ) := uiM sgn(ui ), |ui /uiM | ≥ 1,
(8)
is often used to model actuator saturation. However, it is discontinuous. As shown in Wen et al. (2011), the hyperbolic tangent function is smooth and can be useful for modeling actuator saturation when using integrator backstepping techniques. Thus, a convenient form for τ (u) could be ui τi (ui ) ≈ uiM tanh (9) uiM (see Fig. 1). Assumption 5. The disturbance vector d in (2) consists of two types of disturbances d = de + da , where de is a vector of unknown external disturbances, such as wind or current. As in Wen et al. (2011), the term da is modeled as a bounded disturbance-like error that arises from approximating the true actuator saturation with the function τ (u) (Fig. 1). The components of d and their time derivatives are unknown and time-varying, yet bounded. There exists a positive constant ρ, such that
IFAC CAMS 2018 264 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
Vectorial backstepping Step 1: Define the virtual control signal (14) η˙ = J (η)v := s + α1 , where α1 is a smoothly continuous function that stabilizes the system at the origin, which can be chosen as α1 = η˙ r = η˙ d − Λ˜ η (15) where Λ > 0 is a diagonal matrix. Then, (14) can be written as η ˜˙ = −Λ˜ η + s. (16) Next, consider the Lyapunov function candidate 1 T ˜ K pη ˜. V1 = η (17) 2 Its time derivative is ˜ T K pη ˜ T K ps ˜˙ = −˜ η T K p Λ˜ η+η (18) V˙ 1 = η where K p = K Tp > 0 is a design matrix.
Fig. 1. Use of the hyperbolic tangent function to model actuator saturation. ˙ d(t) ≤ ρ, (10) where · represents the 2-norm of a vector or matrix. 2.2 Control design The tracking controller is designed using the maneuvering equations (2) transformed into an Earth-fixed coordinate system, which can be written as M η (η)¨ η + C η (v, η)η˙ + D η (v, η)η˙ + g η (η) (11) = J −T (η)[τ (u) + d], where M η (η) = M Tη (η) > 0, T 1 ˙ (12) M η (η) − C η (v, η) s = 0, ∀v, η, s s 2 D η (v, η) > 0, and the relationships between the terms M , C(v) and D(v) in the Earth-fixed and body-fixed coordinate systems are shown in Table 3 Fossen (2011). The product Table 3. Relations between Earth-fixed and body-fixed systems. M η (η) := J −T (η)M J −1 (η)
+d − M v˙ r − C(v)v r − D(v)v r −g(η) .
(19)
(20)
(21)
Select the virtual control law ˆ τ (u) := z + α2 − d, (22) where z is a new state variable, α2 is a stabilizing function ˆ is an estimate of the disturbances. Let and d α2 = M v˙ r + C(v)v r + D(v)v r + g(η) (23) ˜. −J T (η)K d s − J T (η)K p η Then V˙ 2 becomes V˙ 2 = −˜ η T K p Λ˜ η − sT [D η (v, η) + K d ] s (24) ˜ , +sT J −T (η) z − d
where
˜ := d ˆ−d d is the disturbance estimation error.
C η (v, η) := J −T (η)[C(v) −1 ˙ (η) − M J −1 (η)J(η)]J
D η (v, η) := J −T (η)D(v)J −1 (η) g η (η) := J −T (η)g(η)
M η (η)s˙ is an important quantity that will be used in the controller derivation. It can be written as M η (η)s˙ = −C η (v, η)s − D η (v, η)s +J −T (η) [τ (u) + d − M v˙ r
Step 2: Consider the Lyapunov function candidate 1 V2 = V1 + sT M η (η)s. 2 Its time derivative is ηT K η V˙ 2 = −˜ p Λ˜ 1 ˙ T ˜ + M η (η)s˙ + M η (η)s . +s K p η 2 Using (12) and (13), this can be rewritten as η T K p Λ˜ η − sT D η (v, η)s V˙ 2 = −˜ ˜ + τ (u) +sT J −T (η) J T (η)K p η
(13)
−C(v)v r − D(v)v r − g(η)] . 264
Step 3: Consider the Lyapunov function candidate 1 1 ˜T ˜ V3 = V2 + z T z + d d. 2 2 Taking its time derivative gives η T K p Λ˜ η − sT [D η (v, η) + K d ] s V˙ 3 = −˜ +z T z˙ + J −1 (η)s ˜ T −J −1 (η)s + d ˜˙ . +d
(25)
(26)
(27)
We will design a control law for z˙ and a disturbance ˜˙ such that V˙ 3 ≤ 0. observer d
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
Stabilizing control law Taking the time derivative of (22) we have ∂τ d ˆ˙ ˙ 2 − d. [τ (u)] = u˙ = z˙ + α (28) dt ∂u Using (7), (28) can be rewritten as ∂τ −1 ˆ˙ = −u + uc . ˙2−d z˙ + α T u˙ = T ∂u Solving for z˙ gives ∂τ −1 ˆ˙ ˙ 2 + d. T (uc − u) − α z˙ = ∂u In order to make the term z T z˙ + J −1 (η)s ≤ 0 in we take z˙ = −J −1 (η)s − K z z, where K z = K Tz > 0 is a design matrix. Equating and (31), the stabilizing controller is found to be ∂τ −1 ˆ˙ − J −1 (η)s − K z z . ˙2−d α uc = u + T ∂u
(29)
(30)
(31) (30) (32)
and −g(η) + τ (u) + K 0 M v] ,
(34)
where K 0 = K T0 > 0. Taking the derivative of (33) and using equations (34) and (2) gives ˜ ˆ˙ = −K 0 d. (35) d Using the derivative of (25), (35) can be written as ˜˙ = −K 0 d ˜ − d. ˙ d (36) Select the Lyapunov function candidate 1 ˜T ˜ 1 ˜ 2 Vd0 = d . (37) d = d 2 2 Taking the derivative of (37) and using (36) gives ˜˙ = −d ˜T K 0d ˜T d ˜−d ˜ T d. ˙ V˙ d0 = d Let λmin (·) be the minimum eigenvalue of a matrix and Cd := ρ2 /2. Using (10) and applying Youngs Product Inequality for vectors to the last term in V˙ d0 yields ˜T d ˜T K 0d ˜ + 1d ˜ + 1 d˙ T d, ˙ V˙ d0 ≤ −d 2 2 ˜ 2 + 1 d ˙ 2, ˜ 2 + 1 d ≤ −λmin (K 0 ) d 2 (38) 2 1 ˜ 2 ρ2 d + , ≤ − λmin (K 0 ) − 2 2 ≤ −2αVd0 + Cd , where (39) λmin (K 0 ) > 1/2 is selected to ensure that α > 0. As proven in Do (2010) and Du et al. (2016), the practical stability of the ˜ disturbance observer is ensured since both Vd0 (t) and d are globally uniformly ultimately bounded such that Cd Cd −2αt 0 ≤ Vd0 (t) ≤ + Vd0 (0) − e , 2α 2α 265
Cd −2αt Cd + 2 Vd0 (0) − e . α 2α The disturbance error settles to within a compact set that can be made arbitrarily small by appropriately selecting K 0 , so that (39) is satisfied. ˜ ≤ d
System stabilization Using (31) and (36), (27) can be reduced to η T K p Λ˜ η − sT [D η (v, η) + K d ] s V˙ 3 = −˜ ˜T K 0d ˜ −z T K z z − d ˜ T J −1 (η)s + d˙ . −d
(27)
Disturbance observer A disturbance observer is constructed using the approach presented in Do (2010) and Du et al. (2016). Its derivation is repeated here so that the presentation of the results to follow is complete. Let ˆ = q(t) + K 0 M v (33) d ˙ q(t) = −K 0 q(t) − K 0 [−C(v)v − D(v)v
and
265
(40)
The last term in this expression can be rewritten using Young’s Product Inequality for vectors, so that η T K p Λ˜ η − zT K z z V˙ 3 ≤ −˜ −sT [D η (v, η) + K d ] s
1 + sT J −T (η)J −1 (η)s 2 T ˜ ˜ + 1 d˙ T d, ˙ −d [K 0 − 1] d 2 η 2 − λmin (K z ) z 2 ≤ −λmin (K p Λ) ˜ −λmin [D η (v, η) + K d ] s 2 1 + λmax J −T (η)J −1 (η) s 2 2 ˜ 2 + Cd , −λmin (K 0 − 1) d
