Stabilization of nonlinear homogeneous systems

Stabilization of nonlinear homogeneous systems Abstract In this article, we study the stabilization problem of a class of homogeneous multi-input systems. First we give a result for driftless homogeneous systems, then we study the stabilization problem of homogeneous affine in control systems. We introduce some hypothesis for the desired system on a submanifold of ℝn diffeomorphic to Sn−1 and we construct a stabilizing homogeneous feedback which preserve the homogeneity of the closed loop system. 1. Introduction For smooth multi-input systems that are affine in the control   x˙=f(x)+∑i=1kuigi(x), (1) where the state x∈ℝn, the input (u1,…,uk)⊤∈ℝk, f(0)=0 and f, g1, g2,…,gk are continuously differentiable vector fields, the basic stabilization Lyapunov condition provided in Artstein (1983), Tsinias (1988), Sontag (1989) and Tsinias (1989) can be expressed as follows: There exists a positive definite real function V: ℝn→ℝ (i.e., V(0)=0 and V(x)>0 for x≠0 near zero) such that for any x≠0 near zero with   {∇V.g1(x)=0⋮∇V.gk(x)=0} it holds ∇V.f(x)<0 In Artstein (1983), it was shown that if the above condition is fulfilled, then the system (1) is stabilizable at the origin by means of a nonlinear feedback law which is smooth for x≠0. The same result was proved independently in Tsinias (1988), Sontag (1989) and Tsinias (1989), where the corresponding stabilizing feedback laws are more explicitly identified. Further progress was provided in Sontag (1989) where an explicit proof of Artstein's theorem is presented. Among other things, Sontag proved that if the system (1) satisfies the above Lyapunov condition, then stabilization is possible by means of a feedback law that depends directly on the dynamics of the system. For homogeneous systems, in Nakamura et al. (2009), the authors design homogeneous controllers using homogeneous control Lyapunov functions. However, there is no a constructive method for a control Lyapunov function. For this, in Jerbi (2002) and Jerbi et al. (2008), the authors give some hypothesis to construct a submanifold of ℝn diffeomorphic to Sn−1 and under some hypothesis on this submanifold, they study the stabilization problem of homogeneous systems. In this article, first we study the stabilization problem of driftless homogeneous systems. We construct a homogeneous stabilizing feedback that preserve the homogeneity of the closed loop system. Second, we study the stabilization problem for homogeneous nonlinear affine systems in the form (1). We assume that f, gi, for i∈{1,…,k}, are continuously differentiable and homogeneous with respect to the same dilation. We use results given in Jerbi (2002), Jerbi & Kharrat (2005) and Jerbi et al. (2008) to prove under some conditions the stabilization of affine homogeneous systems (1) by means of a homogeneous feedback law. We conclude by applying the proven results to a class of bilinear systems in ℝ2 and ℝ3. 