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Input-to-State Stabilization of Differentially Flat Systems⁎
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IFAC PapersOnLine 51-29 (2018) 256–261
Input-to-State Stabilization of Input-to-State Stabilization of Input-to-State Stabilization of ⋆⋆ Differentially Flat Systems Input-to-State Stabilization of Differentially Flat Systems Differentially Flat Systems ⋆⋆ Differentially Flat Systems∗∗ Yasuhiro Fujii ∗∗ Hisakazu Nakamura ∗∗
Yasuhiro Fujii ∗∗ Hisakazu Nakamura ∗∗ Yasuhiro Fujii ∗ Hisakazu Nakamura ∗∗ ∗∗ Yasuhiro Fujii Hisakazu Nakamura ∗ University of Science, Noda, Chiba 278-8510 Japan (e-mail: ∗ Tokyo Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: ∗ Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: [email protected]) ∗ [email protected]) ∗∗Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: [email protected]) Noda, Chiba 278-8510 Japan (e-mail: ∗∗ Tokyo University of Science, Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: ∗∗ [email protected]) Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: [email protected]). ∗∗ [email protected]). Tokyo University of [email protected]). Science, Noda, Chiba 278-8510 Japan (e-mail: [email protected]). Abstract: Abstract:∞We We consider consider input-to-state input-to-state stabilization stabilization of of aa differentially differentially flat flat system. system. Firstly, Firstly, We We Abstract: consider input-to-state stabilization of a differentially flat system. Firstly, We design a C differential strict control Lyapunov function (CLF) for the linear augmented system ∞We design a C differential strict control Lyapunov function (CLF) for the linear augmented system ∞We consider input-to-state stabilization of a differentially flat system. Firstly, We Abstract: design a C differential strict control Lyapunov function (CLF) for the linear augmented system on the extended state space by solving Riccati equation. Then, we show that the designed on the aextended state space by solving Riccatifunction equation. Then, we linear show augmented that the designed design C ∞ satisfies differential strict control Lyapunov (CLF) forfunction the system on the extended state by solving Riccaticontrol equation. Then, we show(ISS-CLF) that the property. designed CLF always the input-to-state-stability Lyapunov CLF always satisfies thespace input-to-state-stability control Lyapunov function (ISS-CLF) property. on the extended state space by solving Riccati equation. Then, we show that the designed CLF always satisfies the input-to-state-stability Lyapunov function Finally, we a and differentially flat by applying dynamic Finally, we derive derive a CLF CLF and an an ISS-CLF ISS-CLF for for the the control differentially flat system system by(ISS-CLF) applying aaproperty. dynamic CLF always satisfies the input-to-state-stability Lyapunov function Finally, weand derive a CLF and an ISS-CLF for the control differentially flat system by(ISS-CLF) applying aproperty. dynamic extension a minimum projection method. extension and a minimum projection method. Finally, weand derive a CLF and an ISS-CLF for the differentially flat system by applying a dynamic extension a minimum projection method. extension and(International a minimum Federation projectionofmethod. © 2018, IFAC Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear Nonlinear control, control, Lyapunov Lyapunov function, function, Stabilization, Stabilization, Robust Robust performance performance Keywords: Keywords: Nonlinear control, Lyapunov function, Stabilization, Robust performance Keywords: Nonlinear control, Lyapunov function, Stabilization, Robust performance 1. INTRODUCTION INTRODUCTION design 1. design method method is is effective effective at at attenuating attenuating disturbance disturbance for for 1. INTRODUCTION design method is effective at attenuating disturbance for aa vessel trajectory tracking problem. vessel trajectory tracking problem. 1. INTRODUCTION design method is effective at attenuating disturbance for Every stabilizable stabilizable control control system system on on Euclidean Euclidean space space perper- a vessel trajectory tracking problem. Every a vessel trajectory2.tracking problem. space perEvery stabilizable control system(CLF). on Euclidean mits control control Lyapunov function (CLF). In particular, particular, dis2. PRELIMINARIES PRELIMINARIES mits Lyapunov function In dis2. PRELIMINARIES Every stabilizable control system on Euclidean space permits control Lyapunovfor (CLF). In particular, turbance attenuation forfunction nonlinear control system can candisbe 2. PRELIMINARIES turbance attenuation aa nonlinear control system be In mits control Lyapunov function (CLF). In particular, disIn this this paper, paper, we we consider consider the the disturbance disturbance attenuation attenuation turbance attenuation for a nonlinear control system can be estabilished by designing input-to-state-stability control estabilished by designing input-to-state-stability control In this paper, we consider the disturbance attenuation control of differentially flat systems. We introduce basic turbance attenuation for a nonlinear control system can be control of differentially flat systems. We introduce basic estabilished by designing input-to-state-stability control Lyapunov function (ISS-CLF). In this of paper, we consider theindisturbance attenuation Lyapunov function (ISS-CLF). control differentially flat systems. We introduce basic definitions and properties used this paper. estabilished by designing input-to-state-stability control definitions and properties used in this We paper. Lyapunov function (ISS-CLF). control of differentially flat systems. introduce basic definitions and properties used in this paper. Solving a disturbance attenuation problem has an advanLyapunov function (ISS-CLF). Solving a disturbance attenuation problem has an advan- definitions and properties used in this paper. 2.1 Nonlinear systems an advanSolving disturbance attenuation problem has tage to to aguarantee guarantee control performance under distur- 2.1 Nonlinear systems tage aa control performance under distur2.1 Nonlinear systems Solving aguarantee disturbance attenuation problem has an advantage to a control performance under disturbance. Krstic and Li (1998) have proposed a disturbance 2.1 Nonlinear systems we consider the following nonlinear bance. Krstic and Lia (1998) have proposed under a disturbance tage to guarantee control performance disturThroughout the the paper, paper, we consider the following nonlinear bance. Krsticcontrol and Li method (1998) have proposed a disturbance attenuation control method by solving solving inverse optimal Throughout attenuation by inverse optimal Throughout the paper, we consider the following nonlinear control system: bance. andwith Li method (1998) have proposed afor disturbance system: attenuation control by solving inverse optimal control control Krstic problems a designed designed ISS-CLF nonlinear Throughout the paper, we consider the following nonlinear control problems with a ISS-CLF for nonlinear control system: x + (1) attenuation control method by solving inverse optimal x˙˙ = = ff (x) (x) + g(x)u g(x)u (1) control problems with a designed ISS-CLF for nonlinear control system: systems. n m control systems. x ˙ = f (x) + g(x)u (1) and u ∈ R denote a state and an input where x ∈ R control problems with a designed ISS-CLF for nonlinear n m control systems. R denote a state and an input where x ∈ Rn and ux˙ ∈ =f fR:(x) + g(x)u m n n×n n (1) Sassanosystems. and Astolfi Astolfi (2012) (2012) have have proposed proposed aa dynamic dynamic ISSISS- where and u ∈ denote a state and an input x ∈ R respectively. Mapping R → R and g : R → control n n×n n Sassano and n respectively. f R:mRdenote and : Rinput n → Ran×n n → n×m x ∈ R Mapping and to u ∈ state andgg an Sassano andmethod Astolfi for (2012) have proposed dynamic ISS- where CLF design design method for a differentially differentially flat asystem system by solvsolvrespectively. Mapping f : R → R and : R → R be locally Lipschitz continuous with n×m are supposed CLF a flat by n n×n n to be Lipschitz continuous Sassano andmethod Astolfisolution (2012) have proposed asystem dynamic ISS- R n×m are supposed respectively. Mapping f flocally :(0) R =→ Rthe origin. and g : R with → CLF design for a differentially flat by solving the algebraic of Hamilton-Jacobi-Inequality R are supposed to be locally Lipschitz continuous with respect to x, and satisfy 0 at ing the algebraic solution of Hamilton-Jacobi-Inequality n×m to x, and satisfy f (0) = 0 at the origin. respect CLF design method for a differentially flat system by solvR are supposed to be locally Lipschitz continuous with ing theA solution Hamilton-Jacobi-Inequality (HJI). Aalgebraic differentially flat of system is aa class class of of nonlinear nonlinear respect to x, and satisfy f (0) = 0 at the origin. (HJI). differentially flat system is ing theAalgebraic solution Hamilton-Jacobi-Inequality respect to x, and satisfy f (0) = 0 at the origin. Differentially Flat (HJI). differentially flat of system is a class of nonlinear systems that are linearizable linearizable by a diffeomorphism diffeomorphism with aa 2.2 2.2 Differentially Flat System System systems that are by a with (HJI). Athat differentially flat system is a class of nonlinear 2.2 Differentially Flat System systems are linearizable by a diffeomorphism with a dynamic compensator. By investigation of Levine (2009), dynamic compensator. By investigation of Levine (2009), Differentially Flat System systems that are linearizable by of a diffeomorphism with a 2.2 We dynamic compensator. By investigation of Levine (2009), most of of all all controllable systems trajectory generation, We derive derive an an augmented augmented system system defined defined on on the the virtual virtual most controllable systems of trajectory generation, dynamic compensator. By investigation of Levine (2009), We derive an augmented system defined on the virtual space by applying a dynamic extension for nonlinear most of all controllable systems of trajectory generation, motion planning, or tracking can be considered as differenspace by applying a dynamic extension for nonlinear motion planning, or tracking can be considered generation, as differen- We derive an introduce augmented system defined flat on the virtual most of all controllable systems of trajectory space by applying a dynamic extension for nonlinear systems and the differentially system as motion planning, or tracking can be considered as differentially flat systems. The control design of unmanned vessels introduce the differentially as tially flat systems.orThe control design of unmanned vessels systems space byand applying a dynamic extension flat for system nonlinear motion planning, tracking can be considered as differensystems and introduce the differentially flat system as follows. tially flat systems. The control design of unmanned vessels is one of the interesting applications (Rigatos (2015)). follows. is one of the interesting applications (Rigatos (2015)). systems and introduce the differentially flat system as tially flat systems. The control design of unmanned vessels follows. 