(41)
λmin (K p Λ) λmax (K p ) ˜ η 2 λmax (K p ) λmin [D η (v, η) + K d ] λmax (M η ) s 2 − λmax (M η ) λmax J −T (η)J −1 (η) λmax (M η ) s 2 + 2λmax (M η )
≤ −
−λmin (K z ) z 2
˜ 2 + Cd . −λmin (K 0 − 1) d Since M and D(v) are real symmetric tensors they can be diagonalized to determine the eigenvalues corresponding their principal axes. The maximum eigenvalue and the minimum eigenvalue of each diagonalized tensor will correspond to the maximum and minimum possible eigenvalues of each tensor under rotations produced by combinations of transformations with J (η), J T (η), J −1 (η) and J −T (η). Let M principal and D principal be the inertial and drag tensors expressed in their respective principal axes coordinate systems. Also let λM min := λmin (M principal ), λM max := λmax (M principal ), λDmin := λmin (D principal ) and λDmax := λmax (D principal ). Then, λM min ≤ λmin (M η ), λM max ≥ λmax (M η ),
λDmin ≤ λmin [D η (v, η)] ,
λDmax ≥ λmax [D η (v, η)] .
(42)
IFAC CAMS 2018 266 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
Since |θ| < π/2, of J (η) it can be from the definition −T −1 seen that λmax J (η)J (η) = 1 + sin(θ) < 2 (see Assumption 1.Using (42), (41) can be rewritten as λmin (K p Λ) λmax (K p ) ˜ η 2 V˙ 3 ≤ − λmax (K p ) λDmin + λmin (K d ) − 1 − λM max s 2 (43) λM max −λmin (K z ) z
2
˜ 2 + Cd . −λmin (K 0 − 1) d From (26), the inequality 1 1 η 2 + λM max s 2 V3 ≤ λmax (K p ) ˜ 2 2 (44) 1 1 ˜ 2 2 + z + d 2 2 can be constructed. Using this inequality for V3 , (41) can be written as (45) V˙ 3 ≤ −2µV3 + Cd , where λmin (K p Λ) , λmin (K z ), µ := min λmax (K p ) λmin (K 0 − 1) , (46) λDmin + λmin (K d ) − 1 . λM max Consider the compact set ˜ : V3 ≤ B0 , ∀B0 > 0} ∈ R24 . Π = {(˜ η , s, z, d)
The design parameters K p , Λ, K z , K 0 , K d can be chosen such that λmin (K 0 ) > 1, (47) (48) λmin (K d ) > 1 − λDmin , and Cd . (49) µ> 2B0 ˜ = B0 . From (45), V˙ 3 ≤ −2µB0 + Cd on V3 (˜ η , s, z, d) Further, applying (49) gives the stronger inequality V˙ 3 < 0. ˜ : V3 ≤ B0 } is Thus, the compact set Π = {(˜ η , s, z, d) invariant. If V3 (0) ≤ B0 , then V3 (t) ≤ B0 and (45) holds for all t > 0. Remark 3. Inequality (47) imposes a consistent, but stronger, condition than (39) on the selection of K 0 . Theorem 1. Consider the closed-loop system consisting of: a) the plant (1)–(2) with unknown time-varying disturbances, input saturation and input rate limits, as characterized under Assumptions 1–5; b) the virtual control law (22) and the stabilizing controller (32); and c) the disturbance observer (33)–(34). For all V3 (0) ≤ B0 , where ˜ = η − ηd B0 is a positive constant, the tracking error η settles to within the compact set η ∈ R6 | ˜ η ≤ ζη˜ , ζη˜ > Cd /µ}. Πη˜ = {˜ The set Πη˜ can be made arbitrarily small by appropriately adjusting the design parameters K p , Λ, K z , K 0 , and K d such that (47)–(49) are satisfied. All signals in the trajectory tracking closed-loop control system are guaranteed to be uniformly ultimately bounded. 266
Proof. The solution to (45) is Cd Cd −2µt + V3 (0) − e 0 ≤ V3 (t) ≤ . (50) 2µ 2µ Thus, V3 (t) is uniformly ultimately bounded for all V3 (0) ≤ ˜ ˜ , s, z, d B0 . From (17), (19) and (26) it can be seen that η are also uniformly ultimately bounded for all V3 (0) ≤ B0 .
From (16), (18), (20), (25), (35) and the boundedness of ˆ are uniformly ultimately d, it can be seen that η and d bounded ∀V3 (0) ≤ B0 . Thus, all signals in the closed loop control system are uniformly ultimately bounded for all V3 (0) ≤ B0 . 3. SIMULATIONS The performance of the control law (32) is investigated using a 6 DOF simulation of an AUV tracking two periods of a sine-shaped trajectory (7.3 m East-West lateral deviations and 111 m Northward travel) at constant depth. Initially, the vehicle is positioned at the origin of a North-East-Down coordinate system with its nose pointing North; the controller is designed to keep the longitudinal axis of the AUV tangent to the trajectory. The manuevering coefficients and physical characteristics of the AUV were obtained from Prestero (2001). The only differences are that here it is assumed the maximum propeller thrust is 7 N, the center of gravity is [xg yg zg ]T = [0 0 0]T and the center of buoyancy is [xb yb zb ]T = [0 0 − 2]T cm. The saturation limits of the actuators are τsat = ±[7.00 9.49 9.49 0.95 4.75 4.75]T (forces are in N and moments are in N-m – see Fig. 4). All six actuators have a time constant of 0.25 secs. Disturbances are modeled as first order Markov processes of the form b˙ = −Tb b + an wn , where b ∈ R6 is a vector of bias forces and moments, Tb ∈ R6×6 is a diagonal time constant matrix, wn ∈ R6 is a vector of zero-mean Guassian white noise, and an ∈ R6×6 is a diagonal matrix that scales the amplitude of wn . The integral of time multiplied by absolute error (ITAE) is provided in Table 4 for the following cases: (1) the backstepping controller with no disturbance; (2) the backstepping controller with disturbances characterized by Tb = 103 · 16×6 , an = [0.35 0.47 0.47 0.05 0.24 0.24]T and b0 = −an ; (3) the backstepping controller with the same disturbances scaled by a factor of 5 (see Figs. 2–4); and (4) a manually-tuned, nonlinear proportional integral derivative (PID) controller (with disturbance observer). The PID controlled system is unstable for the same disturbances used in cases 2 and 3 above. The backstepping controller is fairly robust and outperforms the PID controller, even in the presence of strong disturbances. Table 4. ITAE of N, E, D position in m-s2 and φ, θ, ψ orientation in rad-s2 . Controller
N
E
D
φ
θ
ψ
BS (case 1)
3.80
1.02
0.00
0.00
0.00
0.19
BS (case 2)
4.87
1.49
0.35
0.08
0.45
0.62
BS (case 3)
33.5
24.5
1.80
0.53
2.28
12.4
PID (case 4)
166
643
0.00
0.00
0.00
340
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
observer and nonlinear backstepping. The result can be easily applied to systems with fewer degrees of freedom, such as 3 DOF surge-sway-yaw, or 4 DOF surge-sway-yawroll models of unmanned surface vessels.
