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Adaptive control for the PVTOL system via Minimum Projection Method
Preprints, Preprints, 1st 1st IFAC IFAC Conference Conference on on Modelling, Modelling, Identification Identification and and Preprints, 1st IFAC on Modelling, Identification and Control Nonlinear Systems Preprints, IFAC Conference Conference Control of of 1st Nonlinear Systems on Modelling, Identification and Control of Nonlinear Systems June 24-26, 2015. Saint Saint Petersburg, Russia Russia Available online at www.sciencedirect.com Control of Nonlinear Systems June 24-26, 2015. Petersburg, June June 24-26, 24-26, 2015. 2015. Saint Saint Petersburg, Petersburg, Russia Russia
ScienceDirect IFAC-PapersOnLine 48-11 (2015) 216–221
Adaptive Adaptive control control via Minimum via Minimum
for for the the PVTOL PVTOL system system Projection Method Projection Method
∗∗ ∗∗ Ysuyuki Satoh ∗∗∗ ∗∗∗ Hisakazu Nakamura Hisakazu Nakamura ∗∗ Ysuyuki Satoh ∗∗∗ Hisakazu Nakamura Hisakazu Nakamura ∗∗ Ysuyuki Ysuyuki Satoh Satoh ∗∗∗ ∗ ∗ Tokyo University of Science, University of Science, ∗ ∗ Tokyo Tokyo University of Tokyo University of Science, Science, Yamazaki 2641, 2641, Noda, Chiba 278-8150, 278-8150, Japan (e-mail: (e-mail: Yamazaki Noda, Chiba Japan Yamazaki Noda, Chiba 278-8150, Japan Yamazaki 2641, 2641,[email protected]). Noda, Chiba 278-8150, Japan (e-mail: (e-mail: [email protected]). ∗∗ [email protected]). ∗∗ Tokyo [email protected]). of Science (e-mail: [email protected]) Tokyo University of Science (e-mail: [email protected]) ∗∗ ∗∗∗∗∗ Tokyo University of Science (e-mail: [email protected]) Tokyo University of of Science (e-mail: [email protected]) ∗∗∗ Tokyo University Science ([email protected]) Tokyo University of Science ([email protected]) ∗∗∗ ∗∗∗ Tokyo University of Science ([email protected]) Tokyo University of Science ([email protected])
Soki Soki Soki Soki
∗ ∗ Kuga Kuga ∗ Kuga Kuga ∗
Abstract: “Dynamic “Dynamic extension” extension” is is commonly used for for stabilization stabilization of of planar planar vertical vertical take take Abstract: commonly used Abstract: “Dynamic extension” commonly used stabilization planar vertical Abstract: “Dynamic extension” isMost commonly used for for stabilization ofmethod planar are vertical take off and and landing landing (PVTOL) system. is Most of controllers controllers designed by this thisof method are basedtake on off (PVTOL) system. of designed by based on off and (PVTOL) system. Most off and landing landing (PVTOL) system. Most of of controllers controllers designed designed by by this this method method are are based based on on “dynamic” control Lyapunov functions(CLFs). “dynamic” control Lyapunov functions(CLFs). “dynamic” Lyapunov “dynamic” control Lyapunov functions(CLFs). functions(CLFs). ∞ differentiable We design design aacontrol C∞ “static” strict CLF for the PVTOL system by dynamic extension We C “static” strict CLF for the PVTOL system by dynamic extension ∞ ∞ differentiable We design a C differentiable “static” strict CLF for the PVTOL system by dynamic extension We design a C differentiable “static” strict CLF for the PVTOL system by dynamic extension and minimum projection method. Then we propose an adaptive control law with the an inverse and minimum projection method. Then we propose an adaptive control law with the an inverse and minimum projection method. Then we propose an adaptive control law with the an inverse and minimum projection method. Then we propose an adaptive control law with the an inverse optimal controller based on the static CLF. optimal controller based on the static CLF. optimal controller controller based based on on the the static static CLF. CLF. optimal © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear control, Control Lyapunov function, Adaptive control, Inverse optimal Keywords: Nonlinear control, Control Lyapunov function, Adaptive control, Inverse optimal Keywords: Nonlinear control, Control Control Lyapunov Lyapunov function, function, Adaptive Adaptive control, control, Inverse Inverse optimal optimal Keywords: Nonlinear control, control, PVTOL system. control, PVTOL system. control, PVTOL PVTOL system. system. control, 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. VTOL aircrafts such as V-22 or quadrotors attract much VTOL aircrafts such as V-22 or quadrotors attract much VTOL aircrafts aircrafts such years. as V-22 V-22Planar or quadrotors quadrotors attract much VTOL such as or attract much attentions in recent vertical take off and attentions in recent years. Planar vertical take off and attentions in recent recentsystem years. introduced Planar vertical vertical take off offetand and attentions in years. Planar take landing (PVTOL) by Hauser al landing (PVTOL) system introduced by Hauser et al landingis (PVTOL) (PVTOL) system introduced by Hauser et al al landing system introduced by Hauser et (1992) a useful model to study position control of VTOL (1992) is a useful model to study position control of VTOL (1992) is a useful model to study position control of VTOL (1992) is a useful model to study position control of VTOL aircrafts. aircrafts. aircrafts. aircrafts. A common and useful control strategy is one of proposed A common and useful control strategy is one of proposed A common common and useful control strategy is one one of ofextension. proposed A and useful control strategy is proposed by Hauser et al (1992) based on dynamic by Hauser et al (1992) based on dynamic extension. by Hauser et al (1992) based on dynamic extension. by Hauser et al (1992) based on dynamic extension. Researchers including Hauser et al (1992), Turker et al Researchers including Hauser et al (1992), Turker et al Researchers including Hauseretet et al (1992), Turker(2002) et al al Researchers including Hauser al (1992), Turker et (2012), Teel (1996), Fantoni al (2002) and Saber (2012), Teel (1996), Fantoni et al (2002) and Saber (2002) (2012), Teel Teel (1996), Fantonibased et al al (2002) (2002) and Saber Saber (2002) (2012), (1996), Fantoni et and (2002) proposed many controllers on dynamic extension. proposed many controllers based on dynamic extension. proposed many controllers based on on dynamicpoint extension. proposed many controllers based dynamic extension. The method stabilizes an arbitrary operating with a The method stabilizes an arbitrary operating point with The method stabilizes an arbitrary operating point with aaa The method stabilizes an arbitrary operating point with large domain of attraction of the state space. large domain of attraction of the state space. large domain domain of of attraction attraction of of the the state state space. space. large However, these controllers are based on nonsmooth “dyHowever, these controllers are based on nonsmooth “dyHowever,control these Lyapunov controllers functions are based based(CLFs), on nonsmooth nonsmooth “dyHowever, these controllers are on “dynamic” and lost ronamic” control Lyapunov functions (CLFs), and lost ronamic” control control Lyapunov functions (CLFs), and lost lost See ronamic” Lyapunov functions (CLFs), and robustness due to their dynamic input transformations. bustness due to their dynamic input transformations. See bustnessetdue due to their dynamic dynamic input transformations. See bustness to their input transformations. See Hauser al (1992), Teel (1996) and Uuc et al (2013). Hauser et al (1992), Teel (1996) and Uuc et al (2013). Hauser et et al al (1992), Teel Teel (1996) and and Uuc strategy et al al (2013). (2013). Hauser (1992), (1996) Uuc et Moreover, differentiable CLF-based control can Moreover, differentiable CLF-based control strategy can Moreover, differentiable CLF-based control strategy can Moreover, differentiable CLF-based control strategy can not be applied. not be applied. not be be applied. applied. not ∞ ∞ differentiable “static” In this paper, we propose aa C In this paper, we propose C ∞ differentiable “static” In this this paper, we for propose a C C∞ differentiable “static” In paper, we propose a differentiable “static” strict CLF (SCLF) stabilization of the PVTOL system. strict CLF (SCLF) for stabilization of the PVTOL system. strict CLF (SCLF) for stabilization of the PVTOL system. strict CLF (SCLF) for stabilization of the PVTOL system. The proposed SCLF is designed by following two steps. The proposed SCLF is designed by following two steps. The proposed SCLF is designed by following two steps. The proposed SCLF is designed by following two steps. First, we design a “dynamic” SCLF for the augmented sysFirst, we design a “dynamic” SCLF for the augmented sysFirst, we design a “dynamic” SCLF for the augmented sysFirst, we design a “dynamic” SCLF for the augmented system obtained obtained by by dynamic dynamic extension extension of of the the PVTOL PVTOL system. system. tem tem obtained by dynamic extension of the PVTOL system. tem obtained by dynamic extension of the PVTOL system. Then, we design a static SCLF for the PVTOL system by Then, we design aa static SCLF for the PVTOL system by Then, we design static SCLF for the PVTOL system by Then, we design a static SCLF for the PVTOL system by minimum projection method proposed by Yamazaki et al minimum projection method proposed by Yamazaki et al minimum projection method proposed by Yamazaki et al minimum projection method proposed by Yamazaki et al (2013). (2013). (2013). (2013). Our proposed SCLF is a smooth function. We propose Our proposed SCLF is a smooth function. We propose Our proposed SCLF is a smooth function. We propose Our proposed SCLF is a smooth function. We propose an adaptive control law with an inverse optimal controller an adaptive control law with an inverse optimal controller an adaptive control law with an inverse optimal controller an adaptive control law with an inverse optimal controller based on on the the designed designed differentiable differentiable static static SCLF. SCLF. Finally, Finally, based based on designed differentiable static Finally, based on the the designed differentiable static SCLF. SCLF. Finally, we show computer simulation to confirm that our conwe show computer simulation to confirm that our conwe we show show computer computer simulation simulation to to confirm confirm that that our our concon1 1 1 1