1. (Dynamic compensator). Consider (1) and is one ofaa the interesting (Rigatos (2015)). Definition However static ISS-CLF applications design still still remains remains a problem. problem. Definition 1. (Dynamic compensator). Consider (1) and However static ISS-CLF design a follows. is one ofa the interesting (Rigatos (2015)). Definition 1. (Dynamic compensator). Consider (1) and the following dynamic compensator: However static ISS-CLF applications design still remains a problem. following dynamic compensator: [ [ A static static smooth smooth CLF designdesign method forremains differentially flat the Definition 1. (Dynamic compensator). However a staticCLF ISS-CLF still a problem. [ p˙ ] ]compensator: [a(x, p, v)] ] Consider (1) and the following dynamic A design method for aa differentially flat [ p˙ ]compensator: [a(x, p, v)] , A static smooth CLF design method for a differentially flat system by a minimum projection method was proposed by the following dynamic = (2) system by a minimum projection method was proposedflat by (2) [u p˙ ] = [a(x, p, v) v)] , b(x, A static smooth CLFThe design method for CLF a differentially method was proposed by system by a minimum projection Kuga et al. (2016). static smooth enables us to u b(x, p, = , (2) p ˙ a(x, p, v) Kuga etbyal.a (2016). The static smooth CLF enables us by to u b(x, p, v) system minimum projection method was proposed = denote a virtual , (2) Kuga (2016). static smooth CLF enables us to where p ∈ Rll and v ∈uRm designetaa al. static stateThe feedback controller that can avoid avoid state and input m b(x, p, v) design static state feedback controller that can p ∈ Rl and v ∈ Rm denote a virtual state and input Kuga etasingularities al. (2016). The static smooth CLF enables us to where design static state feedback controller that can avoid delays, points or losing robustness. and v ∈ R denote a virtual state and input where p ∈ R of the dynamic compensator, respectively. delays, singularities or losing robustness. of the pdynamic compensator, respectively. design static statepoints feedback controller that can avoid where v ∈ Rm denote a virtual state and input ∈ Rl and delays, asingularities points or losing robustness. of the dynamic compensator, respectively. In this paper, we propose a static ISS-CLF design method delays, singularities pointsa or losing robustness. the dynamic compensator, respectively. Then we introduce an augmented In this paper, we propose static ISS-CLF design method of we introduce an augmented system system as as follows. follows. In paper, we propose a static ISS-CLF design method forthis a differentially differentially flat system. system. Our proposed method can Then Then we introduce an augmented system as follows. for a flat Our proposed method can Definition 2. (Augmented system). On the state In this paper, we propose a static ISS-CLF design method Definition 2. (Augmented system). On the state space space an augmented system as follows. for a differentially flat an system. Our from proposed methodCLF can Then systematically design an ISS-CLF from a dynamic dynamic CLF n+l we introduce systematically design ISS-CLF a Definition 2. (Augmented system). On the state space , system (1) in combination with dynamic compenR n+l for a linear differentially flat an system. Our from proposed method can R , system (1) in combination with dynamic compensystematically design ISS-CLF a dynamic CLF for a augmented system. Finally, we show computer n+l Definition 2.called (Augmented system). On dynamic the state space for a linear augmented system. Finally, wea show computer , system (1) in combination with compenR sator (2) is an augmented system: systematically design an ISS-CLF from dynamic CLF n+l (2) is called sator an[combination augmented system: for a linear to augmented system. we static show computer [ ] simulation to confirm that that ourFinally, proposed ISS-CLF R , (2) system (1)] in with dynamic compen[ ] [ ] simulation confirm our sator is called an augmented system: proposed static ISS-CLF for a linear to augmented system. we static show computer ff (x) p, [x proposed ISS-CLF sator (2) is called simulation confirm that ourFinally, x˙˙ ] = (x) + + g(x)b(x, g(x)b(x, p, v) v)] an[augmented system: (3) (3) [x simulation confirm that our proposed static ISS-CLF ˙p˙ ] = [f (x) +a(x, g(x)b(x, p, v)] ,, p p, v) ⋆ This work to a(x, p, v) was partially supported by JSPS KAKENHI Grant = , (3) ⋆ This work was partially supported by JSPS KAKENHI Grant x ˙p˙ f (x) +a(x, g(x)b(x, p, v) p, v) ⋆ = , (3) Number 17H03282. where the origin is (x, p) = (0, 0). This work was partially supported by JSPS KAKENHI Grant p˙ is (x, p) = a(x, p, v) Number 17H03282. where the origin (0, 0). ⋆ This work was partially supported by JSPS KAKENHI Grant Number 17H03282. where the origin is (x, p) = (0, 0). Number 17H03282. where the originLtd. is (x, = (0, 0). 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Allp) rights reserved.
Copyright © 2018 IFAC 256 Copyright 2018 IFAC 256 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 256 10.1016/j.ifacol.2018.09.512 Copyright © 2018 IFAC 256
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We consider a differentially flat system having a diffeomorphism such that the augmented system (3) can be represented by a linearized augmented system as follows. Definition 3. (Linearized augmented system). For system (3), consider the following diffeomorphism Φ : Rn+l → Rn+l : ϕ = Φ(x, p), Φ(0) = 0, (4) where we assume that the diffeomorphism satisfy the following equation: dΦ ϕ˙ = (x, p) (5) dt ∂Φ ∂Φ (x, p)x˙ + p˙ (6) = ∂x ∂p = Aϕ + Bv. (7) The augmented system (3) is represented by the following linearized augmented system ϕ ∈ Rn+l : ϕ˙ = Aϕ + Bv, (8) where the matrix pair (A, B) is controllable. The system (8) is said to be a linearized augmented system． Definition 4. (Differentially flat system). If system (1) is linearizable via such diffeomorphism (4) by dynamic compensator (2), system (1) is said to be a differentially flat system. 2.3 Control Lyapunov functions (CLF) In this paper, we define a static CLF as follows. Definition 5. (Control Lyapunov Function). [Kuga et al. (2016)] Consider system (1). A proper positive definite C ∞ differentiable function V : Rn → R satisfying the following condition is said to be a static smooth control Lyapunov function (CLF) for (1): (9) Lf V (x) < 0, ∀x ∈ {x ∈ X ⊆ Rn |Lg V = 0} , where Lf V = (∂V /∂x)f (x) and Lg V = (∂V /∂x)g(x). Then V˙ (x, u) can be written as follows: (10) V˙ (x, u) = Lf V (x) + Lg V (x)u. 2.4 Dynamic smooth CLF design for augmented systems A static CLF for a linearized augmented system is designed by the solution of the Riccati equation. By applying the designed diffeomorphism to the static CLF, we can derive a dynamic CLF for the differentially flat system as follows. Proposition 6. [Kuga et al. (2016)] Let P be a symmetric positive definite solution of the following Riccati equation for system (8): (11) AT P + P A − P BR−1 B T P + Q = 0, where Q and R are arbitrary positive definite matrices. Then, function V˜ (ϕ) : Rn+l → R defined by the following function is a CLF for (8): V˜ (ϕ) = ϕT P ϕ. (12) n+l ¯ Moreover the following function V (Φ(x, p)) : R → R is a CLF for Φ(x, p): (13) V¯ (x, p) = ΦT (x, p)P Φ(x, p) where function V¯ (x, p) is a dynamic smooth CLF including virtual states p. 257
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2.5 Static smooth CLF design We derive a static smooth CLF as follows. Theorem 7. [Kuga et al. (2016)] Let the function V¯ : Rn+l → R obtained in (13) be a dynamic C ∞ differentiable CLF for system (3). Then the following function V : Rn → R is a static C ∞ differentiable CLF for (1): (14) V (x) = ΦT (x, p(x))P0 Φ(x, p(x)), where a function p(x) : Rn → Rl is uniquely determined by the following equation: ∂ V¯ (x, p) = 0. (15) ∂p The relation between CLF V (x) for (1) and CLF V˜ (ϕ) for (8) satisfy the following lemma. Lemma 8. [Kuga et al. (2016)] If there exist CLF V˜ (ϕ) for linear augmented system (8), then static CLF V (x) for (1) derived by a minimum projection method satisfies the following equation: (16) V (x) = min V˜ (ϕ). ϕ∈Φ(x)
2.6 Sontag type CLF based controller In this paper, we design Sontag-type CLF based controller. The controller is represented as follows. Theorem 9. [Sepulchre et al. (2012)] Consider system (1). V (x) is a CLF for the system. Then the control input (17) u = αs (x) 1 Lg V T (x) (18) =− R(x) is asymptotically stabilize at the origin and minimize the following cost function. ) ∫ ∞( 1 T J= l(x) + u R(x)u dt, (19) 2 0 where l(x) and R(x) are functions defined as follows: 1 l(x) = ∥Lg V ∥22 − Lf V (20) 2R(x) ∥L V ∥2 √ g 2 (Lg V ̸= 0) (21) R(x) = Lf V + Lf V 2 + ∥Lg V ∥42 1 (Lg V = 0).