0.5 N [m] 0.4
E [m] D [m]
0.3
The use of nonlinear backstepping for marine vehicle control has been in use for quite some time. With only slight modifications, the approach developed here can be applied to existing systems.
[rad] [rad] [rad]
0.2 0.1
The implementation of the control law requires use of the derivative of α2 . Noisy measurement of the states η, v and u can cause problems in the calculation of this derivative. A computational approach that can be used to circumvent this issue for some surface vessels can be found in Fossen and Berge (1997). Further, many recent commercially produced inertial measurement units provide direct measurements of several of the terms required for the computation of α2 .
0 -0.1 -0.2 -0.3 0
20
40
60
80
100
267
120
Fig. 2. Case 3 Earth-fixed tracking error.
REFERENCES 2
1
0
-1
-2
-3
-4
-5 0
20
40
60
80
100
120
100
120
Fig. 3. Case 3 estimated disturbances.
8 6 4 2 0 -2 -4 -6 -8 0
20
40
60
80
Fig. 4. Case 3 actuator outputs. 4. CONCLUDING REMARKS Here a 6 DOF nonlinear control law for the trajectory tracking of marine vehicles that operate in the presence of unknown time-varying disturbances, input saturation and actuator rate limits is developed using a disturbance 267
Bertaska, I.R. and von Ellenrieder, K.D. (2018). Experimental evaluation of supervisory switching control for unmanned surface vehicles. IEEE J. Oceanic Engineering, 1–22. doi:10.1109/JOE.2018.2802019. Cui, R., Yang, C., Li, Y., and Sharma, S. (2017). Adaptive neural network control of auvs with control input nonlinearities using reinforcement learning. IEEE Trans. Syst., Man, Cybern., Syst., 47(6), 1019–1029. Do, K.D. (2010). Practical control of underactuated ships. Ocean Engineering, 37(13), 1111–1119. Du, J., Hu, X., Krsti´c, M., and Sun, Y. (2016). Robust dynamic positioning of ships with disturbances under input saturation. Automatica, 73, 207–214. Fossen, T.I. (2011). Handbook of marine craft hydrodynamics and motion control. John Wiley & Sons. Fossen, T.I. and Berge, S.P. (1997). Nonlinear vectorial backstepping design for global exponential tracking of marine vessels in the presence of actuator dynamics. In Proc. 36th IEEE Conf. Decis. Control, 4237–4242. Prestero, T.J. (2001). Verification of a six-degree of freedom simulation model for the REMUS autonomous underwater vehicle. Ph.D. thesis, Massachusetts Institute of Technology. Rezazadegan, F., Shojaei, Sheikholeslam, and Chatraei, A. (2015). A novel approach to 6-DOF adaptive trajectory tracking control of an AUV in the presence of parameter uncertainties. Ocean Engineering, 107, 246–258. Sarda, E.I., Qu, H., Bertaska, I.R., and von Ellenrieder, K.D. (2016). Station-keeping control of an unmanned surface vehicle exposed to current and wind disturbances. Ocean Engineering, 127, 305–324. Su, Y., Zheng, C., and Mercorelli, P. (2017). Nonlinear PD fault-tolerant control for dynamic positioning of ships with actuator constraints. IEEE/ASME Trans. Mechatronics, 22(3), 1132–1142. Wen, C., Zhou, J., Liu, Z., and Su, H. (2011). Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance. IEEE Trans. Autom. Control, 56(7), 1672–1678. Zou, A.M., Kumar, K.D., and Ruiter, A.H. (2016). Robust attitude tracking control of spacecraft under control input magnitude and rate saturations. Int. J. Robust Nonlin. Control, 26(4), 799–815.