2. Preliminaries We start by recalling the following definitions and results. Definition 1 (Sepulchre & Aeyels 1996) Let {r1,r2,…,rn} a family of fixed positive reals and r=(r1,r2,…,rn). Let δε(x)=(εr1x1,…,εrnxn), x∈ℝn, ε>0, a dilation on ℝn. (i) We say that a function h:ℝn→ℝ is homogenous of degree k with respect to δ, if   h(δε(x))=εkh(x), ∀x∈ℝn, ∀ε>0. (ii) We say that f:ℝn→ℝn is homogenous of degree k, if each fi,  i∈{1,…,n}, is homogeneous of degree k+ri. Definition 2 Let M be a submanifold of ℝn of dimension n−1. We say that M is Φ−diffeomorphic to Sn−1, if the map   Φ: M→Sn−1  x↦x||x|| is a diffeomorphism of C∞. Notations For x∈ℝn∖{0}, we denote Dx+={λx,  λ>0} and Dx−={λx,  λ<0}. ⟨..∣..⟩ denotes the Euclidean inner product and ||x||=⟨x∣x⟩ for x∈ℝn. For x∈ℝn we define ⟨x⟩={λx ; λ∈ℝ}. M⊤ denotes the transpose matrix of M. Let θ: ℝn→ℝ be a smooth map, we denote ∇θ(x)=(∂θ∂x1(x),….,∂θ∂xn(x)). Now, we recall the following definitions and results. Theorem 1 (Jerbi, 2002) Let M be a submanifold of ℝn of dimension n−1. M is Φ−diffeomorphic to Sn−1 if and only if the following holds (1) ∀x∈M one has ℝn=TxM⊕⟨x⟩, where TxM is the tangent space at the point x to the submanifold M, (2) for all x∈ℝn∖{0}, M∩Dx+ is a unique point. The main results of this article are based on the existence of a submanifold of ℝn Φ-diffeomorphic to Sn−1. The following theorem is useful for the construction of such submanifold. Theorem 2 (Jerbi & Kharrat, 2005) Let θ:ℝn→ℝ a map of class C1 and denote X(x)=(∇θ(x))⊤. If θ is proper and the vector field X satisfies the following condition (P) there exists R>0 and ρ>0 such that for ∥x∥>R one has ⟨X(x)∣x⟩≥ρ, then there exists k∈ℝ such that θ−1{k}=M is a sub manifold of ℝn, Φ-diffeomorphic to Sn−1. Moreover Nx=X(x)||X(x)||. Lemma 1 (Rosier, 1992) (1) The map   α:(0,+∞)×Sn−1→ℝn∖{0} (t,(y1,…,yn))↦(tr1y1,…,trnyn) is a bijection, and its inverse function α−1, which we write β=(β0,β1,…,βn), is of class C∞. (2) The function β0 satisfies   limx→0 x≠0β0(x)=0 and lim||x||→+∞β0(x)=+∞ Proposition 1 (Jerbi et al., 2008) Let M be a sub manifold of ℝn, Φ−diffeomorphic to Sn−1. The map   σ:(0,+∞)×M→ℝn∖{0}     (t,z)↦(tr1z1||z||,…,trnzn||z||) is a bijection of class C∞. Its inverse function σ−1:=γ=(γ0,γ1,…,γn) satisfies γ0(x)=β0(x). Remark 1 If M is a submanifold of ℝn, Φ−homeomorphic to Sn−1, then α is an homeomorphism. We recall the following theorem which is fundamental to conclude the stability results given in this article. Theorem 3 (Andriano, 1993) Consider the system   x˙=h(x),   x∈ℝn, where h is a continuously differentiable function satisfying h(0)=0. Suppose that there exist compact subsets {Dλ}λ∈ℝ+ such that (i) ∩λ∈ℝ+Dλ={0}, (ii) for all λ1<λ2, Dλ1⊂Dλ2°, (iii) for all x∈ℝn, there exists λ>0 such that x∈∂Dλ, (iv) for all λ>0 and x0∈∂Dλ one has Xt(x0)∈Dλ°, ∀t>0; where Xt(x0) is the solution of the system x˙=h(x) starting at x0∈∂Dλ, then the system x˙=h(x) is G.A.S. Theorem 4 (Jerbi et al., 2008) Let M be a submanifold of ℝn of dimension n−1, Φ−diffeomorphic to Sn−1. For λ>0, we denote δλr the map defined by    δλ:M→δλ(M)(x,…,xn)→(λr1x1,…,λrnxn), where ri,  i∈{1,…,n} are fixed positive reals. δλ is a diffeomorphism and for all y∈δλ(M), there exists a unique x∈M such that y=δλ(x) and Nδλ(x)=CλAλ−1Nx, where Cλ=∥∇xφn∥∥Aλ−1∇xφn∥, Aλ=diag(λr1,…,λrn) and φn is the n-th component of φ; ((Ux,φ) is a chart of M at the point x; ( φ:Ux→φ(Ux)⊂ℝn−1×{0}) is a diffeomorphism)). 