2.7 Input-to-state stability
We introduce input-to-state stability and an input-to-state stable control Lyapunov function for disturbed nonlinear systems. Comparison functions are used to define inputto-state stability. Definition 10. (Class K function). [Sepulchre et al. (2012)] If a C 0 function γ : R≥0 → R≥0 is strictly increasing and γ(0) = 0, the function γ is said to belong class K. In addition, if γ(r) → ∞ as r → ∞, the function γ is said to belong to class K∞ . Definition 11. (Class KL function). [Sepulchre et al. (2012)] A C 0 function β : R≥0 × R≥0 → R≥0 is said to belong to class KL if for each fixed s the function β(·, s) belong to class K, and for each fixed r, function β(r, ·) is decreasing and β(r, s) → 0 as s → 0.
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We define input-to-state stability as follows. Definition 12. (Input-to-state stability(ISS)). [Krstic and Li (1998)] System (32) is said to be input-to-state stable (ISS) if there exists a controller which guarantees that ( ) ∥x(t)∥ ≤ β (∥x(0)∥ , t) + χ sup ∥w(τ )∥ , (22) 0≤τ ≤t
where β ∈ KL, χ ∈ K. Theorem 13. [Sontag and Wang (1995)] System (32) is ISS if and only if there exist a smooth positive definite radially unbounded function V (x) and class K∞ functions ν and µ such that the following dissipation inequality holds: V˙ (x) ≤ −µ (∥x∥) + ν (∥w∥) .
(23)
2.8 Input-to-state stable CLF (ISS-CLF)
2.10 ISS-CLF design for linear systems CLF V˜ (ϕ) for linear system (8) is an ISS for disturbed linear system (34) according to the following lemma. Lemma 17. We consider CLF V˜ (ϕ) = ϕT P ϕ for linear system (8). Then the function V˜ (ϕ) is an ISS-CLF for linear system (34). Proof. While due to page limit the detailed proof is omitted, the following inequality holds: ( ) 1 (31) V˜˙ (ϕ) ≤ − ϕT Q − γ 2 J ϕ + 2 ∥d∥2 . γ where J := P HH T P . In equation (31), for arbitrary positive matrix Q, there exits γ ∈ R>0 such as Q − γ 2 J is positive definite. As (23) is held, V˜ (ϕ) is an ISS-CLF for (34).
We define an ISS-CLF for nonlinear systems as follows. 3. PROBLEM STATEMENT Definition 14. (ISS Control Lyapunov Function). [Krstic and Li (1998)] A smooth positive definite radially unbounded function V : Rn → R≥0 and class K∞ is called an ISS-CLF Differentially flat systems are generally defined as no disfor system (32) if there exits a class K∞ function ρ such turbed systems. However, we consider the following disthat the following implication holds for all x ̸= 0 and all turbed differentially flat systems to deal with disturbance attenuation problems. w ∈ Rr : x˙ = f (x) + g(x)u + h(x)w (32) ∥x(t)∥ ≥ ρ (∥w∥) r n ⇒ infm {Lf V (x) + Lg V (x)u + Lh V (x)w} < 0. is bounded disturbance. Mapping h : R → where w ∈ R u∈R n×l R is supposed to be locally Lipschitz continuous with (24) respect to x, and satisfies f (0) = 0 at the origin. The following theorem establishes equivalence between ISS Differentially flat system (32) can be linearizable to the and the existence of an ISS-CLF. following linear augmented system by a diffeomorphism Theorem 15. (Krstic and Li, 1998, Theorem 2.1) System and dynamic compensator. (32) is ISS if and only if there exits an ISS-CLF. ∂Φ ϕ˙ = Aϕ + Bv + (x, p)h(x)d, (33) ∂x 2.9 Disturbance attenuation controller r where d ∈ R is disturbance. We use the following Sontag-type controller proposed by We assume here that (33) is represented as the linearized Krstic and Li (1998) to disturbance attenuation problems. augmented system as follows. Theorem 16. (Krstic and Li, 1998, Theorem 3.2) Consider Assumption 1. System (33) satisfies the following equasystem (32). If (32) is ISS, then there exist ISS-CLF tion. V (x) and the following Sontag-type control law solves the ϕ˙ = Aϕ + Bv + Hd, (34) inverse optimal gain assignment problem. (n+l)×r is a matrix as follows: where H ∈ R ∂Φ (25) u = αd (x) (x, p)h(x) (35) H= ∂x 2 T − (Lg V ) (Lg V ̸= 0) (26) αd (x) = Rd (x) 4. ISS-CLF DESIGN FOR DIFFERENTIALLY FLAT 0 (Lg V = 0) , SYSTEMS where ˜ Lg V = [Lg1 V (x) Lg2 V (x)] . (27) We consider the designed ISS-CLF V (ϕ) for linear augmented system. Then we show that CLF V (x) is an ISSThen ) is ∫ ∞ ( level ∫ ∞the achieved disturbance attenuation CLF for disturbed differentially flat systems. As the main [ ] |ω| dt. (28) theorem in this paper, it is shown in Theorem 19. Before l(x) + uT Rd (x)u dt ≤ βλ γ λ 0 0 that, we show the following lemma to support the main where β ≥ 2, λ ∈ (0, 2] and γ ∈ K∞ . ω and Rd (x) > 0 are theorem. represented as follows: Lemma 18. Consider CLF V (x) for differentially flat sys(29) tem (1) and CLF V˜ (ϕ) for linear augmented system (8). ω = Lf V (x) + |Lh V (x)| ρ−1 (|x|) T For these CLF, the following inequality is satisfied: 2L V (Lg V ) √ g (Lg V ̸= 0) (36) V (x) ≤ V˜ (Φ(x, p)) , ∀p ∈ Rl Rd (x) = ω + ω 2 + (L V (L V )T )2 g g 1 (Lg V = 0). Proof. According to Lemma 8, the following inequality (30) holds: 258
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V (x) = min V˜ (ϕ) ϕ∈Φ(x)
≤ V¯ (x, p), ∀p ∈ Rl ≤ V˜ (ϕ), ∀ϕ ∈ Φ(x, p)
259
(37) (38) (39)
We now give the main theorem in this paper. Theorem 19. We assumed that the static CLF V (x) for differentially flat system (1) is derived from the CLF V˜ (ϕ) for linearized augmented system (8) according to Proposition 6 and Theorem 7. Then the static CLF V (x) is the static ISS-CLF for disturbed differentially flat systems (32).