ScienceDirect
IFAC PapersOnLine 51-29 (2018) 262–267
Stable Backstepping Control of Marine Stable Stable Backstepping Backstepping Control Control of of Marine Marine Vehicles with Actuator Rate Limits and Stable Backstepping Control of Marine Vehicles with Actuator Rate Limits Vehicles with Saturation Actuator Rate Limits and and Vehicles with Saturation Actuator Rate Limits and Saturation KarlSaturation D. von Ellenrieder ∗∗
Karl D. von Ellenrieder ∗∗ Karl D. von Ellenrieder ∗ ∗ D. von Libera Ellenrieder a di Scienze eeKarl Technologie, Universit` a ∗ Facult` Facult` a di Scienze Technologie, Libera Universit` a di di Bolzano, Bolzano, 39100 39100 ∗ ∗ BZ Italy (e-mail: [email protected]). Facult` a di Scienze e Technologie, Libera Universit` a di Bolzano, 39100 BZ Italy (e-mail: [email protected]). ∗ Facult` a di Scienze e Technologie, Libera Universit` a di Bolzano, 39100 BZ Italy (e-mail: [email protected]). BZ Italy (e-mail: [email protected]). Abstract: Abstract: A A six six degree degree of of freedom freedom nonlinear nonlinear control control law law for for the the trajectory trajectory tracking tracking of of marine marine vehicles that operate in the presence of unknown time-varying disturbances, input saturation Abstract: A six degree of freedom nonlinear control law for the trajectory tracking of marine vehicles that operate in the presence of unknown time-varying disturbances, input saturation vehicles thatA rate operate in is the presencenonlinear of unknown time-varying input saturation Abstract: six degree of developed freedom control law observer for thedisturbances, trajectory tracking of marine and actuator actuator limits using disturbance and nonlinear backstepping. and rate limits is developed using aaa disturbance observer and backstepping. and actuator limits using disturbance observer and nonlinear nonlinear backstepping. vehicles that rate operate in is thedeveloped presenceestimates of unknown time-varying disturbances, input saturation The disturbance observer provides of the unknown time-varying disturbances and The disturbance observer provides estimates of the unknown time-varying disturbances and and actuator rate limits is developed using a disturbance observer andsaturation. nonlinear backstepping. a continuously differentiable function is employed to model input The uniform The disturbance observer provides estimates of the unknown time-varying disturbances and a continuously differentiable function is employed to model input saturation. The uniform The disturbance observer provides estimates of the tounknown time-varying disturbances and ultimate boundedness of all all signals in isthe the closed-loop controlinput system is proved. Trajectorya continuously differentiable function employed model saturation. The uniform ultimate boundedness of signals in closed-loop control system is Trajectoryultimate boundedness of all signals closed-loop control system is proved. proved. Trajectoryatracking continuously differentiable functionin underwater isthe employed to model input saturation. The uniform simulations of an autonomous vehicle demonstrate the performance of the tracking simulations of an autonomous underwater vehicle demonstrate the performance of tracking simulations of of an all autonomous vehicle demonstrate performance of the the ultimate boundedness signals in underwater the closed-loop control system the is proved. Trajectoryproposed controller. proposed controller. tracking proposedsimulations controller. of an autonomous underwater vehicle demonstrate the performance of the © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. proposed Keywords:controller. Actuator Saturation, Saturation, Actuator Actuator Rate Rate Limits, Limits, Backstepping, Backstepping, Marine Marine Vehicle Vehicle Control Control Keywords: Actuator Keywords: Actuator Saturation, Actuator Rate Limits, Backstepping, Marine Vehicle Control Keywords: Actuator Saturation, Actuator Rate Limits, Backstepping,functions Marine Vehicle Control 1. INTRODUCTION Hyperbolic Hyperbolic tangent tangent functions are are used used to to model model both both actuactu1. 1. INTRODUCTION INTRODUCTION Hyperbolic tangent functions are used to model both actuator saturation and rate limits. The authors remark ator saturation and rate limits. The authors remark that that ator saturation andfunctions rate limits. authors remark that INTRODUCTION tangent areThe used model both actutheir can to the control of Control design design for for1.marine marine vessels is is often often done done under under the the Hyperbolic their approach approach can be be extended extended to thetotracking tracking control of Control vessels saturation andsuch rate limits. The authors remark that nonlinear systems, as robotic manipulators. Control design foractuator marine vessels is often done under can the ator their approach can be extended to the tracking control of assumption that saturation and dynamics nonlinear systems, such as robotic manipulators. assumption that actuator saturation and dynamics can their approach can be extended to the tracking control of Control design foractuator marineinvessels is often done under can the nonlinear systems, such as robotic manipulators. be neglected. However, a real-world implementation assumption that saturation and dynamics non handling be neglected. However, in a real-world implementation The The use use of ofsystems, non adaptive adaptive control formanipulators. handling input input magmagsuch ascontrol roboticfor assumption that actuatorin saturation andimplementation dynamics can nonlinear the neglected. commanded control inputs calculated by aa tracking tracking be However, a real-world nitude and rate saturation in nonlinear MIMO systems the commanded control inputs calculated by The use of non adaptive control for handling input magnitude and rate saturation in nonlinear MIMO systems the commanded control inputs calculated by a tracking be neglected. However, in a real-world implementation controller are are constrained constrained by by the the maximum maximum forces forces and and The useand of non adaptive control for use handling input magis less common. Du et al. (2016) dynamic surface controller nitude rate saturation in nonlinear MIMO systems is less common. Du et (2016) use surface the commanded control such inputs calculated by a tracking moments that propellers, thrusters and controller are actuators, constrained by as maximum forces is less and common. Ducoupled et al. al. with (2016) use dynamic dynamic surface rate saturation in nonlinear MIMO systems moments that actuators, such asthe propellers, thrusters and nitude control techniques, a nonlinear disturbance techniques, coupled with a nonlinear disturbance controller areproduce constrained byand maximum forces rudders can can (Fossen Berge, 1997). This and can control moments that actuators, such asthe propellers, thrusters is less common. et al. with (2016) use dynamic surface observer, to aacoupled controller for dynamic positioning rudders produce (Fossen and Berge, 1997). This can control techniques, nonlinear disturbance observer, to design designDu controller for athe the dynamic positioning moments thatproduce actuators, such as possibly propellers, thrusters and cause degraded performance and instability of the rudders can (Fossen and Berge, 1997). This can control techniques, coupled with a nonlinear disturbance of surface vessels with actuator saturation and exogenous cause degraded performance and possibly instability of the observer, to design a controller for the dynamic positioning cause degraded performance possibly of can the of surface vessels with actuator saturation and exogenous rudders can produce (Fossenand Berge, instability 1997). the This controlled system, especially inand situations where forces observer, design aal. controller for the dynamic disturbances. Su (2017) propose aa nonlinear proporcontrolled system, especially in situations where the forces of surfaceto vessels with actuator saturation andpositioning exogenous disturbances. Su et et al. (2017) propose nonlinear proporcause degraded performance and possibly instability of the generated by environmental disturbances approach controlled system, especially in situations where the forces disturbances. Su (PD) et al. controller (2017) propose a dynamic nonlinear proporsurface