troller can stabilize origin troller can stabilize the the origin of of the the PVTOL PVTOL system system under under troller can the troller can stabilize stabilize the origin origin of of the the PVTOL PVTOL system system under under parameter uncertainty. parameter uncertainty. parameter parameter uncertainty. uncertainty. 2. PRELIMINARIES PRELIMINARIES 2. 2. 2. PRELIMINARIES PRELIMINARIES 2.1 Control Control System System on on Extended Extended Space Space 2.1 2.1 2.1 Control Control System System on on Extended Extended Space Space Consider the following nonlinear control system: Consider the following nonlinear control system: Consider nonlinear control system: Consider the the following following nonlinear control system: ˙x˙ = x f (x) + g(x)u, (1) = f (x) + g(x)u, (1) x ˙ = f (x) + g(x)u, (1) x ˙ = f (x) + g(x)u, (1) n m n m where, x x∈ ∈R Rn and and u u∈ ∈R Rm denote denote aa state state and and an an input input where, n m n n n n×m where, x and state an input where, x∈ ∈R RMappings and u u∈ ∈ff R R: Rndenote denote aand state and an R input na n×m respectively. →R gg :: and Rnn → respectively. Mappings R R n n n×m n → n and n → n×m respectively. Mappings ff ::: R R → R and ggwith :: R R → R respectively. Mappings R → R and R → R are supposed to be Lipschitz continuous respect to are supposed to be Lipschitz continuous with respect to are supposed to be Lipschitz continuous with respect are supposed be Lipschitz continuous with respect to both x and and u, u,to and and satisfy ff (0, (0, 0) = = 0.(Bacciotti 0.(Bacciotti et to al both x satisfy 0) et al both both x x and and u, u, and and satisfy satisfy ff (0, (0, 0) 0) = = 0.(Bacciotti 0.(Bacciotti et et al al (2013)) (2013)) (2013)) (2013)) This paper paper also also considers considers the the following following augmented augmented system system This n+lfollowing This paper also considers considers the following augmented system system This paper also the augmented n+l of (1) on extended space R : of (1) on extended space R : n+l n+l : ] ] of space R of (1) (1) on on extended extended [ space R[ : [ ] [ ] [ ] ˙˙ ] + g(x)u [x ] [ [ ff (x) ] x + g(x)u x ˜˜˙˙ = (2) x ˙˙ = ff (x) (x) + g(x)u x = = (2) x (x) + g(x)u p v ˙x x = p˙ = = (2) v ˜˜˙ = (2) ˙p˙ p vv ˜(x) + g˜(x)˜ = f u , (3) ˜ = f (x) + g˜(x)˜ u , (3) = u (3) = ff˜˜(x) (x) + + gg˜˜(x)˜ (x)˜ u,, (3) ll ll denotes a virtual state, v ∈ R a virtual input where p ∈ R a virtual state, v ∈ R a virtual input where p ∈ R ll denotes l l a virtual input T denotes a virtual state, v ∈ R where p ∈ R denotes a virtual state, v ∈ R a virtual input where p ∈[u,Rv] T. and u ˜˜ = and = and u u = [u, [u, v] v]TT ... and u ˜˜ = [u, v] 2.2 Control 2.2 Control Lyapunov Lyapunov Function(CLF) Function(CLF) 2.2 2.2 Control Control Lyapunov Lyapunov Function(CLF) Function(CLF) In this paper, we design aa controller based on In this paper, we design controller based on strict strict In paper, design controller based In this this Lyapunov paper, we we functions design aa (SCLFs). controller Throughout based on on strict strict control Lyapunov functions (SCLFs). Throughout this control this control Lyapunov functions this control Lyapunov functions (SCLFs). Throughout this ´´ X ˜˜ (SCLFs). ¯¯ areThroughout paper, state spaces X, X, and X defined in the paper, state spaces X, X, X defined in the ´´ X ˜˜ and ¯¯ are paper, state spaces X, X, X and X are defined in the paper, state spaces X, X, X and X are defined in the neighborhood of of the the origin. origin. neighborhood neighborhood of the origin. neighborhood of the origin. Definition 1. (Strict Control Lyapunov function (SCLF)). Definition 1. (Strict Control Lyapunov function (SCLF)). Definition 1. (Strict Control Lyapunov function (SCLF)). Definition 1. (Strict Control Lyapunov function (SCLF)). (Nami et al (2013)) Consider system (1). Then, aa proper (Nami et al (2013)) Consider system (1). Then, proper 1 n (Nami et al (2013)) Consider system (1). Then, proper (Nami et al (2013)) Consider system (1). Then, aa proper positive definite definite C C 11 function function V V :: R Rnn ⊃ ⊃X X → →R R satisfying satisfying positive 1 n 1→ positive definite C function V :: R ⊃ X R satisfying positive definite C function V R ⊃ X → R satisfying 1 the following condition is said to be a C strict control the following condition is said to be aa C 1 strict control the condition is to the following following condition is said said to be be a C C 1 strict strict control control Lyapunov function (SCLF) for (1): Lyapunov function (SCLF) for (1): Lyapunov function (SCLF) for (1): Lyapunov function (SCLF) for (1):
Copyright © 2015, IFAC 220 2405-8963 © IFAC (International Federation of Automatic Control) Copyright IFAC 2015 2015 220 Hosting by Elsevier Ltd. All rights reserved. Copyright IFAC 2015 220 Copyright ©under IFAC responsibility 2015 220Control. Peer review© of International Federation of Automatic 10.1016/j.ifacol.2015.09.186
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Lf V (x) < 0, ∀x ∈ {x ∈ X|Lg V = 0}, (4) where Lf V = (∂V /∂x)f (x) and Lg V = (∂V /∂x)g(x). Freeman et al (1996) proposed an SCLF design method for nonlinear control system. For feedback linearizable systems, the following proposition holds: Proposition 1. Suppose there exists a diffeomorphism φ = Φ(x) with Φ(0) = 0 which transforms system (1) into φ˙ = Aφ + B [l0 (φ) + l1 (φ)u] , (5) where, the matrix pair (A, B) is controllable and mappings l0 : Rn → Rn and l1 : Rn → Rn×m are continuous, and l1 (Φ) is nonsingular for all Φ. Let P be a symmetric positive definite solution of the following Riccati equation: AT P + P A − P BR−1 B T P + Q = 0 (6) for arbitrary positive definite matrices Q and R. Then, function V˜ (x) : Rn ⊃ X → R defined by the following function is an SCLF for (1): (7) V˜ (x) = ΦT (x)P Φ(x). 2.3 Static CLF Design via Dynamic Extension Yamazaki et al (2013) proposed a nonsmooth CLF design method via minimum projection method. A CLF for the system (1) is heavily related to a CLF for the augmented system (2) as shown in the following theorem. ¯ →R Theorem 1. Let a continuous function V¯ : Rn+l ⊃ X be a CLF for (2). Then, the following function V : Rn ⊃ X → R is a CLF for (1): V (x) = min V¯ (x, p). (8) p∈Rl
Note that Theorem 1 does not guarantee the differentiability of V . 2.4 Inverse Optimal Control with SCLF Nami et al (2013) proposed an inverse optimal controller with SCLF. One of advantages of the inverse optimal control is a sector margin defined below. Definition 2. (Sector nonlineariry). A continuous mapping ϕ : Rm → Rm is said to be a sector nonlinearity in [α, β) with respect to u if ϕ(0) = 0 and αu2 ≤ uϕ(u) < βu2 , ∀u ̸= 0. Definition 3. (Sector margin). Consider a state feedback control u = k(x). A function ϕ(u) is supposed to have a sector nonlinearity [α, β) with respect to u. Then, the control input u for system (1) is said to have a sector margin [α, β) if the origin of the closed-loop system x˙ = f (x) + g(x)ϕ(u(x)) is asymptotically stable. With an SCLF, we can design an inverse optimal controller achieving a sector margin by the following lemma: Lemma 1. Let aj > 0 be a constant for all j ∈ {1, . . . , m}, V : Rn ⊃ X → R an SCLF for systems (1). Then, we define γ as follows: m ∑ 1 1 · |Lgj V (x)|aj +1 − Lf V (x), (9) γ(x) = a + 1 R(x) j=1 j 221
Fig. 1. The PVTOL aircraft where, R : X → R>0 is a positive-valued function on X\{0} and γ(x) is a positive definite function. Then, the following input uj : Rn ⊃ X → Rm asymptotically stabilizes the origin of the system (1): 1 |Lg V (x)|aj sgn(Lgj V (x)), uj (x) = − (10) R(x) j (j = 1, . . . , m), and minimizes the following cost function: ∫ ∞ m ∑ aj γ(x) + J= R1/aj (x)|uj |(aj +1)/aj dt. a + 1 j 0 j=1
(11) Further the input is continuous on X\{0} and achieves at least a sector margin [1/(min1≤j≤l aj + 1), ∞). 3. PROBLEM STATEMENT
The following equation denotes a normalized equation of motion of the PVTOL aircraft with gravitational acceleration g = 1 introduced by Hauser et al (1992) (Fig. 1): x˙ = [x˙ 1 , x˙ 2 , x˙ 3 , x˙ 4 , x˙ 5 , x˙ 6 ]T 0 0 x2 0 − sin x5 ε cos x5 [ ] 0 0 x u1 , = 4 + −1 cos x5 ε sin x5 u2 x6 0 0 0 1 0
(12)
where, x = [x1 , x2 , ..., x6 ]T ∈ R6 and u = [u1 , u2 ]T ∈ R2 denote respective a state variable and a control input. u1 , u2 are the vertical thrust attached to the aircraft and rolling moment respectively. Michel et al (1999) comment that the normalized lengths x1 , x3 represent actual lengths divides by the intensity g of the gravitation field, hence a normalized length of 1 represents an actual length of about 10 m. ε denotes a coupling coefficient between u2 and lateral acceleration εu2 . When small ε, the system is a strong nonminimum phase system. In many VTOL aircrafts, thrust spindles are vertically attached to the body; i.e., the inputs coupling ε = 0 structurally. Hence in this paper we suppose ε = 0; then the system becomes an under-actuated system.