Fig. 1. Maneuvering kinematics coordinate system
Proof. The CLF V˜ (ϕ) is an ISS-CLF for differentially flat system if and only if (24) is satisfied. The conclusion part is rewritten as follows: } { V (x(t + ∆t)) − V (x(t)) <0 (40) infm lim ∆t→0 u∈R ∆t According to Lemma 18, both CLF V˜ (ϕ) and V (x) for the linear augmented system and differentially flat system hold the following inequality for arbitrary time t ∈ R≥0 :
Therefore, there exist input u satisfying (49) for differentially flat system as follows: ˜ p0 )) u = b(x, p0 ) ≤ k(Φ(x, (52) ≤ −R−1 B T P Φ(x, p0 )
V (x(t + ∆t)) − V (x(t)) ∆t (41) V¯ (x(t + ∆t), p) − V¯ (x(t), p) ≤ lim , ∀p ∈ Rl ∆t→0 ∆t then there exist p0 satisfying (16) as follows: { } p0 ∈ argmin V¯ (x(t), p(t)) , ∀t ∈ R≥0 (42)
In this section, we design the static smooth CLF and ISS-CLF for a vessel trajectory tracking problem. Then we design a static state feedback controller and a static disturbance attenuation controller. We compare state feedback controllers with or without disturbance attenuation to confirm the effectiveness of our static ISS-CLF design method.
lim
∆t→0
p
The initial state of (41) is represented as follows: V (x(∆t)) − V (x(0)) lim ∆t→0 ∆t (43) V¯ (x(∆t), p(∆t)) − V¯ (x(0), p(0)) ≤ lim ∆t→0 ∆t Now we can choose p(0) = p0 , then the left hand side of (43) is rewritten as follows: V (x(∆t)) − V (x(0)) V˙ = lim ∆t→0 ∆t V (x(∆t)) − V¯ (x(0), p0 ) = lim ∆t→0 ∆t V¯ (x(∆t), p(∆t)) − V¯ (x(0), p0 ) ≤ lim ∆t→0 ∆t ˙ ¯ =V
(44) (45) (46) (47)
(44) - (47) are satisfied for all time t ∈ R≥0 and x(0) ∈ Rn . According to (31), we can obtain the following inequality: V˙ (x) ≤ V¯˙ (x, p0 )
(48) ( ) 1 ≤ −ΦT (x, p0 ) Q − γ 2 J Φ(x, p0 ) + 2 ∥d∥2 (49) γ ≤ −α1 (∥x∥) + α2 (∥w∥) (50)
where α1 , α2 ∈ K∞ , α1 : r �→ λmax (Q−γ 2 J)∥Φ(r, p0 )∥ and α2 : r �→ (1/γ 2 )∥r∥2 . Then there exist input v satisfying (31) for the linearized augmented system as follows: ˜ (51) v = k(ϕ) ≤ −R−1 B T P ϕ 259
As (40) is held, the CLF for differentially flat system is always the ISS-CLF for (32). 5. EXAMPLE
5.1 Design of differentially flat systems We consider the following simple maneuvering model represented as Fig.1. x˙ = u1 cos θ + wx y˙ = u1 sin θ + wy (53) ˙ θ = u2 ,
where x, y ∈ R, θ ∈ S and u = [u1 , u2 ] denote respective state variables and control inputs. x and y are the position. θ is the azimuth angle. u1 and u2 are respective an input of forward movement and a yawing movement. wx , wy ∈ R is the total disturbance force in the direction to x and y, wind force being one such disurbance. We assume the bounded random signal such as Fig.2. We consider constant desired velocity vd ∈ R>0 , and we give a desired trajectory xd (t) = vd t, yd (t) = 0, θd (t) = 0. The tracking error e := [ex , ey , eθ ]T is written as ex = x − xd , ey = y − yd and eθ = θ − θd . Then the trajectory tracking problem is represented as the following error system: { e˙ x = u1 cos eθ + wx − vd e˙ y = u1 sin eθ + wy (54) e˙ θ = u2 .
We assumed that regular feedback ud = [u1d , u2d ]T for System (53) is defined on the origin, where u1d = vd , u2d = 0. In addition, the new input is represented as u ˜1 = u1 −vd . Then the trajectory tracking problem is rewritten as the following error system:
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{
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u1 + vd ) cos eθ − vd + wx e˙ x = (˜ e˙ y = (˜ u1 + vd ) sin eθ + wy e˙ θ = u2 .
5.5 Simulation results (55)
5.2 Deriving of linearized augmented systems We introduce an augmented system by applying dynamic extension for (55). We consider the following diffeomorphic mapping: T
˜1 (e)) (56) φ = [φ1 φ2 φ3 φ4 ] = Φ(e, u ex ˜ cos eθ + vd cos eθ − vd u (57) = 1 ey (˜ u1 + vd ) sin eθ Moreover, we apply the following input transformation: [ ][ ] [ ] cos eθ −(˜ v u1 + vd ) sin eθ u ˜˙ 1 = 1 (58) sin eθ (˜ u1 + vd ) cos eθ v2 u2
Then we can obtain the following linearized augmented system for (55): φ˙ = φ2 + wx 1 φ˙ 2 = v1 (59) φ˙ 3 = φ4 + wy φ˙ 4 = v2 .
We show a simulation result of (55) applying the ISSCLF based disturbance attenuation controller (25) and the static CLF based controller (17). These controller are derived from (63). A static CLF based controller for (54) is given by u(e) = αs (e) + ud . On the otherhand, a static ISS-CLF based disturbance attenuation controller for (54) is given by u(e) = αd (e) + ud . Figuare.3 shows the time response of the vessel trajectory every 2 seconds. The disturbance norm is represented by the arrow on the center of vessel. The circle region shows the position keeping region of the radius ∥e∥ and the controller is assumed ρ(r) : r �→ |r| in (29). Figuare.4, Fig.5 and Fig.6 show state response, velocity and input response respectively. We can confirm that the designed controller based on a static ISS-CLF design method achieves the desired velocity and yaw rate within the desired disturbance attenuation performance. In the input history of Fig.6, u ˜1 ISS-CLF demonstrates five local maximum points; instead eight local maximum points in CLF. The worst case of heading angle θ with the ISS-CLF’s input is smaller than that with the CLF’s input in Fig.4. However, the amount of input u2 with ISS-CLF’s input is smaller than that of CLF’s input in the worst case.
5.3 ISS-CLF design for linearized augmented systems
We design CLF V˜ (ϕ) for (59) at w = 0. Following P roposition 6, we substitute the following positive matrices Q and R: 1000 [ ] 10 0 2 0 0 Q= , R= . (60) 0010 01 0002 Then we can obtain the following positive symmetric matrix P0 and the CLF for (59) at ω = 0 are obtained as follows: 2100 1 2 0 0 . (61) V˜ (φ) = φT P0 φ, P0 = 0 0 2 1 0012 According to Lemma 17, (61) is the ISS-CLF for (59).
Fig. 2. Assumed speed disturbance
5.4 CLF design for differentially flat systems A dynamic CLF V˜ (Φ (e, u ˜1 (e))) for the differentially flat system is represented as follows: ) {( u1 + vd )] V˜ (Φ (e, u ˜1 (e))) = 2 e2x + e2y + ey [sin eθ · (˜ 2 2 + (cos eθ · u ˜1 + cos eθ · vd − vd ) + (sin eθ · (˜ u1 + vd )) + ex (cos eθ · u ˜1 + cos eθ · vd − vd )} . (62) We apply a minimum projection method to (62). Then a static CLF is represented as follows: ) ( 2 ex − e2y sin2 eθ − ex ey · sin 2eθ 3 2 2 ˆ V (e) = ex + 2ey + 2 2) ( +ey vd · sin 2eθ + 2 vd2 − ex vd · sin2 eθ . (63) According to Theorem 18, (63) is a static ISS-CLF for (55). 260
6. CONCLUSION We showed that an ISS-CLF design method for a differentially flat system. Our proposed method derives the ISS-CLF from the CLF for linear augmented system by applying a dynamic extension and a minimum projection method. Then we confirm that the proposed ISS-CLF design method is successfully applicable to disturbance attenuation problems. The advantage of our proposed method has a diffeomorphic mapping related to two ISS-CLF for the differentially flat system and linear augmented system. Analysing H∞ gain for the linear system and ISS-gain for the differentially flat system remain future works.
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Fig. 5. Velocity and yaw rate
Fig. 3. Trajectory tracking of vessel under disturbance
Fig. 6. Input response
Fig. 4. State response REFERENCES Krstic, M. and Li, Z.H. (1998). Inverse optimal design of input-to-state stabilizing nonlinear controllers. IEEE Transactions on Automatic Control, 43(3), 336–350. Kuga, S., Nakamura, H., and Satoh, Y. (2016). Static smooth control lyapunov function design for differentially flat systems. IFAC-PapersOnLine, 49(18), 241– 246. 261
Levine, J. (2009). Analysis and control of nonlinear systems: A flatness-based approach. Springer Science & Business Media. Rigatos, G.G. (2015). Differential flatness theory and flatness-based control. In Nonlinear Control and Filtering Using Differential Flatness Approaches, 47–101. Springer. Sassano, M. and Astolfi, A. (2012). Dynamic approximate solutions of the hj inequality and of the hjb equation for input-affine nonlinear systems. IEEE Transactions on Automatic Control, 57(10), 2490–2503. Sepulchre, R., Jankovic, M., and Kokotovic, P.V. (2012). Constructive nonlinear control. Springer Science & Business Media. Sontag, E.D. and Wang, Y. (1995). On characterizations of the input-to-state stability property. Systems & Control Letters, 24(5), 351–359.