vessels with actuator saturation and exogenous generated by environmental disturbances approach the of tional derivative for the positiontional derivative (PD) controller for the dynamic positioncontrolled system, especially indisturbances situations where the forces output capabilities of thrusters (Sarda et al., 2016) or when generated by environmental approach the disturbances. Su et al. (2017) propose a nonlinear proporing of surface vessels, including both actuator saturation output capabilities of thrusters (Sarda et al., 2016) or when tional derivative (PD) controller for the dynamic positionof surface vessels, including both actuator saturation et in generated by environmental disturbances approach the a vehicle’s vehicle’s mission requires it to to operate wide range range of ing output capabilities of thrusters (Sarda al.,aa2016) or when tional derivative (PD)input controller for theactuator dynamic positionand faults. However, rate limits and the exogenous a mission requires it operate in wide of ing of surface vessels, including both saturation adifferent vehicle’s mission requires it to(Sarda operate wide range of and faults. However, input rate limits and the exogenous output capabilities of thrusters et in al.,a2016) or when orientations for which which its actuator configuration ing of surface vessels, including both actuator disturbances are not different orientations for its actuator configuration and faults. However, input rate limits and the saturation exogenous disturbances are not addressed. addressed. a vehicle’s mission requires it to operate in a wide range of may not be optimal (Bertaska and von Ellenrieder, 2018). different orientations for which its actuator configuration disturbances are not addressed. faults. However, input rate limits and the exogenous may not be optimal (Bertaska and von Ellenrieder, 2018). and different orientations for whichand its von actuator configuration may not be optimal (Bertaska Ellenrieder, 2018). Here, Here, non-adaptive non-adaptive backstepping and aa nonlinear nonlinear disturdisturare notbackstepping addressed. and Mostnot recent work in in(Bertaska which actuator actuator saturation and rate disturbances may be optimal and vonsaturation Ellenrieder, 2018). bance observer are used to develop a control law takes Most recent work which and rate Here, non-adaptive backstepping and a nonlinear bance observer are used to develop a control law that thatdisturtakes response are included during nonlinear multiple input mulMost recent work in which actuator saturation and rate bance observerunknown are used to develop and a control law thatdisturtakes non-adaptive backstepping a nonlinear response are included during nonlinear multiple input mul- Here, into account time-varying disturbances, input into account unknown time-varying disturbances, input Most recent work in during which actuator saturation and rate tiple output (MIMO) control design involves the use of response are included nonlinear multiple input mulobserver are used to develop a The control law that takes saturation, and actuator rate limits. approach is novel tiple output (MIMO) control design involves the use of bance into account unknown time-varying disturbances, input and actuator rate limits. The approach is novel response are included during nonlinear multiple input muladaptive and/or neural network control techniques. For tiple output (MIMO) control design involves the use of saturation, into account unknown time-varying disturbances, input in that it does not rely on adaptive control approaches adaptive and/or neural network control techniques. For saturation, and actuator rate limits. The approach is novel that it does not rely on adaptive control approaches adaptive and/or neuralcontrol control techniques. tiple output (MIMO) design involves useFor of in example, Rezazadegan etnetwork al. (2015) (2015) explore the the trajectory in that it does not rely rate on adaptive control approaches saturation, andtoactuator limits. The approach is while novel for robustness disturbances/unmodeled dynamics, example, Rezazadegan et al. explore the trajectory for robustness to disturbances/unmodeled dynamics, while adaptive and/or neuraletnetwork control techniques. For tracking of an underactuated autonomous underwater veexample, Rezazadegan al. (2015) explore the trajectory for that robustness tonot disturbances/unmodeled dynamics, while it does rely on adaptive control approaches also handling both input magnitude and rate limits. tracking of an underactuated autonomous underwater ve- in handling both input magnitude and rate limits. example, Rezazadegan et al.of(2015) explore the trajectory hicle (AUV) (AUV) inunderactuated six degrees degrees freedom withunderwater input magnitracking of anin autonomous ve- also for to disturbances/unmodeled dynamics, hicle six of freedom with input magnialsorobustness handling both input magnitude and rate limits. while hicle (AUV) six degrees of autonomous freedom withunderwater input magnitracking of aninunderactuated ve- also handling both input magnitude and rate limits. tude saturation using adaptive backstepping (exogenous 2. PROBLEM FORMULATION tude saturation using adaptive backstepping (exogenous 2. tude saturation using backstepping hicle (AUV) in degrees with disturbances do six not actadaptive on of thefreedom system). Cui input et(exogenous al. magni(2017) 2. PROBLEM PROBLEM FORMULATION FORMULATION disturbances do not act on the system). Cui et al. (2017) tude saturation using adaptive backstepping (exogenous propose a combination of neural network, reinforcement disturbances do not act on the system). Cui et al. (2017) 2. PROBLEM FORMULATION propose a combination of neural network, reinforcement 2.1 Governing equations equations and and assumptions assumptions disturbances do notand act adaptive on neural the system). Cuithe et trajectory al. (2017) 2.1 Governing learning techniques techniques control for propose a combination of network, reinforcement learning and adaptive control for the trajectory 2.1 Governing equations and assumptions learning techniques and adaptive control for the trajectory propose aofcombination of neural network, reinforcement tracking a fully-actuated AUV in the horizontal plane 2.1 Governing equations of andmotion assumptions tracking of a fully-actuated AUV in the horizontal plane Consider the equations tracking of a fully-actuated AUV inboth the actuator horizontal plane Consider the equations of motion for for aa marine marine vehicle vehicle learning techniques and adaptive control the trajectory with disturbances, and saturawith exogenous exogenous disturbances, and both for actuator satura- Consider maneuvering in (mostly) calm, flat water, which be the equations of motion for a vehicle which can can be tracking ofdeadzone. a fully-actuated AUV inboth theuse horizontal plane maneuvering in (mostly) calm, flat water,marine tion and Zou et al. (2016) backstepping with exogenous disturbances, and actuator saturamaneuvering in (mostly)ofcalm, flat for water, which can be tion and deadzone. Zou et al. (2016) use backstepping Consider the equations motion a marine vehicle represented as represented as with exogenous disturbances, and both actuator saturacoupled with robust and adaptive control techniques to tion and deadzone. Zou et al. (2016) use backstepping coupled with robust and adaptive control techniques to maneuvering in (mostly) calm, flat water, which can be ˙ η = J (η)v (1) represented as coupled with robusttracking and et adaptive control to η˙ = J (η)v (1) tion deadzone. Zou al. (2016) use backstepping designand an attitude controller for techniques spacecraft in design an attitude tracking controller for spacecraft in represented as ˙ η = J (η)v (1) coupled with robust and adaptive control to three of with exogenous disturbances. design an attitude tracking controller for techniques spacecraft in and and three degrees degrees of freedom freedom with exogenous disturbances. η˙ = J (η)v (1) design an attitude tracking controller for spacecraft in and M (2) three degrees of freedom with exogenous disturbances. This work was sponsored, in part, by the US National Science Mv v˙˙ + + C(v)v C(v)v + + D(v)v D(v)v + + g(η) g(η) = =τ τ (u) (u) + +d d (2) and ˙ M v + C(v)v + D(v)v + g(η) = τ (u) + d (2) This degrees work was of sponsored, part, exogenous