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“−1” appeared in both Fig. 1 and the first term of the right-hand side of (12) denotes the normalized gravitational acceleration. Note that the origin of (12) is an equilibrium when u1 = 1. Let a new vertical control input u ˜1 = u1 − 1, and then we obtain the following control system: 0 x2 − sin x5 − sin x5 0 x4 x˙ = + cos x5 − 1 cos x5 x6 0 0 0 = f (x) + g1 (x)˜ u1 + g2 (x)u2 .
0 0[ ] ˜1 0 u 0 u2 0 1
(13)
(14)
In this paper, we consider a static state feedback controller design problem for asymptotic stabilization of the origin of (13). 4. CONVENTIONAL DYNAMIC EXTENSION TO THE PVTOL SYSTEM In this section, we apply conventional dynamic extension to PVTOL system (13), and obtain a dynamic SCLF for the linear augmented system of (13). In accordance with Hauser et al (1992), we consider the following coordinate and input transformation with a new ˜ ⊂ R8 and v ∈ R2 : state variable φ ∈ X T ˜1 , u ˜˙ 1 ) φ = [φ1 , φ2 , . . . , φ8 ] = Φ(x, u x1 x2 −(1 + u ˜1 ) sin x5 ˜1 )x6 cos x5 − u ˜˙ 1 sin x5 −(1 + u (15) = , x3 x4 (1 + u ˜1 ) cos x5 − 1 ˜˙ 1 cos x5 −(1 + u ˜1 )x6 sin x5 + u v=
[
[
v1 v2
]
x26 sin x5 = −x26 cos x5 [ − sin x5 + cos x5
(16) ] u ˜1 + 1 u ˜˙ 1 ][ ] ¨ −(˜ u1 + 1) cos x5 u ˜1 . −(˜ u1 + 1) sin x5 u2 −2x6 cos x5 −2x6 sin x5
][
(17)
Then, we can transform the system (1) into the following linear augmented control system:
˜˙ 1 ) : = V˜ (Φ(x, u ˜1 , u ˜˙ 1 )) V¯ (x, u ˜1 , u T ˜1 , u ˜˙ 1 )P0 Φ(x, u ˜1 , u ˜˙ 1 ). = Φ (x, u
(20)
(21) Remark 1. Note that the input transformation (17) is valid locally in the neighborhood of the operating point x1 , x3 = 0 due to the following inverse mapping of (17) has a singularity at u ˜ = −1: 12 [ ] u1 + 1) x6 (˜ u ˜¨1 2x6 u ˜˙ 1 = u2 − (˜ u1 + 1) [ ] − sin x5 cos x5 v1 − cos x sin x 5 5 . (22) + v2 (˜ u1 + 1) (˜ u1 + 1)
Uuc et al (2013) introduce this singularity due to dynamic input transformation. Here, V¯ is valid in the following set due to the singularity: ¯ = {(x, u ¯ max }, X ˜1 , u ˜˙ 1 )|V¯ (x, u ˜1 , u ˜˙ 1 ) < R (23) ¯ ¯ ˙ ˜1 ) = 2.44. ˜1 , u where, Rmax = minu˜1 =−1 V (x, u Remark 2. Dynamic control input v designed by dynamic extension may lost the robustness due to the dynamic input transformation (22). 5. PROPOSED MINIMUM PROJECTION TO THE PVTOL SYSTEM 5.1 Static SCLF Design for the PVTOL system In this section, we develop a C ∞ “static” SCLF for the PVTOL system (13) based on minimum projection method Theorem 1 proposed by Yamazaki et al (2013). ¯ → R denoted Theorem 2. Let the function V¯ : R8 ⊃ X in (21) be a C ∞ SCLF for the system (18). Then, the following function V´ : R6 ⊃ X → R is a C ∞ SCLF for the system (1): V´ (x) =ΦT (x, p1 (x), p2 (x))P0 Φ(x, p1 (x), p2 (x)) (24) =ΦT (x, p(x))P0 Φ(x, p(x)), where, the function p(x) = [p1 , p2 ] : R6 → R2 is uniquely determined by the folloing equations: ∂ V¯ ∂Φ (x, p1 , p2 ) = 2ΦT P0 = 0, ∂p1 ∂p1 (25) ∂ V¯ ∂Φ T (x, p1 , p2 ) = 2Φ P0 = 0, ∂p2 ∂p2 and uniquely minimizing V¯ with respect to u ˜1 and u ˜˙ 1 .
(18)
Note that Theorem 2 guarantees differentiability of V , though Theorem 1 does not consider the differentiability of the CLF obtained by minimum projection method.
˜ →R We can easily design a dynamic SCLF V˜ : R8 ⊃ X for the differentially flat system (18) as follows: (19) V˜ (φ) =φT P0 φ, where, P0 is a solution of the Riccati equation denoted in Proposition 1 for (18) with R = I,Q = I. According to Proposition 1, the following function V¯ : ¯ → R is an SCLF for (18) with respect to R8 ⊃ X ˜˙ 1 ) ⊂ R8 : Φ(x, u ˜1 , u
In order to proof Theorem 2, we employ following three lemmas. Lemma 2. There exist mappings C : R6 → R2×2 , D : R6 → R2 that (25) is equivalent to the following equation: C(x)p = D(x), (26) ¯ ˙ ˜1 , u ˜1 ) and Proof 1. Derivatives (∂ V /∂ u ˜1 )(x, u ˜1 , u ˜˙ 1 ) consist of linear functions with respect (∂ V¯ /∂ u ˜˙ 1 )(x, u ˜˙ 1 for every fixed x according to (15). to u ˜1 or u
T
φ˙ = [φ2 , φ3 , φ4 , v1 , φ6 , φ7 , φ8 , v2 ] .
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Lemma 3. The implicit function p = [p1 , p2 ] : R6 → R2 defined by (25) is uniquely determined. Proof 2. V¯ is proper and bounded below; there must exist p minimizing V¯ for an x. Properness of V¯ guarantees that a set L(x) = argminp V¯ (x, p) is compact. This implies that the matrix C(x) is nonsingular for all x. Therefore there exists a unique minimizer p of V¯ . Owing to nonsingularity of C(x), smoothness of function V´ is confirmed by implicit function theorem introduced by Isidori (1995). Lemma 4. The following function V´ : R6 ⊃ X → R is a C ∞ differentiable function: V´ (x) =ΦT (x, p1 (x), p2 (x))P0 Φ(x, p1 (x), p2 (x)) (27) =ΦT (x, p(x))P0 Φ(x, p(x)). Proof 3. We define the following mapping F with mappings C, D in (26): F (x, p) = C(x)p − D(x). Note that ∂F = C(x) ∂p is nonsingular in x. Therefore, according to implicit function theorem, there exist a C ∞ differentiable mapping p(x) satisfying F (x, p(x)) = 0. (28) Then, the following function V´ (x) = ΦT (x, p(x))P0 Φ(x, p(x)) (29) is a C ∞ differentiable function. We show the proof of Theorem 2. Proof 4. Let x ¯ = (x, p(x)) with p(x) in Lemma 3, u ¯0 = [u0 , v0 ]T be a constant number satisfying V¯˙ (¯ x, u0 , v0 ) < 0. Then, the following inequality holds for sufficiently small δ > 0: ¯ V´ (ψ(δt, x; u0 )) ≤ V¯ (ψ(δt, x ¯; u ¯0 )) < V¯ (¯ x), (30) where, ψ(t, x; u) is a unique solution of the differential equation (13) with Lebesgue measurable input u(t) ∈ R2 ¯ for t starting at x, ψ(δt, x ¯; u ¯0 )) is also a unique solution of the differential equation (18) with Lebesgue measurable input u ¯(t) ∈ R4 for t starting at x ¯.Bacciotti et al (2013) ˙´ Therefore, V (x, u0 ) < 0. This implies the function V´ is a SCLF in X ⊂ R6 . Remark 3. The static SCLF V´ obtained by the Theorem 2 is valid in the following subset: (31) X = {x|V´ (x) < Rmax }, where, Rmax = minx {V´ (x)|Lg V´ (x) = 0, Lf V´ (x) ≥ 0} = 3.57.