ScienceDirect
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Input-to-State Stabilization of Input-to-State Stabilization of Input-to-State Stabilization of ⋆⋆ Differentially Flat Systems Input-to-State Stabilization of Differentially Flat Systems Differentially Flat Systems ⋆⋆ Differentially Flat Systems∗∗ Yasuhiro Fujii ∗∗ Hisakazu Nakamura ∗∗
Yasuhiro Fujii ∗∗ Hisakazu Nakamura ∗∗ Yasuhiro Fujii ∗ Hisakazu Nakamura ∗∗ ∗∗ Yasuhiro Fujii Hisakazu Nakamura ∗ University of Science, Noda, Chiba 278-8510 Japan (e-mail: ∗ Tokyo Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: ∗ Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: [email protected]) ∗ [email protected]) ∗∗Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: [email protected]) Noda, Chiba 278-8510 Japan (e-mail: ∗∗ Tokyo University of Science, Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: ∗∗ [email protected]) Tokyo University of Science, Noda, Chiba 278-8510 Japan (e-mail: [email protected]). ∗∗ [email protected]). Tokyo University of [email protected]). Science, Noda, Chiba 278-8510 Japan (e-mail: [email protected]). Abstract: Abstract:∞We We consider consider input-to-state input-to-state stabilization stabilization of of aa differentially differentially flat flat system. system. Firstly, Firstly, We We Abstract: consider input-to-state stabilization of a differentially flat system. Firstly, We design a C differential strict control Lyapunov function (CLF) for the linear augmented system ∞We design a C differential strict control Lyapunov function (CLF) for the linear augmented system ∞We consider input-to-state stabilization of a differentially flat system. Firstly, We Abstract: design a C differential strict control Lyapunov function (CLF) for the linear augmented system on the extended state space by solving Riccati equation. Then, we show that the designed on the aextended state space by solving Riccatifunction equation. Then, we linear show augmented that the designed design C ∞ satisfies differential strict control Lyapunov (CLF) forfunction the system on the extended state by solving Riccaticontrol equation. Then, we show(ISS-CLF) that the property. designed CLF always the input-to-state-stability Lyapunov CLF always satisfies thespace input-to-state-stability control Lyapunov function (ISS-CLF) property. on the extended state space by solving Riccati equation. Then, we show that the designed CLF always satisfies the input-to-state-stability Lyapunov function Finally, we a and differentially flat by applying dynamic Finally, we derive derive a CLF CLF and an an ISS-CLF ISS-CLF for for the the control differentially flat system system by(ISS-CLF) applying aaproperty. dynamic CLF always satisfies the input-to-state-stability Lyapunov function Finally, weand derive a CLF and an ISS-CLF for the control differentially flat system by(ISS-CLF) applying aproperty. dynamic extension a minimum projection method. extension and a minimum projection method. Finally, weand derive a CLF and an ISS-CLF for the differentially flat system by applying a dynamic extension a minimum projection method. extension and(International a minimum Federation projectionofmethod. © 2018, IFAC Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear Nonlinear control, control, Lyapunov Lyapunov function, function, Stabilization, Stabilization, Robust Robust performance performance Keywords: Keywords: Nonlinear control, Lyapunov function, Stabilization, Robust performance Keywords: Nonlinear control, Lyapunov function, Stabilization, Robust performance 1. INTRODUCTION INTRODUCTION design 1. design method method is is effective effective at at attenuating attenuating disturbance disturbance for for 1. INTRODUCTION design method is effective at attenuating disturbance for aa vessel trajectory tracking problem. vessel trajectory tracking problem. 1. INTRODUCTION design method is effective at attenuating disturbance for Every stabilizable stabilizable control control system system on on Euclidean Euclidean space space perper- a vessel trajectory tracking problem. Every a vessel trajectory2.tracking problem. space perEvery stabilizable control system(CLF). on Euclidean mits control control Lyapunov function (CLF). In particular, particular, dis2. PRELIMINARIES PRELIMINARIES mits Lyapunov function In dis2. PRELIMINARIES Every stabilizable control system on Euclidean space permits control Lyapunovfor (CLF). In particular, turbance attenuation forfunction nonlinear control system can candisbe 2. PRELIMINARIES turbance attenuation aa nonlinear control system be In mits control Lyapunov function (CLF). In particular, disIn this this paper, paper, we we consider consider the the disturbance disturbance attenuation attenuation turbance attenuation for a nonlinear control system can be estabilished by designing input-to-state-stability control estabilished by designing input-to-state-stability control In this paper, we consider the disturbance attenuation control of differentially flat systems. We introduce basic turbance attenuation for a nonlinear control system can be control of differentially flat systems. We introduce basic estabilished by designing input-to-state-stability control Lyapunov function (ISS-CLF). In this of paper, we consider theindisturbance attenuation Lyapunov function (ISS-CLF). control differentially flat systems. We introduce basic definitions and properties used this paper. estabilished by designing input-to-state-stability control definitions and properties used in this We paper. Lyapunov function (ISS-CLF). control of differentially flat systems. introduce basic definitions and properties used in this paper. Solving a disturbance attenuation problem has an advanLyapunov function (ISS-CLF). Solving a disturbance attenuation problem has an advan- definitions and properties used in this paper. 2.1 Nonlinear systems an advanSolving disturbance attenuation problem has tage to to aguarantee guarantee control performance under distur- 2.1 Nonlinear systems tage aa control performance under distur2.1 Nonlinear systems Solving aguarantee disturbance attenuation problem has an advantage to a control performance under disturbance. Krstic and Li (1998) have proposed a disturbance 2.1 Nonlinear systems we consider the following nonlinear bance. Krstic and Lia (1998) have proposed under a disturbance tage to guarantee control performance disturThroughout the the paper, paper, we consider the following nonlinear bance. Krsticcontrol and Li method (1998) have proposed a disturbance attenuation control method by solving solving inverse optimal Throughout attenuation by inverse optimal Throughout the paper, we consider the following nonlinear control system: bance. andwith Li method (1998) have proposed afor disturbance system: attenuation control by solving inverse optimal control control Krstic problems a designed designed ISS-CLF nonlinear Throughout the paper, we consider the following nonlinear control problems with a ISS-CLF for nonlinear control system: x + (1) attenuation control method by solving inverse optimal x˙˙ = = ff (x) (x) + g(x)u g(x)u (1) control problems with a designed ISS-CLF for nonlinear control system: systems. n m control systems. x ˙ = f (x) + g(x)u (1) and u ∈ R denote a state and an input where x ∈ R control problems with a designed ISS-CLF for nonlinear n m control systems. R denote a state and an input where x ∈ Rn and ux˙ ∈ =f fR:(x) + g(x)u m n n×n n (1) Sassanosystems. and Astolfi Astolfi (2012) (2012) have have proposed proposed aa dynamic dynamic ISSISS- where and u ∈ denote a state and an input x ∈ R respectively. Mapping R → R and g : R → control n n×n n Sassano and n respectively. f R:mRdenote and : Rinput n → Ran×n n → n×m x ∈ R Mapping and to u ∈ state andgg an Sassano andmethod Astolfi for (2012) have proposed dynamic ISS- where CLF design design method for a differentially differentially flat asystem system by solvsolvrespectively. Mapping f : R → R and : R → R be locally Lipschitz continuous with n×m are supposed CLF a flat by n n×n n to be Lipschitz continuous Sassano andmethod Astolfisolution (2012) have proposed asystem dynamic ISS- R n×m are supposed respectively. Mapping f flocally :(0) R =→ Rthe origin. and g : R with → CLF design for a differentially flat by solving the algebraic of Hamilton-Jacobi-Inequality R are supposed to be locally Lipschitz continuous with respect to x, and satisfy 0 at ing the algebraic solution of Hamilton-Jacobi-Inequality n×m to x, and satisfy f (0) = 0 at the origin. respect CLF design method for a differentially flat system by solvR are supposed to be locally Lipschitz continuous with ing theA solution Hamilton-Jacobi-Inequality (HJI). Aalgebraic differentially flat of system is aa class class of of nonlinear nonlinear respect to x, and satisfy f (0) = 0 at the origin. (HJI). differentially flat system is ing theAalgebraic solution Hamilton-Jacobi-Inequality respect to x, and satisfy f (0) = 0 at the origin. Differentially Flat (HJI). differentially flat of system is a class of nonlinear systems that are linearizable linearizable by a diffeomorphism diffeomorphism with aa 2.2 2.2 Differentially Flat System System systems that are by a with (HJI). Athat differentially flat system is a class of nonlinear 2.2 Differentially Flat System systems are linearizable by a diffeomorphism with a dynamic compensator. By investigation of Levine (2009), dynamic compensator. By investigation of Levine (2009), Differentially Flat System systems that are linearizable by of a diffeomorphism with a 2.2 We dynamic compensator. By investigation of Levine (2009), most of of all all controllable systems trajectory generation, We derive derive an an augmented augmented system system defined defined on on the the virtual virtual most controllable systems of trajectory generation, dynamic compensator. By investigation of Levine (2009), We derive an augmented system defined on the virtual space by applying a dynamic extension for nonlinear most of all controllable systems of trajectory generation, motion planning, or tracking can be considered as differenspace by applying a dynamic extension for nonlinear motion planning, or tracking can be considered generation, as differen- We derive an introduce augmented system defined flat on the virtual most of all controllable systems of trajectory space by applying a dynamic extension for nonlinear systems and the differentially system as motion planning, or tracking can be considered as differentially flat systems. The control design of unmanned vessels introduce the differentially as tially flat systems.orThe control design of unmanned vessels systems space byand applying a dynamic extension flat for system nonlinear motion planning, tracking can be considered as differensystems and introduce the differentially flat system as follows. tially flat systems. The control design of unmanned vessels is one of the interesting applications (Rigatos (2015)). follows. is one of the interesting applications (Rigatos (2015)). systems and introduce the differentially flat system as tially flat systems. The control design of unmanned vessels follows. 