by the US National Science three freedomin with disturbances. in body-fixed coordinates (Fossen, 2011). The terms apFoundation (Award #1526016). This work was sponsored, in part, by the US National Science in body-fixed coordinates (Fossen, 2011). The terms apFoundation (Award #1526016). ˙ M v + C(v)v + D(v)v + g(η) = τ (u) + d (2) pearing in equations are in 1. n in body-fixed coordinates 2011). The terms The author also sponsored, with the Dept. Ocean Mechanical Engineering, Foundation (Award #1526016). This workis in part, by& US National Science pearing in these these equations (Fossen, are defined defined in Table Table 1. Here Hereapn The author is was also with the Dept. Ocean &the Mechanical Engineering, in body-fixed coordinates (Fossen, 2011). The terms apFlorida Atlantic University, DaniaOcean Beach,&FL 33004 USA is the number of degrees of freedom (DOF) of the motion pearing in these equations are defined in Table 1. Here n The author is also with the Dept. Mechanical Engineering, Foundation (Award #1526016). Florida Atlantic University, Dania Beach, FL 33004 USA is the number of degrees of freedom (DOF) of the motion Florida Atlantic University, DaniaOcean Beach,&FL 33004 USA is the number of equations degrees of are freedom (DOF) of the pearing in these defined in Table 1. motion Here n The author is also with the Dept. Mechanical Engineering, Florida Atlantic University, Dania Beach, FL 33004 USA is the number ofLtd. degrees of freedom 2405-8963 © 2018 2018, IFAC (International Federation of Automatic Control) by Elsevier All rights reserved. (DOF) of the motion Copyright © IFAC 262 Hosting Copyright 2018 IFAC 262 Control. Peer review© responsibility of International Federation of Automatic Copyright © under 2018 IFAC 262 10.1016/j.ifacol.2018.09.513 Copyright © 2018 IFAC 262
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
Definition 2. Let the measure of tracking be ˜˙ + Λ˜ η. (6) s := η˙ − η˙ r = η Assumption 3. The plant is intput to state stable (ISS). Assumption 4. The system of actuators has the following properties:
Table 1. Variables used in (1) and (2). Term
Dimension
Description
M
Rn
Inertia tensor
C(v)
Rn × Rn
Coriolis and centripetal matrix Hydrodynamic damping matrix
g(η)
Rn × R n
Rn
Gravity and buoyancy forces
J(η)
Transformation matrix
v
Rn × Rn
Rn
Velocity/angular rate vector
η
Rn
Position/attitude vector
u
Rn
Vector of actuator states
τ (u)
Rn
Actuator forces/moments
d
Rn
Vector of disturbances
D(v)
×
Rn
(including added mass effects) (including added mass effects)
and r is the number of actuators. In general, a marine craft with actuation in all DOFs, such as an underwater vehicle, requires a n = 6 DOF model for model-based controller and observer design, while ship and semi-submersible control systems can be designed using an n = 3, or 4 DOF model. In 6 DOF η is a composite vector of translations in 3 DOF and Euler angle rotations in 3 DOF. Thus, in 6 DOF, its dimensions are often denoted as R3 × S 3 . Assumption 1. In the following formulation, the coordinate transformation matrix J (η), which is used to convert the representation of vectors between a body-fixed coordinate system and an Earth-fixed North-East-Down coordinate system, is based on the use of Euler angles. The transformation matrix J (η) has singularities at pitch angles of θ = ±π/2. Here, it is assumed that |θ| < π/2. Remark 1. The singularities at θ = ±π/2 are not generally a problem for surface vessels. However, an underwater vehicle may occassionally approach these singularities if performing extreme maneuvers. In such cases, as suggested in Fossen (2011), the kinematic equations could be described by two Euler angle representations with different singularities and the singular points can be avoided by switching between the representations. The terms M , C(v) and D(v) have the useful mathematical properties shown in Table 2. The control objective is to make the system track a desired trajectory η d . (3)
Assumption 2. The trajectory η d and its derivatives η d , ¨ d , and η˙ d are smooth and bounded. η Definition 1. Let ˜ := η − η d (3) η be the Earth-fixed tracking error. Then define the reference trajectories in body-fixed and Earth-fixed coordinates as η (4) η˙ r := η˙ d − Λ˜ and (5) v r := J −1 (η)η˙ r , where Λ > 0 is a diagonal design matrix. Table 2. Mathematical properties of M , C(v) and D(v). M = M T > 0 ⇒ xT M x > 0, ∀x = 0
C(v) = −C T (v) ⇒ xT C(v)x = 0, ∀x D(v) > 0 ⇒
1 T x [D(v) 2
263
+ D T (v)]x > 0, ∀x = 0
263
a) The vehicle is fully-actuated or overactuated (r ≥ n). b) The vector τ (u) ∈ Rn is the result of the combined effects of a control allocation scheme and r actuators. A control allocation scheme would generally accept a control input uc ∈ Rn and then send a control signal to each of the r actuators. Each actuator would have its own rate limit and saturation response. When r > n it is assumed that the combined effects of control allocation and overactuation can be modeled as a set of n control inputs uc , which are rate-limited to a set of n actuator states u. Owing to saturation, the vector of forces and moments output by the actuators is τ (u). c) Integrator backstepping, requires smoothly differentiable functions. Each component of τ (u) is modeled by a smooth function bounded by uiMax := supui τi (ui ) and uiMin := inf ui τi (ui ). The Jacobian of τ (u) is smooth and nonsingular, such that its inverse exists. Each component of τ (u) is only dependent on the corresponding component of the actuator states, i.e. ∂τi /∂uj = 0, ∀i = j such that ∂τ /∂u is diagonal. d) The saturation limits and the rate limit of each component of τ (u) are independent. The time response of the combined set of actuators is governed by the first order differential equation (7) T u˙ = −u + uc where T = T T > 0 ∈ Rn × Rn is a diagonal matrix of time constants (see Fossen and Berge (1997), for example). e) The error between the forces/moments commanded by the controller and the actual forces/moments produced by the system of combined actuators, uc − τ (u), and its time derivative, are bounded.
Remark 2. The saturation function ui , |ui /uiM | < 1, sat(ui /uiM ) := uiM sgn(ui ), |ui /uiM | ≥ 1,
(8)
is often used to model actuator saturation. However, it is discontinuous. As shown in Wen et al. (2011), the hyperbolic tangent function is smooth and can be useful for modeling actuator saturation when using integrator backstepping techniques. Thus, a convenient form for τ (u) could be ui τi (ui ) ≈ uiM tanh (9) uiM (see Fig. 1). Assumption 5. The disturbance vector d in (2) consists of two types of disturbances d = de + da , where de is a vector of unknown external disturbances, such as wind or current. As in Wen et al. (2011), the term da is modeled as a bounded disturbance-like error that arises from approximating the true actuator saturation with the function τ (u) (Fig. 1). The components of d and their time derivatives are unknown and time-varying, yet bounded. There exists a positive constant ρ, such that
IFAC CAMS 2018 264 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
Vectorial backstepping Step 1: Define the virtual control signal (14) η˙ = J (η)v := s + α1 , where α1 is a smoothly continuous function that stabilizes the system at the origin, which can be chosen as α1 = η˙ r = η˙ d − Λ˜ η (15) where Λ > 0 is a diagonal matrix. Then, (14) can be written as η ˜˙ = −Λ˜ η + s. (16) Next, consider the Lyapunov function candidate 1 T ˜ K pη ˜. V1 = η (17) 2 Its time derivative is ˜ T K pη ˜ T K ps ˜˙ = −˜ η T K p Λ˜ η+η (18) V˙ 1 = η where K p = K Tp > 0 is a design matrix.
Fig. 1. Use of the hyperbolic tangent function to model actuator saturation. ˙ d(t) ≤ ρ, (10) where · represents the 2-norm of a vector or matrix. 2.2 Control design The tracking controller is designed using the maneuvering equations (2) transformed into an Earth-fixed coordinate system, which can be written as M η (η)¨ η + C η (v, η)η˙ + D η (v, η)η˙ + g η (η) (11) = J −T (η)[τ (u) + d], where M η (η) = M Tη (η) > 0, T 1 ˙ (12) M η (η) − C η (v, η) s = 0, ∀v, η, s s 2 D η (v, η) > 0, and the relationships between the terms M , C(v) and D(v) in the Earth-fixed and body-fixed coordinate systems are shown in Table 3 Fossen (2011). The product Table 3. Relations between Earth-fixed and body-fixed systems. M η (η) := J −T (η)M J −1 (η)
+d − M v˙ r − C(v)v r − D(v)v r −g(η) .