In domain of attraction Rmax , we proposed a new SCLF for the PVTOL system: Theorem 3. The following function V : R6 ⊃ X → R is a static SCLF; V´ (x)Rmax V (x) = . (32) Rmax − V´ (x) In Fig. 2, we illustrate V on the surface x1 = 1, x2 = x4 = x6 = 0. 223
0 Fig. 2. The obtained SCLF of the PVTOL aircraft system on x1 = 1, x2 = x4 = x6 = 0 Remark 4. Our proposed SCLF V´ , V are smooth function. This means that we can apply many control strategy based on differentiability CLFs. 5.2 Static State Feedback Controller Design for the PVTOL System We design an inverse optimal controller for the PVTOL system (13) with the smooth static SCLF V by Lemma 1 as follows: Proposition 2. Consider (13) and SCLF V designed by Theorem 3. Then, the following controller u : R6 ⊃ X → R2 asymptotically stabilizes the origin of the PVTOL system (13): ] u ˜1 (x) = u= u2 (x) ] )[ ( Lf V + |Lf V | Lg1 V − + C L g2 V (Lg1 V )2 + (Lg2 V )2 (33) V = ̸ 0) (L g 0 (Lg V = 0) where, Lg1 V = (∂V /∂x)g1 (x), Lg2 V = (∂V /∂x)g2 (x) and C is an arbitrary positive constant number. [
Moreover, the controller guarantees a sector margin (1/2, +∞). Proof 5. For all x ∈ X, Lf V < 0 if Lg V = 0.
Then, let a1 , a2 = 1 and (Lg1 V )2 + (Lg2 V )2 R(x) = Lf V + |Lf V | + C{(Lg1 V )2 + (Lg2 V )2 } Then, V˙ (x) < 0, ∀x ∈ X.
(34)
Hence, assumptions in Lemma 1 hold for an arbitrary C > 0. Therefore, the input (33) asymptotically stabilizes the origin of the system (13) for all x ∈ X, and guarantees the sector margin (1/2, +∞). Remark 5. Controllers designed by our proposed SCLF V´ , V don’t suffer from singularity depicted in Remark 1 because these controllers do not use dynamic input transformation (22)
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5
6 x1 x
4
x
x
x (m),(rad)
x(m),(rad)
x
5
4
2 1
3 2 1
0
0
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4
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0
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−1 0
20
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u=
−2 −2.5 6
8
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8
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14
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We propose an adaptive control law with the smooth SCLF V which stabilize the origin of the system (35). Theorem 4. Consider the system (35). The following controller u(x, w) ˆ asymptotically stabilizes the origin of the system (35) for all x ∈ X:
1
u2
4
6
Fig. 5. State response, x(0) = (0.5, 0, 0.5, 0, 0, 0) under parameter uncertainty w = 0.1.
tilde u
1
2
4
0
Fig. 3. State response, x(0) = (0.5, 0, 0.5, 0, 0, 0).
u(x)
x
3
5
3
−3 0
1
5
3
20
[
u ´1 (x) u ´2 (x)
]
=
[
ˆ u ˜1 (x) − w u2 (x)
w ˆ˙ =Lg1 V (x),
]
(36) (37)
where, V is a smooth SCLF for the PVTOL system (13) designed by Theorem 3, u ˜1 , u2 are control inputs for the nominal PVTOL system (13) designed by Proposition 2, w ˆ is estimated value of w.
Fig. 4. Input response, x(0) = (0.5, 0, 0.5, 0, 0, 0). 5.3 Computer Simulation Result With the proposed static SCLF V , let C = 0.7 and we can easily design an inverse optimal controller for the PVTOL system. We show a simulation result in the case of x(0) = (0.5, 0, 0.5, 0, 0, 0) . We can confirm that states can be stabilized at the origin by Fig. 3, and control inputs smoothly change with respect to time by Fig. 4. By these figures we can confirm that the proposed static state feedback controller successfully stabilizes the PVTOL system (13) at the origin. 6. ADAPTIVE CONTROL OF THE PVTOL SYSTEM In this section, we consider the following system which is changed by the parameter uncertainty w from the PVTOL system (13):
We show a simple proof of the Theorem 4 Proof 6. Consider the following proper positive definite function Vˆ : X × R → R≥0 : 1 ˆ 2. Vˆ (x, w) ˆ = V (x) + (w − w) 2
(38)
Here, the following equations hold: ˙ Vˆ (x, w) ˆ =V˙ (x) − w(w ˆ˙ − w) ˆ ∂V (x) [f (x) + g1 (x)(´ u1 + w) + g2 (x)´ u2 ] = ∂x − w(w ˆ˙ − w) ˆ u1 + w) =Lf V (x) + Lg1 V (x)(´ u2 − w(w ˆ˙ − w) ˆ + Lg V (x)´ 2
u1 + w) + g2 (x)´ u2 , (35) x˙ = f (x) + g1 (x)(´ where, u = [´ u1 , u ´2 ] is a new control input, the origin of the system is x = 0. We propose a controller for asymptotic stabilization of the origin of (35). In the system (35), parameter uncertainty w occurs offset error in equilibrium point. Fig. 5 illustrate state response of the system (35) with w = 0.1 and a controller designed in (33) for the nominal PVTOL system (13). We can confirm a offset error due to the parameter uncertainty. 224
u1 + w − w) ˆ =Lf V (x) + Lg1 V (x)(˜ u2 − Lg1 V (x)(w − w) ˆ + Lg2 V (x)´ u1 + Lg2 V (x)u2 =Lf V (x) + Lg1 V (x)˜ ≤0.
(39)
Then, the function Vˆ is a weak Lyapunov function for the system (35). Therefore, the controller asymptotically stabilize the origin of the system (35). Here, we show a simulation of the the system (35) under w = 0.1 and a controller designed by Theorem 4. State response converges to the origin in Fig. 6. Estimated value
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CLFs for the PVTOL system via minimum projection method and dynamic extension. We apply inverse optimal control and adaptive control based on smooth CLF.
x
1
x
5
3
x5
We confirm that the proposed controller successfully stabilizes a desired operating point.
x (m),(rad)
4 3
Note that the proof Theorem 2 are not limited to the case of the PVTOL system. The results in the paper would extended to general differentially flat control systems. This remains future study.
2 1 0 −1 0
2
4
6
0
8
10 12 time (sec)
14
16
18
REFERENCES
20
Fig. 6. State response with adaptive control x(0) = (0.5, 0, 0.5, 0, 0, 0) under parameter uncertainty w = 0.1. 1.5
1 0.5
u(x)
0 −0.5 −1 −1.5 −2
tilde u −hat w 1
−2.5 −3 0
u2 5
0
10 time (sec)
15
20
Fig. 7. Input response with adaptive control x(0) = (0.5, 0, 0.5, 0, 0, 0) under parameter uncertainty w = 0.1. 0.35 0.3
hat w
0.25 0.2 0.15 0.1 0.05 0 0
0
221
5
10 time (sec)
15
20
Fig. 8. Estimated value w, ˆ x(0) = (0.5, 0, 0.5, 0, 0, 0) under parameter uncertainty w = 0.1. w ˆ converges to w = 0.1. We can confirm that the controller compensates the offset error depicted in Fig. 5. 7. CONCLUSION This paper considers a static state feedback controller design problem for the PVTOL system. We proposed smooth 225
J. Hauser, S. Sastry and G. Meyer. Nonlinear control design for slightly nonminimum phase systems: Application to V/STOL Aircraft. Automatica, volume 28, pages 665–679, 1992. Turker Turker, Tugce Oflaz, Haluk Gorgun and Galip Cansever. A Stabilizing Controller for PVTOL Aircraft. American Control Conference, pages 909–913, 2012. A. R. Teel. A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Transactions on Automatic Control, volume 41, pages 1256–1270, 1996. I. Fantoni and R. Lozano. Non-linear Control for Underactuated Mechanical Systems. Springer, 2002. O. Saber. Global configuration stabilization for VTOL aircraft with strong input coupling. IEEE Transaction on Automatic Control, volume 47, pages 1949–1952, 2002. Minh-Uuc Hua, Tarek Hamel, Pascal Monrin and Claude Samason. Introduction to Feedback Control of Underactuated VTOL Vehicles. IEEE Control Systems Magazine, volume 33, 2013. T. Yamazaki, Y. Yamashita and H. Nakamura. Static nonsmooth control Lyapunov function design via dynamic extension. 50th IEEE Conference on Decision and Control and European Control Conference (CDCECC), pages 7573–7578, 2011. Andrea Bacciotti and Lionel Rosier. Liapunov Functions and Stability in Control Theory. Springer, 2001. R. A. Freeman, P. V. Kokotovi. Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. 1996. Nami. Nakamura, H. Nakamura and H. Nishitani. Global inverse optimal control with guaranteed convergence rates of input affine nonlinear systems. IEEE Transactions on Automatic Control, 2011. A. Isidori Nonlienar Control Systems. Springer, 1995. Michel Fliess, Jean L´evine, Philippe Martin, and Pierre Rouchon. A LieBacklund Approach to Equivalence and Flatness of Nonlinear Systems. IEEE TRANSACTIONS ON AUTOMATIC CONTROL, volume 44, pages 922–937, 1999. I. Fantoni, A. Zavala and R. Lonzano Global stabilization of a PVTOL aircraft with bounded thrust. 41th IEEE Conference on Decision and Control, volume 76, pages 1833–1844, 2002.