1. (Dynamic compensator). Consider (1) and is one ofaa the interesting (Rigatos (2015)). Definition However static ISS-CLF applications design still still remains remains a problem. problem. Definition 1. (Dynamic compensator). Consider (1) and However static ISS-CLF design a follows. is one ofa the interesting (Rigatos (2015)). Definition 1. (Dynamic compensator). Consider (1) and the following dynamic compensator: However static ISS-CLF applications design still remains a problem. following dynamic compensator: [ [ A static static smooth smooth CLF designdesign method forremains differentially flat the Definition 1. (Dynamic compensator). However a staticCLF ISS-CLF still a problem. [ p˙ ] ]compensator: [a(x, p, v)] ] Consider (1) and the following dynamic A design method for aa differentially flat [ p˙ ]compensator: [a(x, p, v)] , A static smooth CLF design method for a differentially flat system by a minimum projection method was proposed by the following dynamic = (2) system by a minimum projection method was proposedflat by (2) [u p˙ ] = [a(x, p, v) v)] , b(x, A static smooth CLFThe design method for CLF a differentially method was proposed by system by a minimum projection Kuga et al. (2016). static smooth enables us to u b(x, p, = , (2) p ˙ a(x, p, v) Kuga etbyal.a (2016). The static smooth CLF enables us by to u b(x, p, v) system minimum projection method was proposed = denote a virtual , (2) Kuga (2016). static smooth CLF enables us to where p ∈ Rll and v ∈uRm designetaa al. static stateThe feedback controller that can avoid avoid state and input m b(x, p, v) design static state feedback controller that can p ∈ Rl and v ∈ Rm denote a virtual state and input Kuga etasingularities al. (2016). The static smooth CLF enables us to where design static state feedback controller that can avoid delays, points or losing robustness. and v ∈ R denote a virtual state and input where p ∈ R of the dynamic compensator, respectively. delays, singularities or losing robustness. of the pdynamic compensator, respectively. design static statepoints feedback controller that can avoid where v ∈ Rm denote a virtual state and input ∈ Rl and delays, asingularities points or losing robustness. of the dynamic compensator, respectively. In this paper, we propose a static ISS-CLF design method delays, singularities pointsa or losing robustness. the dynamic compensator, respectively. Then we introduce an augmented In this paper, we propose static ISS-CLF design method of we introduce an augmented system system as as follows. follows. In paper, we propose a static ISS-CLF design method forthis a differentially differentially flat system. system. Our proposed method can Then Then we introduce an augmented system as follows. for a flat Our proposed method can Definition 2. (Augmented system). On the state In this paper, we propose a static ISS-CLF design method Definition 2. (Augmented system). On the state space space an augmented system as follows. for a differentially flat an system. Our from proposed methodCLF can Then systematically design an ISS-CLF from a dynamic dynamic CLF n+l we introduce systematically design ISS-CLF a Definition 2. (Augmented system). On the state space , system (1) in combination with dynamic compenR n+l for a linear differentially flat an system. Our from proposed method can R , system (1) in combination with dynamic compensystematically design ISS-CLF a dynamic CLF for a augmented system. Finally, we show computer n+l Definition 2.called (Augmented system). On dynamic the state space for a linear augmented system. Finally, wea show computer , system (1) in combination with compenR sator (2) is an augmented system: systematically design an ISS-CLF from dynamic CLF n+l (2) is called sator an[combination augmented system: for a linear to augmented system. we static show computer [ ] simulation to confirm that that ourFinally, proposed ISS-CLF R , (2) system (1)] in with dynamic compen[ ] [ ] simulation confirm our sator is called an augmented system: proposed static ISS-CLF for a linear to augmented system. we static show computer ff (x) p, [x proposed ISS-CLF sator (2) is called simulation confirm that ourFinally, x˙˙ ] = (x) + + g(x)b(x, g(x)b(x, p, v) v)] an[augmented system: (3) (3) [x simulation confirm that our proposed static ISS-CLF ˙p˙ ] = [f (x) +a(x, g(x)b(x, p, v)] ,, p p, v) ⋆ This work to a(x, p, v) was partially supported by JSPS KAKENHI Grant = , (3) ⋆ This work was partially supported by JSPS KAKENHI Grant x ˙p˙ f (x) +a(x, g(x)b(x, p, v) p, v) ⋆ = , (3) Number 17H03282. where the origin is (x, p) = (0, 0). This work was partially supported by JSPS KAKENHI Grant p˙ is (x, p) = a(x, p, v) Number 17H03282. where the origin (0, 0). ⋆ This work was partially supported by JSPS KAKENHI Grant Number 17H03282. where the origin is (x, p) = (0, 0). Number 17H03282. where the originLtd. is (x, = (0, 0). 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Allp) rights reserved.
Copyright © 2018 IFAC 256 Copyright 2018 IFAC 256 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 256 10.1016/j.ifacol.2018.09.512 Copyright © 2018 IFAC 256
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We consider a differentially flat system having a diffeomorphism such that the augmented system (3) can be represented by a linearized augmented system as follows. Definition 3. (Linearized augmented system). For system (3), consider the following diffeomorphism Φ : Rn+l → Rn+l : ϕ = Φ(x, p), Φ(0) = 0, (4) where we assume that the diffeomorphism satisfy the following equation: dΦ ϕ˙ = (x, p) (5) dt ∂Φ ∂Φ (x, p)x˙ + p˙ (6) = ∂x ∂p = Aϕ + Bv. (7) The augmented system (3) is represented by the following linearized augmented system ϕ ∈ Rn+l : ϕ˙ = Aϕ + Bv, (8) where the matrix pair (A, B) is controllable. The system (8) is said to be a linearized augmented system． Definition 4. (Differentially flat system). If system (1) is linearizable via such diffeomorphism (4) by dynamic compensator (2), system (1) is said to be a differentially flat system. 2.3 Control Lyapunov functions (CLF) In this paper, we define a static CLF as follows. Definition 5. (Control Lyapunov Function). [Kuga et al. (2016)] Consider system (1). A proper positive definite C ∞ differentiable function V : Rn → R satisfying the following condition is said to be a static smooth control Lyapunov function (CLF) for (1): (9) Lf V (x) < 0, ∀x ∈ {x ∈ X ⊆ Rn |Lg V = 0} , where Lf V = (∂V /∂x)f (x) and Lg V = (∂V /∂x)g(x). Then V˙ (x, u) can be written as follows: (10) V˙ (x, u) = Lf V (x) + Lg V (x)u. 2.4 Dynamic smooth CLF design for augmented systems A static CLF for a linearized augmented system is designed by the solution of the Riccati equation. By applying the designed diffeomorphism to the static CLF, we can derive a dynamic CLF for the differentially flat system as follows. Proposition 6. [Kuga et al. (2016)] Let P be a symmetric positive definite solution of the following Riccati equation for system (8): (11) AT P + P A − P BR−1 B T P + Q = 0, where Q and R are arbitrary positive definite matrices. Then, function V˜ (ϕ) : Rn+l → R defined by the following function is a CLF for (8): V˜ (ϕ) = ϕT P ϕ. (12) n+l ¯ Moreover the following function V (Φ(x, p)) : R → R is a CLF for Φ(x, p): (13) V¯ (x, p) = ΦT (x, p)P Φ(x, p) where function V¯ (x, p) is a dynamic smooth CLF including virtual states p. 257
257
2.5 Static smooth CLF design We derive a static smooth CLF as follows. Theorem 7. [Kuga et al. (2016)] Let the function V¯ : Rn+l → R obtained in (13) be a dynamic C ∞ differentiable CLF for system (3). Then the following function V : Rn → R is a static C ∞ differentiable CLF for (1): (14) V (x) = ΦT (x, p(x))P0 Φ(x, p(x)), where a function p(x) : Rn → Rl is uniquely determined by the following equation: ∂ V¯ (x, p) = 0. (15) ∂p The relation between CLF V (x) for (1) and CLF V˜ (ϕ) for (8) satisfy the following lemma. Lemma 8. [Kuga et al. (2016)] If there exist CLF V˜ (ϕ) for linear augmented system (8), then static CLF V (x) for (1) derived by a minimum projection method satisfies the following equation: (16) V (x) = min V˜ (ϕ). ϕ∈Φ(x)
2.6 Sontag type CLF based controller In this paper, we design Sontag-type CLF based controller. The controller is represented as follows. Theorem 9. [Sepulchre et al. (2012)] Consider system (1). V (x) is a CLF for the system. Then the control input (17) u = αs (x) 1 Lg V T (x) (18) =− R(x) is asymptotically stabilize at the origin and minimize the following cost function. ) ∫ ∞( 1 T J= l(x) + u R(x)u dt, (19) 2 0 where l(x) and R(x) are functions defined as follows: 1 l(x) = ∥Lg V ∥22 − Lf V (20) 2R(x) ∥L V ∥2 √ g 2 (Lg V ̸= 0) (21) R(x) = Lf V + Lf V 2 + ∥Lg V ∥42 1 (Lg V = 0).