(19)
(20)
(21)
Select the virtual control law ˆ τ (u) := z + α2 − d, (22) where z is a new state variable, α2 is a stabilizing function ˆ is an estimate of the disturbances. Let and d α2 = M v˙ r + C(v)v r + D(v)v r + g(η) (23) ˜. −J T (η)K d s − J T (η)K p η Then V˙ 2 becomes V˙ 2 = −˜ η T K p Λ˜ η − sT [D η (v, η) + K d ] s (24) ˜ , +sT J −T (η) z − d
where
˜ := d ˆ−d d is the disturbance estimation error.
C η (v, η) := J −T (η)[C(v) −1 ˙ (η) − M J −1 (η)J(η)]J
D η (v, η) := J −T (η)D(v)J −1 (η) g η (η) := J −T (η)g(η)
M η (η)s˙ is an important quantity that will be used in the controller derivation. It can be written as M η (η)s˙ = −C η (v, η)s − D η (v, η)s +J −T (η) [τ (u) + d − M v˙ r
Step 2: Consider the Lyapunov function candidate 1 V2 = V1 + sT M η (η)s. 2 Its time derivative is ηT K η V˙ 2 = −˜ p Λ˜ 1 ˙ T ˜ + M η (η)s˙ + M η (η)s . +s K p η 2 Using (12) and (13), this can be rewritten as η T K p Λ˜ η − sT D η (v, η)s V˙ 2 = −˜ ˜ + τ (u) +sT J −T (η) J T (η)K p η
(13)
−C(v)v r − D(v)v r − g(η)] . 264
Step 3: Consider the Lyapunov function candidate 1 1 ˜T ˜ V3 = V2 + z T z + d d. 2 2 Taking its time derivative gives η T K p Λ˜ η − sT [D η (v, η) + K d ] s V˙ 3 = −˜ +z T z˙ + J −1 (η)s ˜ T −J −1 (η)s + d ˜˙ . +d
(25)
(26)
(27)
We will design a control law for z˙ and a disturbance ˜˙ such that V˙ 3 ≤ 0. observer d
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
Stabilizing control law Taking the time derivative of (22) we have ∂τ d ˆ˙ ˙ 2 − d. [τ (u)] = u˙ = z˙ + α (28) dt ∂u Using (7), (28) can be rewritten as ∂τ −1 ˆ˙ = −u + uc . ˙2−d z˙ + α T u˙ = T ∂u Solving for z˙ gives ∂τ −1 ˆ˙ ˙ 2 + d. T (uc − u) − α z˙ = ∂u In order to make the term z T z˙ + J −1 (η)s ≤ 0 in we take z˙ = −J −1 (η)s − K z z, where K z = K Tz > 0 is a design matrix. Equating and (31), the stabilizing controller is found to be ∂τ −1 ˆ˙ − J −1 (η)s − K z z . ˙2−d α uc = u + T ∂u
(29)
(30)
(31) (30) (32)
and −g(η) + τ (u) + K 0 M v] ,
(34)
where K 0 = K T0 > 0. Taking the derivative of (33) and using equations (34) and (2) gives ˜ ˆ˙ = −K 0 d. (35) d Using the derivative of (25), (35) can be written as ˜˙ = −K 0 d ˜ − d. ˙ d (36) Select the Lyapunov function candidate 1 ˜T ˜ 1 ˜ 2 Vd0 = d . (37) d = d 2 2 Taking the derivative of (37) and using (36) gives ˜˙ = −d ˜T K 0d ˜T d ˜−d ˜ T d. ˙ V˙ d0 = d Let λmin (·) be the minimum eigenvalue of a matrix and Cd := ρ2 /2. Using (10) and applying Youngs Product Inequality for vectors to the last term in V˙ d0 yields ˜T d ˜T K 0d ˜ + 1d ˜ + 1 d˙ T d, ˙ V˙ d0 ≤ −d 2 2 ˜ 2 + 1 d ˙ 2, ˜ 2 + 1 d ≤ −λmin (K 0 ) d 2 (38) 2 1 ˜ 2 ρ2 d + , ≤ − λmin (K 0 ) − 2 2 ≤ −2αVd0 + Cd , where (39) λmin (K 0 ) > 1/2 is selected to ensure that α > 0. As proven in Do (2010) and Du et al. (2016), the practical stability of the ˜ disturbance observer is ensured since both Vd0 (t) and d are globally uniformly ultimately bounded such that Cd Cd −2αt 0 ≤ Vd0 (t) ≤ + Vd0 (0) − e , 2α 2α 265
Cd −2αt Cd + 2 Vd0 (0) − e . α 2α The disturbance error settles to within a compact set that can be made arbitrarily small by appropriately selecting K 0 , so that (39) is satisfied. ˜ ≤ d
System stabilization Using (31) and (36), (27) can be reduced to η T K p Λ˜ η − sT [D η (v, η) + K d ] s V˙ 3 = −˜ ˜T K 0d ˜ −z T K z z − d ˜ T J −1 (η)s + d˙ . −d
(27)
Disturbance observer A disturbance observer is constructed using the approach presented in Do (2010) and Du et al. (2016). Its derivation is repeated here so that the presentation of the results to follow is complete. Let ˆ = q(t) + K 0 M v (33) d ˙ q(t) = −K 0 q(t) − K 0 [−C(v)v − D(v)v
and
265
(40)
The last term in this expression can be rewritten using Young’s Product Inequality for vectors, so that η T K p Λ˜ η − zT K z z V˙ 3 ≤ −˜ −sT [D η (v, η) + K d ] s
1 + sT J −T (η)J −1 (η)s 2 T ˜ ˜ + 1 d˙ T d, ˙ −d [K 0 − 1] d 2 η 2 − λmin (K z ) z 2 ≤ −λmin (K p Λ) ˜ −λmin [D η (v, η) + K d ] s 2 1 + λmax J −T (η)J −1 (η) s 2 2 ˜ 2 + Cd , −λmin (K 0 − 1) d
(41)
λmin (K p Λ) λmax (K p ) ˜ η 2 λmax (K p ) λmin [D η (v, η) + K d ] λmax (M η ) s 2 − λmax (M η ) λmax J −T (η)J −1 (η) λmax (M η ) s 2 + 2λmax (M η )
≤ −
−λmin (K z ) z 2
˜ 2 + Cd . −λmin (K 0 − 1) d Since M and D(v) are real symmetric tensors they can be diagonalized to determine the eigenvalues corresponding their principal axes. The maximum eigenvalue and the minimum eigenvalue of each diagonalized tensor will correspond to the maximum and minimum possible eigenvalues of each tensor under rotations produced by combinations of transformations with J (η), J T (η), J −1 (η) and J −T (η). Let M principal and D principal be the inertial and drag tensors expressed in their respective principal axes coordinate systems. Also let λM min := λmin (M principal ), λM max := λmax (M principal ), λDmin := λmin (D principal ) and λDmax := λmax (D principal ). Then, λM min ≤ λmin (M η ), λM max ≥ λmax (M η ),
λDmin ≤ λmin [D η (v, η)] ,
λDmax ≥ λmax [D η (v, η)] .