ScienceDirect IFAC-PapersOnLine 48-11 (2015) 216–221
Adaptive Adaptive control control via Minimum via Minimum
for for the the PVTOL PVTOL system system Projection Method Projection Method
∗∗ ∗∗ Ysuyuki Satoh ∗∗∗ ∗∗∗ Hisakazu Nakamura Hisakazu Nakamura ∗∗ Ysuyuki Satoh ∗∗∗ Hisakazu Nakamura Hisakazu Nakamura ∗∗ Ysuyuki Ysuyuki Satoh Satoh ∗∗∗ ∗ ∗ Tokyo University of Science, University of Science, ∗ ∗ Tokyo Tokyo University of Tokyo University of Science, Science, Yamazaki 2641, 2641, Noda, Chiba 278-8150, 278-8150, Japan (e-mail: (e-mail: Yamazaki Noda, Chiba Japan Yamazaki Noda, Chiba 278-8150, Japan Yamazaki 2641, 2641,[email protected]). Noda, Chiba 278-8150, Japan (e-mail: (e-mail: [email protected]). ∗∗ [email protected]). ∗∗ Tokyo [email protected]). of Science (e-mail: [email protected]) Tokyo University of Science (e-mail: [email protected]) ∗∗ ∗∗∗∗∗ Tokyo University of Science (e-mail: [email protected]) Tokyo University of of Science (e-mail: [email protected]) ∗∗∗ Tokyo University Science ([email protected]) Tokyo University of Science ([email protected]) ∗∗∗ ∗∗∗ Tokyo University of Science ([email protected]) Tokyo University of Science ([email protected])
Soki Soki Soki Soki
∗ ∗ Kuga Kuga ∗ Kuga Kuga ∗
Abstract: “Dynamic “Dynamic extension” extension” is is commonly used for for stabilization stabilization of of planar planar vertical vertical take take Abstract: commonly used Abstract: “Dynamic extension” commonly used stabilization planar vertical Abstract: “Dynamic extension” isMost commonly used for for stabilization ofmethod planar are vertical take off and and landing landing (PVTOL) system. is Most of controllers controllers designed by this thisof method are basedtake on off (PVTOL) system. of designed by based on off and (PVTOL) system. Most off and landing landing (PVTOL) system. Most of of controllers controllers designed designed by by this this method method are are based based on on “dynamic” control Lyapunov functions(CLFs). “dynamic” control Lyapunov functions(CLFs). “dynamic” Lyapunov “dynamic” control Lyapunov functions(CLFs). functions(CLFs). ∞ differentiable We design design aacontrol C∞ “static” strict CLF for the PVTOL system by dynamic extension We C “static” strict CLF for the PVTOL system by dynamic extension ∞ ∞ differentiable We design a C differentiable “static” strict CLF for the PVTOL system by dynamic extension We design a C differentiable “static” strict CLF for the PVTOL system by dynamic extension and minimum projection method. Then we propose an adaptive control law with the an inverse and minimum projection method. Then we propose an adaptive control law with the an inverse and minimum projection method. Then we propose an adaptive control law with the an inverse and minimum projection method. Then we propose an adaptive control law with the an inverse optimal controller based on the static CLF. optimal controller based on the static CLF. optimal controller controller based based on on the the static static CLF. CLF. optimal © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear control, Control Lyapunov function, Adaptive control, Inverse optimal Keywords: Nonlinear control, Control Lyapunov function, Adaptive control, Inverse optimal Keywords: Nonlinear control, Control Control Lyapunov Lyapunov function, function, Adaptive Adaptive control, control, Inverse Inverse optimal optimal Keywords: Nonlinear control, control, PVTOL system. control, PVTOL system. control, PVTOL PVTOL system. system. control, 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. VTOL aircrafts such as V-22 or quadrotors attract much VTOL aircrafts such as V-22 or quadrotors attract much VTOL aircrafts aircrafts such years. as V-22 V-22Planar or quadrotors quadrotors attract much VTOL such as or attract much attentions in recent vertical take off and attentions in recent years. Planar vertical take off and attentions in recent recentsystem years. introduced Planar vertical vertical take off offetand and attentions in years. Planar take landing (PVTOL) by Hauser al landing (PVTOL) system introduced by Hauser et al landingis (PVTOL) (PVTOL) system introduced by Hauser et al al landing system introduced by Hauser et (1992) a useful model to study position control of VTOL (1992) is a useful model to study position control of VTOL (1992) is a useful model to study position control of VTOL (1992) is a useful model to study position control of VTOL aircrafts. aircrafts. aircrafts. aircrafts. A common and useful control strategy is one of proposed A common and useful control strategy is one of proposed A common common and useful control strategy is one one of ofextension. proposed A and useful control strategy is proposed by Hauser et al (1992) based on dynamic by Hauser et al (1992) based on dynamic extension. by Hauser et al (1992) based on dynamic extension. by Hauser et al (1992) based on dynamic extension. Researchers including Hauser et al (1992), Turker et al Researchers including Hauser et al (1992), Turker et al Researchers including Hauseretet et al (1992), Turker(2002) et al al Researchers including Hauser al (1992), Turker et (2012), Teel (1996), Fantoni al (2002) and Saber (2012), Teel (1996), Fantoni et al (2002) and Saber (2002) (2012), Teel Teel (1996), Fantonibased et al al (2002) (2002) and Saber Saber (2002) (2012), (1996), Fantoni et and (2002) proposed many controllers on dynamic extension. proposed many controllers based on dynamic extension. proposed many controllers based on on dynamicpoint extension. proposed many controllers based dynamic extension. The method stabilizes an arbitrary operating with a The method stabilizes an arbitrary operating point with The method stabilizes an arbitrary operating point with aaa The method stabilizes an arbitrary operating point with large domain of attraction of the state space. large domain of attraction of the state space. large domain domain of of attraction attraction of of the the state state space. space. large However, these controllers are based on nonsmooth “dyHowever, these controllers are based on nonsmooth “dyHowever,control these Lyapunov controllers functions are based based(CLFs), on nonsmooth nonsmooth “dyHowever, these controllers are on “dynamic” and lost ronamic” control Lyapunov functions (CLFs), and lost ronamic” control control Lyapunov functions (CLFs), and lost lost See ronamic” Lyapunov functions (CLFs), and robustness due to their dynamic input transformations. bustness due to their dynamic input transformations. See bustnessetdue due to their dynamic dynamic input transformations. See bustness to their input transformations. See Hauser al (1992), Teel (1996) and Uuc et al (2013). Hauser et al (1992), Teel (1996) and Uuc et al (2013). Hauser et et al al (1992), Teel Teel (1996) and and Uuc strategy et al al (2013). (2013). Hauser (1992), (1996) Uuc et Moreover, differentiable CLF-based control can Moreover, differentiable CLF-based control strategy can Moreover, differentiable CLF-based control strategy can Moreover, differentiable CLF-based control strategy can not be applied. not be applied. not be be applied. applied. not ∞ ∞ differentiable “static” In this paper, we propose aa C In this paper, we propose C ∞ differentiable “static” In this this paper, we for propose a C C∞ differentiable “static” In paper, we propose a differentiable “static” strict CLF (SCLF) stabilization of the PVTOL system. strict CLF (SCLF) for stabilization of the PVTOL system. strict CLF (SCLF) for stabilization of the PVTOL system. strict CLF (SCLF) for stabilization of the PVTOL system. The proposed SCLF is designed by following two steps. The proposed SCLF is designed by following two steps. The proposed SCLF is designed by following two steps. The proposed SCLF is designed by following two steps. First, we design a “dynamic” SCLF for the augmented sysFirst, we design a “dynamic” SCLF for the augmented sysFirst, we design a “dynamic” SCLF for the augmented sysFirst, we design a “dynamic” SCLF for the augmented system obtained obtained by by dynamic dynamic extension extension of of the the PVTOL PVTOL system. system. tem tem obtained by dynamic extension of the PVTOL system. tem obtained by dynamic extension of the PVTOL system. Then, we design a static SCLF for the PVTOL system by Then, we design aa static SCLF for the PVTOL system by Then, we design static SCLF for the PVTOL system by Then, we design a static SCLF for the PVTOL system by minimum projection method proposed by Yamazaki et al minimum projection method proposed by Yamazaki et al minimum projection method proposed by Yamazaki et al minimum projection method proposed by Yamazaki et al (2013). (2013). (2013). (2013). Our proposed SCLF is a smooth function. We propose Our proposed SCLF is a smooth function. We propose Our proposed SCLF is a smooth function. We propose Our proposed SCLF is a smooth function. We propose an adaptive control law with an inverse optimal controller an adaptive control law with an inverse optimal controller an adaptive control law with an inverse optimal controller an adaptive control law with an inverse optimal controller based on on the the designed designed differentiable differentiable static static SCLF. SCLF. Finally, Finally, based based on designed differentiable static Finally, based on the the designed differentiable static SCLF. SCLF. Finally, we show computer simulation to confirm that our conwe show computer simulation to confirm that our conwe we show show computer computer simulation simulation to to confirm confirm that that our our concon1 1 1 1