2.7 Input-to-state stability
We introduce input-to-state stability and an input-to-state stable control Lyapunov function for disturbed nonlinear systems. Comparison functions are used to define inputto-state stability. Definition 10. (Class K function). [Sepulchre et al. (2012)] If a C 0 function γ : R≥0 → R≥0 is strictly increasing and γ(0) = 0, the function γ is said to belong class K. In addition, if γ(r) → ∞ as r → ∞, the function γ is said to belong to class K∞ . Definition 11. (Class KL function). [Sepulchre et al. (2012)] A C 0 function β : R≥0 × R≥0 → R≥0 is said to belong to class KL if for each fixed s the function β(·, s) belong to class K, and for each fixed r, function β(r, ·) is decreasing and β(r, s) → 0 as s → 0.
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We define input-to-state stability as follows. Definition 12. (Input-to-state stability(ISS)). [Krstic and Li (1998)] System (32) is said to be input-to-state stable (ISS) if there exists a controller which guarantees that ( ) ∥x(t)∥ ≤ β (∥x(0)∥ , t) + χ sup ∥w(τ )∥ , (22) 0≤τ ≤t
where β ∈ KL, χ ∈ K. Theorem 13. [Sontag and Wang (1995)] System (32) is ISS if and only if there exist a smooth positive definite radially unbounded function V (x) and class K∞ functions ν and µ such that the following dissipation inequality holds: V˙ (x) ≤ −µ (∥x∥) + ν (∥w∥) .
(23)
2.8 Input-to-state stable CLF (ISS-CLF)
2.10 ISS-CLF design for linear systems CLF V˜ (ϕ) for linear system (8) is an ISS for disturbed linear system (34) according to the following lemma. Lemma 17. We consider CLF V˜ (ϕ) = ϕT P ϕ for linear system (8). Then the function V˜ (ϕ) is an ISS-CLF for linear system (34). Proof. While due to page limit the detailed proof is omitted, the following inequality holds: ( ) 1 (31) V˜˙ (ϕ) ≤ − ϕT Q − γ 2 J ϕ + 2 ∥d∥2 . γ where J := P HH T P . In equation (31), for arbitrary positive matrix Q, there exits γ ∈ R>0 such as Q − γ 2 J is positive definite. As (23) is held, V˜ (ϕ) is an ISS-CLF for (34).
We define an ISS-CLF for nonlinear systems as follows. 3. PROBLEM STATEMENT Definition 14. (ISS Control Lyapunov Function). [Krstic and Li (1998)] A smooth positive definite radially unbounded function V : Rn → R≥0 and class K∞ is called an ISS-CLF Differentially flat systems are generally defined as no disfor system (32) if there exits a class K∞ function ρ such turbed systems. However, we consider the following disthat the following implication holds for all x ̸= 0 and all turbed differentially flat systems to deal with disturbance attenuation problems. w ∈ Rr : x˙ = f (x) + g(x)u + h(x)w (32) ∥x(t)∥ ≥ ρ (∥w∥) r n ⇒ infm {Lf V (x) + Lg V (x)u + Lh V (x)w} < 0. is bounded disturbance. Mapping h : R → where w ∈ R u∈R n×l R is supposed to be locally Lipschitz continuous with (24) respect to x, and satisfies f (0) = 0 at the origin. The following theorem establishes equivalence between ISS Differentially flat system (32) can be linearizable to the and the existence of an ISS-CLF. following linear augmented system by a diffeomorphism Theorem 15. (Krstic and Li, 1998, Theorem 2.1) System and dynamic compensator. (32) is ISS if and only if there exits an ISS-CLF. ∂Φ ϕ˙ = Aϕ + Bv + (x, p)h(x)d, (33) ∂x 2.9 Disturbance attenuation controller r where d ∈ R is disturbance. We use the following Sontag-type controller proposed by We assume here that (33) is represented as the linearized Krstic and Li (1998) to disturbance attenuation problems. augmented system as follows. Theorem 16. (Krstic and Li, 1998, Theorem 3.2) Consider Assumption 1. System (33) satisfies the following equasystem (32). If (32) is ISS, then there exist ISS-CLF tion. V (x) and the following Sontag-type control law solves the ϕ˙ = Aϕ + Bv + Hd, (34) inverse optimal gain assignment problem. (n+l)×r is a matrix as follows: where H ∈ R ∂Φ (25) u = αd (x) (x, p)h(x) (35) H= ∂x 2 T − (Lg V ) (Lg V ̸= 0) (26) αd (x) = Rd (x) 4. ISS-CLF DESIGN FOR DIFFERENTIALLY FLAT 0 (Lg V = 0) , SYSTEMS where ˜ Lg V = [Lg1 V (x) Lg2 V (x)] . (27) We consider the designed ISS-CLF V (ϕ) for linear augmented system. Then we show that CLF V (x) is an ISSThen ) is ∫ ∞ ( level ∫ ∞the achieved disturbance attenuation CLF for disturbed differentially flat systems. As the main [ ] |ω| dt. (28) theorem in this paper, it is shown in Theorem 19. Before l(x) + uT Rd (x)u dt ≤ βλ γ λ 0 0 that, we show the following lemma to support the main where β ≥ 2, λ ∈ (0, 2] and γ ∈ K∞ . ω and Rd (x) > 0 are theorem. represented as follows: Lemma 18. Consider CLF V (x) for differentially flat sys(29) tem (1) and CLF V˜ (ϕ) for linear augmented system (8). ω = Lf V (x) + |Lh V (x)| ρ−1 (|x|) T For these CLF, the following inequality is satisfied: 2L V (Lg V ) √ g (Lg V ̸= 0) (36) V (x) ≤ V˜ (Φ(x, p)) , ∀p ∈ Rl Rd (x) = ω + ω 2 + (L V (L V )T )2 g g 1 (Lg V = 0). Proof. According to Lemma 8, the following inequality (30) holds: 258
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V (x) = min V˜ (ϕ) ϕ∈Φ(x)
≤ V¯ (x, p), ∀p ∈ Rl ≤ V˜ (ϕ), ∀ϕ ∈ Φ(x, p)
259
(37) (38) (39)
We now give the main theorem in this paper. Theorem 19. We assumed that the static CLF V (x) for differentially flat system (1) is derived from the CLF V˜ (ϕ) for linearized augmented system (8) according to Proposition 6 and Theorem 7. Then the static CLF V (x) is the static ISS-CLF for disturbed differentially flat systems (32).