(42)
IFAC CAMS 2018 266 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
Since |θ| < π/2, of J (η) it can be from the definition −T −1 seen that λmax J (η)J (η) = 1 + sin(θ) < 2 (see Assumption 1.Using (42), (41) can be rewritten as λmin (K p Λ) λmax (K p ) ˜ η 2 V˙ 3 ≤ − λmax (K p ) λDmin + λmin (K d ) − 1 − λM max s 2 (43) λM max −λmin (K z ) z
2
˜ 2 + Cd . −λmin (K 0 − 1) d From (26), the inequality 1 1 η 2 + λM max s 2 V3 ≤ λmax (K p ) ˜ 2 2 (44) 1 1 ˜ 2 2 + z + d 2 2 can be constructed. Using this inequality for V3 , (41) can be written as (45) V˙ 3 ≤ −2µV3 + Cd , where λmin (K p Λ) , λmin (K z ), µ := min λmax (K p ) λmin (K 0 − 1) , (46) λDmin + λmin (K d ) − 1 . λM max Consider the compact set ˜ : V3 ≤ B0 , ∀B0 > 0} ∈ R24 . Π = {(˜ η , s, z, d)
The design parameters K p , Λ, K z , K 0 , K d can be chosen such that λmin (K 0 ) > 1, (47) (48) λmin (K d ) > 1 − λDmin , and Cd . (49) µ> 2B0 ˜ = B0 . From (45), V˙ 3 ≤ −2µB0 + Cd on V3 (˜ η , s, z, d) Further, applying (49) gives the stronger inequality V˙ 3 < 0. ˜ : V3 ≤ B0 } is Thus, the compact set Π = {(˜ η , s, z, d) invariant. If V3 (0) ≤ B0 , then V3 (t) ≤ B0 and (45) holds for all t > 0. Remark 3. Inequality (47) imposes a consistent, but stronger, condition than (39) on the selection of K 0 . Theorem 1. Consider the closed-loop system consisting of: a) the plant (1)–(2) with unknown time-varying disturbances, input saturation and input rate limits, as characterized under Assumptions 1–5; b) the virtual control law (22) and the stabilizing controller (32); and c) the disturbance observer (33)–(34). For all V3 (0) ≤ B0 , where ˜ = η − ηd B0 is a positive constant, the tracking error η settles to within the compact set η ∈ R6 | ˜ η ≤ ζη˜ , ζη˜ > Cd /µ}. Πη˜ = {˜ The set Πη˜ can be made arbitrarily small by appropriately adjusting the design parameters K p , Λ, K z , K 0 , and K d such that (47)–(49) are satisfied. All signals in the trajectory tracking closed-loop control system are guaranteed to be uniformly ultimately bounded. 266
Proof. The solution to (45) is Cd Cd −2µt + V3 (0) − e 0 ≤ V3 (t) ≤ . (50) 2µ 2µ Thus, V3 (t) is uniformly ultimately bounded for all V3 (0) ≤ ˜ ˜ , s, z, d B0 . From (17), (19) and (26) it can be seen that η are also uniformly ultimately bounded for all V3 (0) ≤ B0 .
From (16), (18), (20), (25), (35) and the boundedness of ˆ are uniformly ultimately d, it can be seen that η and d bounded ∀V3 (0) ≤ B0 . Thus, all signals in the closed loop control system are uniformly ultimately bounded for all V3 (0) ≤ B0 . 3. SIMULATIONS The performance of the control law (32) is investigated using a 6 DOF simulation of an AUV tracking two periods of a sine-shaped trajectory (7.3 m East-West lateral deviations and 111 m Northward travel) at constant depth. Initially, the vehicle is positioned at the origin of a North-East-Down coordinate system with its nose pointing North; the controller is designed to keep the longitudinal axis of the AUV tangent to the trajectory. The manuevering coefficients and physical characteristics of the AUV were obtained from Prestero (2001). The only differences are that here it is assumed the maximum propeller thrust is 7 N, the center of gravity is [xg yg zg ]T = [0 0 0]T and the center of buoyancy is [xb yb zb ]T = [0 0 − 2]T cm. The saturation limits of the actuators are τsat = ±[7.00 9.49 9.49 0.95 4.75 4.75]T (forces are in N and moments are in N-m – see Fig. 4). All six actuators have a time constant of 0.25 secs. Disturbances are modeled as first order Markov processes of the form b˙ = −Tb b + an wn , where b ∈ R6 is a vector of bias forces and moments, Tb ∈ R6×6 is a diagonal time constant matrix, wn ∈ R6 is a vector of zero-mean Guassian white noise, and an ∈ R6×6 is a diagonal matrix that scales the amplitude of wn . The integral of time multiplied by absolute error (ITAE) is provided in Table 4 for the following cases: (1) the backstepping controller with no disturbance; (2) the backstepping controller with disturbances characterized by Tb = 103 · 16×6 , an = [0.35 0.47 0.47 0.05 0.24 0.24]T and b0 = −an ; (3) the backstepping controller with the same disturbances scaled by a factor of 5 (see Figs. 2–4); and (4) a manually-tuned, nonlinear proportional integral derivative (PID) controller (with disturbance observer). The PID controlled system is unstable for the same disturbances used in cases 2 and 3 above. The backstepping controller is fairly robust and outperforms the PID controller, even in the presence of strong disturbances. Table 4. ITAE of N, E, D position in m-s2 and φ, θ, ψ orientation in rad-s2 . Controller
N
E
D
φ
θ
ψ
BS (case 1)
3.80
1.02
0.00
0.00
0.00
0.19
BS (case 2)
4.87
1.49
0.35
0.08
0.45
0.62
BS (case 3)
33.5
24.5
1.80
0.53
2.28
12.4
PID (case 4)
166
643
0.00
0.00
0.00
340
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Karl D. von Ellenrieder / IFAC PapersOnLine 51-29 (2018) 262–267
observer and nonlinear backstepping. The result can be easily applied to systems with fewer degrees of freedom, such as 3 DOF surge-sway-yaw, or 4 DOF surge-sway-yawroll models of unmanned surface vessels.
0.5 N [m] 0.4
E [m] D [m]
0.3
The use of nonlinear backstepping for marine vehicle control has been in use for quite some time. With only slight modifications, the approach developed here can be applied to existing systems.
[rad] [rad] [rad]
0.2 0.1
The implementation of the control law requires use of the derivative of α2 . Noisy measurement of the states η, v and u can cause problems in the calculation of this derivative. A computational approach that can be used to circumvent this issue for some surface vessels can be found in Fossen and Berge (1997). Further, many recent commercially produced inertial measurement units provide direct measurements of several of the terms required for the computation of α2 .
0 -0.1 -0.2 -0.3 0
20
40
60
80
100
267
120
Fig. 2. Case 3 Earth-fixed tracking error.
REFERENCES 2
1
0
-1
-2
-3
-4
-5 0
20
40
60
80
100
120
100
120
Fig. 3. Case 3 estimated disturbances.
8 6 4 2 0 -2 -4 -6 -8 0
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80
Fig. 4. Case 3 actuator outputs. 4. CONCLUDING REMARKS Here a 6 DOF nonlinear control law for the trajectory tracking of marine vehicles that operate in the presence of unknown time-varying disturbances, input saturation and actuator rate limits is developed using a disturbance 267
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