troller can stabilize origin troller can stabilize the the origin of of the the PVTOL PVTOL system system under under troller can the troller can stabilize stabilize the origin origin of of the the PVTOL PVTOL system system under under parameter uncertainty. parameter uncertainty. parameter parameter uncertainty. uncertainty. 2. PRELIMINARIES PRELIMINARIES 2. 2. 2. PRELIMINARIES PRELIMINARIES 2.1 Control Control System System on on Extended Extended Space Space 2.1 2.1 2.1 Control Control System System on on Extended Extended Space Space Consider the following nonlinear control system: Consider the following nonlinear control system: Consider nonlinear control system: Consider the the following following nonlinear control system: ˙x˙ = x f (x) + g(x)u, (1) = f (x) + g(x)u, (1) x ˙ = f (x) + g(x)u, (1) x ˙ = f (x) + g(x)u, (1) n m n m where, x x∈ ∈R Rn and and u u∈ ∈R Rm denote denote aa state state and and an an input input where, n m n n n n×m where, x and state an input where, x∈ ∈R RMappings and u u∈ ∈ff R R: Rndenote denote aand state and an R input na n×m respectively. →R gg :: and Rnn → respectively. Mappings R R n n n×m n → n and n → n×m respectively. Mappings ff ::: R R → R and ggwith :: R R → R respectively. Mappings R → R and R → R are supposed to be Lipschitz continuous respect to are supposed to be Lipschitz continuous with respect to are supposed to be Lipschitz continuous with respect are supposed be Lipschitz continuous with respect to both x and and u, u,to and and satisfy ff (0, (0, 0) = = 0.(Bacciotti 0.(Bacciotti et to al both x satisfy 0) et al both both x x and and u, u, and and satisfy satisfy ff (0, (0, 0) 0) = = 0.(Bacciotti 0.(Bacciotti et et al al (2013)) (2013)) (2013)) (2013)) This paper paper also also considers considers the the following following augmented augmented system system This n+lfollowing This paper also considers considers the following augmented system system This paper also the augmented n+l of (1) on extended space R : of (1) on extended space R : n+l n+l : ] ] of space R of (1) (1) on on extended extended [ space R[ : [ ] [ ] [ ] ˙˙ ] + g(x)u [x ] [ [ ff (x) ] x + g(x)u x ˜˜˙˙ = (2) x ˙˙ = ff (x) (x) + g(x)u x = = (2) x (x) + g(x)u p v ˙x x = p˙ = = (2) v ˜˜˙ = (2) ˙p˙ p vv ˜(x) + g˜(x)˜ = f u , (3) ˜ = f (x) + g˜(x)˜ u , (3) = u (3) = ff˜˜(x) (x) + + gg˜˜(x)˜ (x)˜ u,, (3) ll ll denotes a virtual state, v ∈ R a virtual input where p ∈ R a virtual state, v ∈ R a virtual input where p ∈ R ll denotes l l a virtual input T denotes a virtual state, v ∈ R where p ∈ R denotes a virtual state, v ∈ R a virtual input where p ∈[u,Rv] T. and u ˜˜ = and = and u u = [u, [u, v] v]TT ... and u ˜˜ = [u, v] 2.2 Control 2.2 Control Lyapunov Lyapunov Function(CLF) Function(CLF) 2.2 2.2 Control Control Lyapunov Lyapunov Function(CLF) Function(CLF) In this paper, we design aa controller based on In this paper, we design controller based on strict strict In paper, design controller based In this this Lyapunov paper, we we functions design aa (SCLFs). controller Throughout based on on strict strict control Lyapunov functions (SCLFs). Throughout this control this control Lyapunov functions this control Lyapunov functions (SCLFs). Throughout this ´´ X ˜˜ (SCLFs). ¯¯ areThroughout paper, state spaces X, X, and X defined in the paper, state spaces X, X, X defined in the ´´ X ˜˜ and ¯¯ are paper, state spaces X, X, X and X are defined in the paper, state spaces X, X, X and X are defined in the neighborhood of of the the origin. origin. neighborhood neighborhood of the origin. neighborhood of the origin. Definition 1. (Strict Control Lyapunov function (SCLF)). Definition 1. (Strict Control Lyapunov function (SCLF)). Definition 1. (Strict Control Lyapunov function (SCLF)). Definition 1. (Strict Control Lyapunov function (SCLF)). (Nami et al (2013)) Consider system (1). Then, aa proper (Nami et al (2013)) Consider system (1). Then, proper 1 n (Nami et al (2013)) Consider system (1). Then, proper (Nami et al (2013)) Consider system (1). Then, aa proper positive definite definite C C 11 function function V V :: R Rnn ⊃ ⊃X X → →R R satisfying satisfying positive 1 n 1→ positive definite C function V :: R ⊃ X R satisfying positive definite C function V R ⊃ X → R satisfying 1 the following condition is said to be a C strict control the following condition is said to be aa C 1 strict control the condition is to the following following condition is said said to be be a C C 1 strict strict control control Lyapunov function (SCLF) for (1): Lyapunov function (SCLF) for (1): Lyapunov function (SCLF) for (1): Lyapunov function (SCLF) for (1):
Copyright © 2015, IFAC 220 2405-8963 © IFAC (International Federation of Automatic Control) Copyright IFAC 2015 2015 220 Hosting by Elsevier Ltd. All rights reserved. Copyright IFAC 2015 220 Copyright ©under IFAC responsibility 2015 220Control. Peer review© of International Federation of Automatic 10.1016/j.ifacol.2015.09.186
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Lf V (x) < 0, ∀x ∈ {x ∈ X|Lg V = 0}, (4) where Lf V = (∂V /∂x)f (x) and Lg V = (∂V /∂x)g(x). Freeman et al (1996) proposed an SCLF design method for nonlinear control system. For feedback linearizable systems, the following proposition holds: Proposition 1. Suppose there exists a diffeomorphism φ = Φ(x) with Φ(0) = 0 which transforms system (1) into φ˙ = Aφ + B [l0 (φ) + l1 (φ)u] , (5) where, the matrix pair (A, B) is controllable and mappings l0 : Rn → Rn and l1 : Rn → Rn×m are continuous, and l1 (Φ) is nonsingular for all Φ. Let P be a symmetric positive definite solution of the following Riccati equation: AT P + P A − P BR−1 B T P + Q = 0 (6) for arbitrary positive definite matrices Q and R. Then, function V˜ (x) : Rn ⊃ X → R defined by the following function is an SCLF for (1): (7) V˜ (x) = ΦT (x)P Φ(x). 2.3 Static CLF Design via Dynamic Extension Yamazaki et al (2013) proposed a nonsmooth CLF design method via minimum projection method. A CLF for the system (1) is heavily related to a CLF for the augmented system (2) as shown in the following theorem. ¯ →R Theorem 1. Let a continuous function V¯ : Rn+l ⊃ X be a CLF for (2). Then, the following function V : Rn ⊃ X → R is a CLF for (1): V (x) = min V¯ (x, p). (8) p∈Rl
Note that Theorem 1 does not guarantee the differentiability of V . 2.4 Inverse Optimal Control with SCLF Nami et al (2013) proposed an inverse optimal controller with SCLF. One of advantages of the inverse optimal control is a sector margin defined below. Definition 2. (Sector nonlineariry). A continuous mapping ϕ : Rm → Rm is said to be a sector nonlinearity in [α, β) with respect to u if ϕ(0) = 0 and αu2 ≤ uϕ(u) < βu2 , ∀u ̸= 0. Definition 3. (Sector margin). Consider a state feedback control u = k(x). A function ϕ(u) is supposed to have a sector nonlinearity [α, β) with respect to u. Then, the control input u for system (1) is said to have a sector margin [α, β) if the origin of the closed-loop system x˙ = f (x) + g(x)ϕ(u(x)) is asymptotically stable. With an SCLF, we can design an inverse optimal controller achieving a sector margin by the following lemma: Lemma 1. Let aj > 0 be a constant for all j ∈ {1, . . . , m}, V : Rn ⊃ X → R an SCLF for systems (1). Then, we define γ as follows: m ∑ 1 1 · |Lgj V (x)|aj +1 − Lf V (x), (9) γ(x) = a + 1 R(x) j=1 j 221
Fig. 1. The PVTOL aircraft where, R : X → R>0 is a positive-valued function on X\{0} and γ(x) is a positive definite function. Then, the following input uj : Rn ⊃ X → Rm asymptotically stabilizes the origin of the system (1): 1 |Lg V (x)|aj sgn(Lgj V (x)), uj (x) = − (10) R(x) j (j = 1, . . . , m), and minimizes the following cost function: ∫ ∞ m ∑ aj γ(x) + J= R1/aj (x)|uj |(aj +1)/aj dt. a + 1 j 0 j=1
(11) Further the input is continuous on X\{0} and achieves at least a sector margin [1/(min1≤j≤l aj + 1), ∞). 3. PROBLEM STATEMENT
The following equation denotes a normalized equation of motion of the PVTOL aircraft with gravitational acceleration g = 1 introduced by Hauser et al (1992) (Fig. 1): x˙ = [x˙ 1 , x˙ 2 , x˙ 3 , x˙ 4 , x˙ 5 , x˙ 6 ]T 0 0 x2 0 − sin x5 ε cos x5 [ ] 0 0 x u1 , = 4 + −1 cos x5 ε sin x5 u2 x6 0 0 0 1 0
(12)
where, x = [x1 , x2 , ..., x6 ]T ∈ R6 and u = [u1 , u2 ]T ∈ R2 denote respective a state variable and a control input. u1 , u2 are the vertical thrust attached to the aircraft and rolling moment respectively. Michel et al (1999) comment that the normalized lengths x1 , x3 represent actual lengths divides by the intensity g of the gravitation field, hence a normalized length of 1 represents an actual length of about 10 m. ε denotes a coupling coefficient between u2 and lateral acceleration εu2 . When small ε, the system is a strong nonminimum phase system. In many VTOL aircrafts, thrust spindles are vertically attached to the body; i.e., the inputs coupling ε = 0 structurally. Hence in this paper we suppose ε = 0; then the system becomes an under-actuated system.