Fig. 1. Maneuvering kinematics coordinate system
Proof. The CLF V˜ (ϕ) is an ISS-CLF for differentially flat system if and only if (24) is satisfied. The conclusion part is rewritten as follows: } { V (x(t + ∆t)) − V (x(t)) <0 (40) infm lim ∆t→0 u∈R ∆t According to Lemma 18, both CLF V˜ (ϕ) and V (x) for the linear augmented system and differentially flat system hold the following inequality for arbitrary time t ∈ R≥0 :
Therefore, there exist input u satisfying (49) for differentially flat system as follows: ˜ p0 )) u = b(x, p0 ) ≤ k(Φ(x, (52) ≤ −R−1 B T P Φ(x, p0 )
V (x(t + ∆t)) − V (x(t)) ∆t (41) V¯ (x(t + ∆t), p) − V¯ (x(t), p) ≤ lim , ∀p ∈ Rl ∆t→0 ∆t then there exist p0 satisfying (16) as follows: { } p0 ∈ argmin V¯ (x(t), p(t)) , ∀t ∈ R≥0 (42)
In this section, we design the static smooth CLF and ISS-CLF for a vessel trajectory tracking problem. Then we design a static state feedback controller and a static disturbance attenuation controller. We compare state feedback controllers with or without disturbance attenuation to confirm the effectiveness of our static ISS-CLF design method.
lim
∆t→0
p
The initial state of (41) is represented as follows: V (x(∆t)) − V (x(0)) lim ∆t→0 ∆t (43) V¯ (x(∆t), p(∆t)) − V¯ (x(0), p(0)) ≤ lim ∆t→0 ∆t Now we can choose p(0) = p0 , then the left hand side of (43) is rewritten as follows: V (x(∆t)) − V (x(0)) V˙ = lim ∆t→0 ∆t V (x(∆t)) − V¯ (x(0), p0 ) = lim ∆t→0 ∆t V¯ (x(∆t), p(∆t)) − V¯ (x(0), p0 ) ≤ lim ∆t→0 ∆t ˙ ¯ =V
(44) (45) (46) (47)
(44) - (47) are satisfied for all time t ∈ R≥0 and x(0) ∈ Rn . According to (31), we can obtain the following inequality: V˙ (x) ≤ V¯˙ (x, p0 )
(48) ( ) 1 ≤ −ΦT (x, p0 ) Q − γ 2 J Φ(x, p0 ) + 2 ∥d∥2 (49) γ ≤ −α1 (∥x∥) + α2 (∥w∥) (50)
where α1 , α2 ∈ K∞ , α1 : r �→ λmax (Q−γ 2 J)∥Φ(r, p0 )∥ and α2 : r �→ (1/γ 2 )∥r∥2 . Then there exist input v satisfying (31) for the linearized augmented system as follows: ˜ (51) v = k(ϕ) ≤ −R−1 B T P ϕ 259
As (40) is held, the CLF for differentially flat system is always the ISS-CLF for (32). 5. EXAMPLE
5.1 Design of differentially flat systems We consider the following simple maneuvering model represented as Fig.1. x˙ = u1 cos θ + wx y˙ = u1 sin θ + wy (53) ˙ θ = u2 ,
where x, y ∈ R, θ ∈ S and u = [u1 , u2 ] denote respective state variables and control inputs. x and y are the position. θ is the azimuth angle. u1 and u2 are respective an input of forward movement and a yawing movement. wx , wy ∈ R is the total disturbance force in the direction to x and y, wind force being one such disurbance. We assume the bounded random signal such as Fig.2. We consider constant desired velocity vd ∈ R>0 , and we give a desired trajectory xd (t) = vd t, yd (t) = 0, θd (t) = 0. The tracking error e := [ex , ey , eθ ]T is written as ex = x − xd , ey = y − yd and eθ = θ − θd . Then the trajectory tracking problem is represented as the following error system: { e˙ x = u1 cos eθ + wx − vd e˙ y = u1 sin eθ + wy (54) e˙ θ = u2 .
We assumed that regular feedback ud = [u1d , u2d ]T for System (53) is defined on the origin, where u1d = vd , u2d = 0. In addition, the new input is represented as u ˜1 = u1 −vd . Then the trajectory tracking problem is rewritten as the following error system:
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{
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u1 + vd ) cos eθ − vd + wx e˙ x = (˜ e˙ y = (˜ u1 + vd ) sin eθ + wy e˙ θ = u2 .
5.5 Simulation results (55)
5.2 Deriving of linearized augmented systems We introduce an augmented system by applying dynamic extension for (55). We consider the following diffeomorphic mapping: T
˜1 (e)) (56) φ = [φ1 φ2 φ3 φ4 ] = Φ(e, u ex ˜ cos eθ + vd cos eθ − vd u (57) = 1 ey (˜ u1 + vd ) sin eθ Moreover, we apply the following input transformation: [ ][ ] [ ] cos eθ −(˜ v u1 + vd ) sin eθ u ˜˙ 1 = 1 (58) sin eθ (˜ u1 + vd ) cos eθ v2 u2
Then we can obtain the following linearized augmented system for (55): φ˙ = φ2 + wx 1 φ˙ 2 = v1 (59) φ˙ 3 = φ4 + wy φ˙ 4 = v2 .
We show a simulation result of (55) applying the ISSCLF based disturbance attenuation controller (25) and the static CLF based controller (17). These controller are derived from (63). A static CLF based controller for (54) is given by u(e) = αs (e) + ud . On the otherhand, a static ISS-CLF based disturbance attenuation controller for (54) is given by u(e) = αd (e) + ud . Figuare.3 shows the time response of the vessel trajectory every 2 seconds. The disturbance norm is represented by the arrow on the center of vessel. The circle region shows the position keeping region of the radius ∥e∥ and the controller is assumed ρ(r) : r �→ |r| in (29). Figuare.4, Fig.5 and Fig.6 show state response, velocity and input response respectively. We can confirm that the designed controller based on a static ISS-CLF design method achieves the desired velocity and yaw rate within the desired disturbance attenuation performance. In the input history of Fig.6, u ˜1 ISS-CLF demonstrates five local maximum points; instead eight local maximum points in CLF. The worst case of heading angle θ with the ISS-CLF’s input is smaller than that with the CLF’s input in Fig.4. However, the amount of input u2 with ISS-CLF’s input is smaller than that of CLF’s input in the worst case.
5.3 ISS-CLF design for linearized augmented systems
We design CLF V˜ (ϕ) for (59) at w = 0. Following P roposition 6, we substitute the following positive matrices Q and R: 1000 [ ] 10 0 2 0 0 Q= , R= . (60) 0010 01 0002 Then we can obtain the following positive symmetric matrix P0 and the CLF for (59) at ω = 0 are obtained as follows: 2100 1 2 0 0 . (61) V˜ (φ) = φT P0 φ, P0 = 0 0 2 1 0012 According to Lemma 17, (61) is the ISS-CLF for (59).
Fig. 2. Assumed speed disturbance
5.4 CLF design for differentially flat systems A dynamic CLF V˜ (Φ (e, u ˜1 (e))) for the differentially flat system is represented as follows: ) {( u1 + vd )] V˜ (Φ (e, u ˜1 (e))) = 2 e2x + e2y + ey [sin eθ · (˜ 2 2 + (cos eθ · u ˜1 + cos eθ · vd − vd ) + (sin eθ · (˜ u1 + vd )) + ex (cos eθ · u ˜1 + cos eθ · vd − vd )} . (62) We apply a minimum projection method to (62). Then a static CLF is represented as follows: ) ( 2 ex − e2y sin2 eθ − ex ey · sin 2eθ 3 2 2 ˆ V (e) = ex + 2ey + 2 2) ( +ey vd · sin 2eθ + 2 vd2 − ex vd · sin2 eθ . (63) According to Theorem 18, (63) is a static ISS-CLF for (55). 260
6. CONCLUSION We showed that an ISS-CLF design method for a differentially flat system. Our proposed method derives the ISS-CLF from the CLF for linear augmented system by applying a dynamic extension and a minimum projection method. Then we confirm that the proposed ISS-CLF design method is successfully applicable to disturbance attenuation problems. The advantage of our proposed method has a diffeomorphic mapping related to two ISS-CLF for the differentially flat system and linear augmented system. Analysing H∞ gain for the linear system and ISS-gain for the differentially flat system remain future works.
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Fig. 5. Velocity and yaw rate
Fig. 3. Trajectory tracking of vessel under disturbance
Fig. 6. Input response
Fig. 4. State response REFERENCES Krstic, M. and Li, Z.H. (1998). Inverse optimal design of input-to-state stabilizing nonlinear controllers. IEEE Transactions on Automatic Control, 43(3), 336–350. Kuga, S., Nakamura, H., and Satoh, Y. (2016). Static smooth control lyapunov function design for differentially flat systems. IFAC-PapersOnLine, 49(18), 241– 246. 261
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