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“−1” appeared in both Fig. 1 and the first term of the right-hand side of (12) denotes the normalized gravitational acceleration. Note that the origin of (12) is an equilibrium when u1 = 1. Let a new vertical control input u ˜1 = u1 − 1, and then we obtain the following control system: 0 x2 − sin x5 − sin x5 0 x4 x˙ = + cos x5 − 1 cos x5 x6 0 0 0 = f (x) + g1 (x)˜ u1 + g2 (x)u2 .
0 0[ ] ˜1 0 u 0 u2 0 1
(13)
(14)
In this paper, we consider a static state feedback controller design problem for asymptotic stabilization of the origin of (13). 4. CONVENTIONAL DYNAMIC EXTENSION TO THE PVTOL SYSTEM In this section, we apply conventional dynamic extension to PVTOL system (13), and obtain a dynamic SCLF for the linear augmented system of (13). In accordance with Hauser et al (1992), we consider the following coordinate and input transformation with a new ˜ ⊂ R8 and v ∈ R2 : state variable φ ∈ X T ˜1 , u ˜˙ 1 ) φ = [φ1 , φ2 , . . . , φ8 ] = Φ(x, u x1 x2 −(1 + u ˜1 ) sin x5 ˜1 )x6 cos x5 − u ˜˙ 1 sin x5 −(1 + u (15) = , x3 x4 (1 + u ˜1 ) cos x5 − 1 ˜˙ 1 cos x5 −(1 + u ˜1 )x6 sin x5 + u v=
[
[
v1 v2
]
x26 sin x5 = −x26 cos x5 [ − sin x5 + cos x5
(16) ] u ˜1 + 1 u ˜˙ 1 ][ ] ¨ −(˜ u1 + 1) cos x5 u ˜1 . −(˜ u1 + 1) sin x5 u2 −2x6 cos x5 −2x6 sin x5
][
(17)
Then, we can transform the system (1) into the following linear augmented control system:
˜˙ 1 ) : = V˜ (Φ(x, u ˜1 , u ˜˙ 1 )) V¯ (x, u ˜1 , u T ˜1 , u ˜˙ 1 )P0 Φ(x, u ˜1 , u ˜˙ 1 ). = Φ (x, u
(20)
(21) Remark 1. Note that the input transformation (17) is valid locally in the neighborhood of the operating point x1 , x3 = 0 due to the following inverse mapping of (17) has a singularity at u ˜ = −1: 12 [ ] u1 + 1) x6 (˜ u ˜¨1 2x6 u ˜˙ 1 = u2 − (˜ u1 + 1) [ ] − sin x5 cos x5 v1 − cos x sin x 5 5 . (22) + v2 (˜ u1 + 1) (˜ u1 + 1)
Uuc et al (2013) introduce this singularity due to dynamic input transformation. Here, V¯ is valid in the following set due to the singularity: ¯ = {(x, u ¯ max }, X ˜1 , u ˜˙ 1 )|V¯ (x, u ˜1 , u ˜˙ 1 ) < R (23) ¯ ¯ ˙ ˜1 ) = 2.44. ˜1 , u where, Rmax = minu˜1 =−1 V (x, u Remark 2. Dynamic control input v designed by dynamic extension may lost the robustness due to the dynamic input transformation (22). 5. PROPOSED MINIMUM PROJECTION TO THE PVTOL SYSTEM 5.1 Static SCLF Design for the PVTOL system In this section, we develop a C ∞ “static” SCLF for the PVTOL system (13) based on minimum projection method Theorem 1 proposed by Yamazaki et al (2013). ¯ → R denoted Theorem 2. Let the function V¯ : R8 ⊃ X in (21) be a C ∞ SCLF for the system (18). Then, the following function V´ : R6 ⊃ X → R is a C ∞ SCLF for the system (1): V´ (x) =ΦT (x, p1 (x), p2 (x))P0 Φ(x, p1 (x), p2 (x)) (24) =ΦT (x, p(x))P0 Φ(x, p(x)), where, the function p(x) = [p1 , p2 ] : R6 → R2 is uniquely determined by the folloing equations: ∂ V¯ ∂Φ (x, p1 , p2 ) = 2ΦT P0 = 0, ∂p1 ∂p1 (25) ∂ V¯ ∂Φ T (x, p1 , p2 ) = 2Φ P0 = 0, ∂p2 ∂p2 and uniquely minimizing V¯ with respect to u ˜1 and u ˜˙ 1 .
(18)
Note that Theorem 2 guarantees differentiability of V , though Theorem 1 does not consider the differentiability of the CLF obtained by minimum projection method.
˜ →R We can easily design a dynamic SCLF V˜ : R8 ⊃ X for the differentially flat system (18) as follows: (19) V˜ (φ) =φT P0 φ, where, P0 is a solution of the Riccati equation denoted in Proposition 1 for (18) with R = I,Q = I. According to Proposition 1, the following function V¯ : ¯ → R is an SCLF for (18) with respect to R8 ⊃ X ˜˙ 1 ) ⊂ R8 : Φ(x, u ˜1 , u
In order to proof Theorem 2, we employ following three lemmas. Lemma 2. There exist mappings C : R6 → R2×2 , D : R6 → R2 that (25) is equivalent to the following equation: C(x)p = D(x), (26) ¯ ˙ ˜1 , u ˜1 ) and Proof 1. Derivatives (∂ V /∂ u ˜1 )(x, u ˜1 , u ˜˙ 1 ) consist of linear functions with respect (∂ V¯ /∂ u ˜˙ 1 )(x, u ˜˙ 1 for every fixed x according to (15). to u ˜1 or u
T
φ˙ = [φ2 , φ3 , φ4 , v1 , φ6 , φ7 , φ8 , v2 ] .
222
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Lemma 3. The implicit function p = [p1 , p2 ] : R6 → R2 defined by (25) is uniquely determined. Proof 2. V¯ is proper and bounded below; there must exist p minimizing V¯ for an x. Properness of V¯ guarantees that a set L(x) = argminp V¯ (x, p) is compact. This implies that the matrix C(x) is nonsingular for all x. Therefore there exists a unique minimizer p of V¯ . Owing to nonsingularity of C(x), smoothness of function V´ is confirmed by implicit function theorem introduced by Isidori (1995). Lemma 4. The following function V´ : R6 ⊃ X → R is a C ∞ differentiable function: V´ (x) =ΦT (x, p1 (x), p2 (x))P0 Φ(x, p1 (x), p2 (x)) (27) =ΦT (x, p(x))P0 Φ(x, p(x)). Proof 3. We define the following mapping F with mappings C, D in (26): F (x, p) = C(x)p − D(x). Note that ∂F = C(x) ∂p is nonsingular in x. Therefore, according to implicit function theorem, there exist a C ∞ differentiable mapping p(x) satisfying F (x, p(x)) = 0. (28) Then, the following function V´ (x) = ΦT (x, p(x))P0 Φ(x, p(x)) (29) is a C ∞ differentiable function. We show the proof of Theorem 2. Proof 4. Let x ¯ = (x, p(x)) with p(x) in Lemma 3, u ¯0 = [u0 , v0 ]T be a constant number satisfying V¯˙ (¯ x, u0 , v0 ) < 0. Then, the following inequality holds for sufficiently small δ > 0: ¯ V´ (ψ(δt, x; u0 )) ≤ V¯ (ψ(δt, x ¯; u ¯0 )) < V¯ (¯ x), (30) where, ψ(t, x; u) is a unique solution of the differential equation (13) with Lebesgue measurable input u(t) ∈ R2 ¯ for t starting at x, ψ(δt, x ¯; u ¯0 )) is also a unique solution of the differential equation (18) with Lebesgue measurable input u ¯(t) ∈ R4 for t starting at x ¯.Bacciotti et al (2013) ˙´ Therefore, V (x, u0 ) < 0. This implies the function V´ is a SCLF in X ⊂ R6 . Remark 3. The static SCLF V´ obtained by the Theorem 2 is valid in the following subset: (31) X = {x|V´ (x) < Rmax }, where, Rmax = minx {V´ (x)|Lg V´ (x) = 0, Lf V´ (x) ≥ 0} = 3.57.
In domain of attraction Rmax , we proposed a new SCLF for the PVTOL system: Theorem 3. The following function V : R6 ⊃ X → R is a static SCLF; V´ (x)Rmax V (x) = . (32) Rmax − V´ (x) In Fig. 2, we illustrate V on the surface x1 = 1, x2 = x4 = x6 = 0. 223
0 Fig. 2. The obtained SCLF of the PVTOL aircraft system on x1 = 1, x2 = x4 = x6 = 0 Remark 4. Our proposed SCLF V´ , V are smooth function. This means that we can apply many control strategy based on differentiability CLFs. 5.2 Static State Feedback Controller Design for the PVTOL System We design an inverse optimal controller for the PVTOL system (13) with the smooth static SCLF V by Lemma 1 as follows: Proposition 2. Consider (13) and SCLF V designed by Theorem 3. Then, the following controller u : R6 ⊃ X → R2 asymptotically stabilizes the origin of the PVTOL system (13): ] u ˜1 (x) = u= u2 (x) ] )[ ( Lf V + |Lf V | Lg1 V − + C L g2 V (Lg1 V )2 + (Lg2 V )2 (33) V = ̸ 0) (L g 0 (Lg V = 0) where, Lg1 V = (∂V /∂x)g1 (x), Lg2 V = (∂V /∂x)g2 (x) and C is an arbitrary positive constant number. [
Moreover, the controller guarantees a sector margin (1/2, +∞). Proof 5. For all x ∈ X, Lf V < 0 if Lg V = 0.
Then, let a1 , a2 = 1 and (Lg1 V )2 + (Lg2 V )2 R(x) = Lf V + |Lf V | + C{(Lg1 V )2 + (Lg2 V )2 } Then, V˙ (x) < 0, ∀x ∈ X.
(34)
Hence, assumptions in Lemma 1 hold for an arbitrary C > 0. Therefore, the input (33) asymptotically stabilizes the origin of the system (13) for all x ∈ X, and guarantees the sector margin (1/2, +∞). Remark 5. Controllers designed by our proposed SCLF V´ , V don’t suffer from singularity depicted in Remark 1 because these controllers do not use dynamic input transformation (22)
MICNON 2015 220 June 24-26, 2015. Saint Petersburg, Russia Soki Kuga et al. / IFAC-PapersOnLine 48-11 (2015) 216–221
5
6 x1 x
4
x
x
x (m),(rad)
x(m),(rad)
x
5
4
2 1
3 2 1
0
0
−1 0
2
4
6
8
0
10 12 time (sec)
14
16
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−1 0
20
2
1.5
0.5 0 −0.5 −1 −1.5
u=
−2 −2.5 6
8
10 12 time (sec)
8
14
16
18
10 12 time (sec)
14
16
18
20
We propose an adaptive control law with the smooth SCLF V which stabilize the origin of the system (35). Theorem 4. Consider the system (35). The following controller u(x, w) ˆ asymptotically stabilizes the origin of the system (35) for all x ∈ X:
1
u2
4
6
Fig. 5. State response, x(0) = (0.5, 0, 0.5, 0, 0, 0) under parameter uncertainty w = 0.1.
tilde u
1
2
4
0
Fig. 3. State response, x(0) = (0.5, 0, 0.5, 0, 0, 0).
u(x)
x
3
5
3
−3 0
1
5
3
20
[
u ´1 (x) u ´2 (x)
]
=
[
ˆ u ˜1 (x) − w u2 (x)
w ˆ˙ =Lg1 V (x),
]
(36) (37)
where, V is a smooth SCLF for the PVTOL system (13) designed by Theorem 3, u ˜1 , u2 are control inputs for the nominal PVTOL system (13) designed by Proposition 2, w ˆ is estimated value of w.
Fig. 4. Input response, x(0) = (0.5, 0, 0.5, 0, 0, 0). 5.3 Computer Simulation Result With the proposed static SCLF V , let C = 0.7 and we can easily design an inverse optimal controller for the PVTOL system. We show a simulation result in the case of x(0) = (0.5, 0, 0.5, 0, 0, 0) . We can confirm that states can be stabilized at the origin by Fig. 3, and control inputs smoothly change with respect to time by Fig. 4. By these figures we can confirm that the proposed static state feedback controller successfully stabilizes the PVTOL system (13) at the origin. 6. ADAPTIVE CONTROL OF THE PVTOL SYSTEM In this section, we consider the following system which is changed by the parameter uncertainty w from the PVTOL system (13):
We show a simple proof of the Theorem 4 Proof 6. Consider the following proper positive definite function Vˆ : X × R → R≥0 : 1 ˆ 2. Vˆ (x, w) ˆ = V (x) + (w − w) 2
(38)
Here, the following equations hold: ˙ Vˆ (x, w) ˆ =V˙ (x) − w(w ˆ˙ − w) ˆ ∂V (x) [f (x) + g1 (x)(´ u1 + w) + g2 (x)´ u2 ] = ∂x − w(w ˆ˙ − w) ˆ u1 + w) =Lf V (x) + Lg1 V (x)(´ u2 − w(w ˆ˙ − w) ˆ + Lg V (x)´ 2
u1 + w) + g2 (x)´ u2 , (35) x˙ = f (x) + g1 (x)(´ where, u = [´ u1 , u ´2 ] is a new control input, the origin of the system is x = 0. We propose a controller for asymptotic stabilization of the origin of (35). In the system (35), parameter uncertainty w occurs offset error in equilibrium point. Fig. 5 illustrate state response of the system (35) with w = 0.1 and a controller designed in (33) for the nominal PVTOL system (13). We can confirm a offset error due to the parameter uncertainty. 224
u1 + w − w) ˆ =Lf V (x) + Lg1 V (x)(˜ u2 − Lg1 V (x)(w − w) ˆ + Lg2 V (x)´ u1 + Lg2 V (x)u2 =Lf V (x) + Lg1 V (x)˜ ≤0.
(39)
Then, the function Vˆ is a weak Lyapunov function for the system (35). Therefore, the controller asymptotically stabilize the origin of the system (35). Here, we show a simulation of the the system (35) under w = 0.1 and a controller designed by Theorem 4. State response converges to the origin in Fig. 6. Estimated value
MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Soki Kuga et al. / IFAC-PapersOnLine 48-11 (2015) 216–221
6
CLFs for the PVTOL system via minimum projection method and dynamic extension. We apply inverse optimal control and adaptive control based on smooth CLF.
x
1
x
5
3
x5
We confirm that the proposed controller successfully stabilizes a desired operating point.
x (m),(rad)
4 3
Note that the proof Theorem 2 are not limited to the case of the PVTOL system. The results in the paper would extended to general differentially flat control systems. This remains future study.
2 1 0 −1 0
2
4
6
0
8
10 12 time (sec)
14
16
18
REFERENCES
20
Fig. 6. State response with adaptive control x(0) = (0.5, 0, 0.5, 0, 0, 0) under parameter uncertainty w = 0.1. 1.5
1 0.5
u(x)
0 −0.5 −1 −1.5 −2
tilde u −hat w 1
−2.5 −3 0
u2 5
0
10 time (sec)
15
20
Fig. 7. Input response with adaptive control x(0) = (0.5, 0, 0.5, 0, 0, 0) under parameter uncertainty w = 0.1. 0.35 0.3
hat w
0.25 0.2 0.15 0.1 0.05 0 0
0
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5
10 time (sec)
15
20
Fig. 8. Estimated value w, ˆ x(0) = (0.5, 0, 0.5, 0, 0, 0) under parameter uncertainty w = 0.1. w ˆ converges to w = 0.1. We can confirm that the controller compensates the offset error depicted in Fig. 5. 7. CONCLUSION This paper considers a static state feedback controller design problem for the PVTOL system. We proposed smooth 225
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