3. Stabilization of driftless homogeneous systems Let the system described by   x˙=∑i=1kuigi(x),   x∈ℝn,  u=(u1⋮uk)∈ℝk, (2) where each gi is continuously differentiable and homogeneous of degree qi for all i∈{1,…,k} with respect to the same dilation δ, where δλ(x)=(λr1x1,…,λrnxn), λ>0 and ri are fixed positive reals. We establish the following lemma which is useful for the proofs of the theorems. Lemma 2 Let the system   x˙=h(x), (3) where h is continuous locally Lipshitz and homogeneous of degree d with respect of the dilation δ. If there exists a submanifold M of ℝn Φ-diffeomorphic to Sn−1 such that ⟨Ny|h(y)⟩<0 for all y∈M, then ⟨Nx|h(x)⟩<0 for all x∈ℝn∖{0}. Proof Let x∈ℝn∖{0}, there exists a unique (λ,y)∈(0,+∞)×M, such that x=δλ(y). A simple computation gives     ⟨Nx∣h(x)⟩=⟨Nδλ(y)∣h(δλ(y))⟩=⟨CλAλ−1Ny∣h(δλ(y))⟩=⟨CλAλ−1Ny∣λdAλh(y)⟩=λdCλ⟨Ny∣h(y)⟩<0.    □ Lemma 3 Let p∈ℝ, the maps ui, i∈{1,…,k}, defined by   {ui(x)=−(β0(x))p−qi⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩u(0)=0 (4) are well defined, continuous over ℝn\{0} and homogeneous of degree p−qi. If in addition p>qi, for all i then ui is continuous at the origin. Proof Let i∈{1,…,k}, we define ui(x)=−(β0(x))p−qi⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩ and ui(0)=0. The map ui is well defined and continuous over ℝn\{0}. The continuity of ui is a trivial consequence of the continuity of the maps Φ−1,β,N,gi and the inner product. The maps   θ:(0,+∞)×M→(0,+∞)×Sn−1    (t,y)↦(t,y||y||) and   α:(0,+∞)×Sn−1→ℝn∖{0}       (t,z)↦(tr1z1,…,trnzn) are bijections of class C∞. Then for all x∈ℝn∖{0}, there exists a unique (ε,y)∈(0,+∞)×M, such that x=α∘θ(ε,y). We have α∘θ(ε,y)=α(ε,Φ(y))=x. So (ε,Φ(y))=β(x)=(β0(x),β1(x),…,βn(x)). Then ε=β0(x) and y=Φ−1(β1(x),…,βn(x)). Thus, we can write ui(x)=−εp−qi⟨Ny∣gi(y)⟩. Let λ>0, we have   δλ(x)=(λr1x1,…,λrnxn)=(λr1tr1y1∥y∥,…,λrntrnyn∥y∥)=((λt)r1y1∥y∥,…,(λε)rnyn∥y∥)=α(λε,y∥y∥)=α(θ(λε,y)) So β0(δλ(x))=λε. Now, we prove that the map ui is homogeneous of degree p−qi. Let x∈ℝn\{0}, there exist ε>0 and y∈M such that y=δε(x). So for λ>0, one has   ui(δλ(x))=−(β0(δλ(x))p−q⟨Ny∣gi(y))⟩=−(λε)p−qi⟨Ny∣gi(y)⟩=−λp−qi(β0(x))p−qi⟨Ny∣gi(y)⟩=λp−qiui(x). If in addition p>qi, then ui is continuous at 0, indeed for x∈ℝn\{0}, there exist ε>0 and y∈M such that y=δε(x). Moreover, by Lemma 1, ε tends to 0 as x tends to zero. By the continuity of ui and the fact that y is in a compact set, one has ui is in a compact set. So limx→0ui(x)=limε→0εp−qiui(y)=0=ui(0).    □ Notation Let M be a submanifold of ℝn, Φ-diffeomorphic to Sn−1. We denote   Mi={x∈M such that ⟨Nx|gi(x)⟩=0}. Now, we establish the following result Theorem 5 Let p>sup{qi, 1≤i≤k}. If ∩i∈{1,…,k}Mi=Ø, then the system (2) is globally asymptotically stabilizable at the origin by a continuous feedback u=(u1,…,uk) satisfying each ui is homogeneous of degree p−qi. Proof We define for i∈{1,…,k},   {ui(x)=−(β0(x))p−qi⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩ for x∈ℝn∖{0}ui(0)=0, where β0,β1,…,βn are the components of the function β given in Lemma 1. It was proven in Lemma 3 that ui is homogeneous of degree p−qi for all i and continuous on ℝn. Let i∈{1,…,k}, y∈M, one has   ui(y)=−⟨Ny∣gi(y)⟩. So   ⟨Ny|∑i=1kui(y)gi(y)⟩=−∑i=1k(⟨Ny|gi(y)⟩)2. By the hypothesis ∩i∈{1,…,k}Mi=Ø, there exists i∈{1,…,k} such that ⟨Ny|gi(y)⟩≠0. We deduce that ⟨Ny|∑i=1kui(y)gi(y)⟩<0. Let x∈ℝn∖{0}, by the hypothesis M is Φ-diffeomorphic to Sn−1, there exists λ>0 and y∈M such that x=δλ(x). So     ⟨Nx|∑i=1kui(x)gi(x)⟩=−∑i=1kui(x)⟨Nx|gi(x)⟩=−∑i=1kλp−qi⟨Ny|gi(y)⟩⟨CλAλ−1Ny|λqiAλgi(y)⟩=−∑i=1kλpCλ(⟨Ny|gi(y)⟩)2<0. By the fact that M is Φ-diffeomorphic to Sn−1, one can write   ℝn∖M=O1∪O2, where O1 and O2 are two connected open sets, we choose 0∈O1. We define Dλ=δλ(O¯1)=δλ(O1∪M). Using the fact that for all x∈ℝn∖{0}  one has  ⟨Nx∣∑i=1kui(x)gi(x)⟩<0, one can deduce that Xt(x0)∈D°λ for all t>0, where Xt(x0) is the solution of the closed loop system (2) starting at x0∈Dλ; that is,   (d/dt)Xt(x0)=∑i=1kui(Xt(x0))gi(Xt(x0)) and X0(x0)=x0. Then, we deduce that Xt(x0)∈Dλ° for all t>0 where Xt(x0) is the solution of the closed loop system (2) starting at x0∈Dλ¯∖D°λ=∂Dλ=δλ(M) and the proof follows from Theorem 3.     □ 4. Stabilization of homogeneous affine in control systems Let us consider the multi-input systems that are affine in control:   x˙=f(x)+∑i=1kuigi(x), (5) where the state x∈ℝn, the input (u1,…,uk)⊤∈ℝk, f(0)=0, f is continuously differentiable homogeneous of degree p and each  gi, i∈{1,…,k}, is continuously differentiable and homogeneous of degree qi. Using the same notations of Section 2, one has the following result. Theorem 6 If there exists a submanifold M of degree n−1 of ℝn, Φ−diffeomorphic to Sn−1 and satisfying   for all x∈∩i∈{1,…,k}Mi,one has ⟨Nx∣f(x)⟩<0, then there exists a feedback u=(u1⋮uk) such that ui is homogeneous of of degree p−qi, for i=1,…,k, which stabilizes the system (5). Proof Suppose that there exists a submanifold M of degree n−1 of ℝn, Φ−diffeomorphic to Sn−1 and satisfying for all x∈∩i∈{1,…,k}Mi one has ⟨Nx∣f(x)⟩<0. We define the set A={x∈M such that ⟨Nx∣f(x)⟩≥0}. We have two cases If A=Ø, then the system x˙=f(x) is globally asymptotically stable. If A≠Ø, we can easily deduce that A is a compact set. Moreover   A⊂{x∈M such that ∑i=1k⟨Nx∣gi(x)⟩2≠0 }. We define ω=infx∈A{∑i=1k⟨Nx∣gi(x)⟩2}>0 and M=supx∈A⟨Nx∣f(x)⟩≥0. Let ρ>Mω>0, we define for i∈{1,…,k},   {ui(x)=−ρ(β0(x))p−qi⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩ if x∈ℝn∖{0}ui(0)=0, where β0,β1,…,βn are the components of the function β given in Lemma 1. According to Lemma 3, the maps ui are well defined, continuous on ℝn∖{0} and homogeneous of degree p−qi. Moreover, the maps ui satisfy ui(x)=−ρ(⟨Nx∣gi(x)⟩) for all x∈M. Under the choice of the constant ρ, we can write for x∈A  ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩=⟨Nx∣f(x)⟩−ρ∑i=1k(⟨Nx∣gi(x)⟩)2≤M−ρω<0, and since for all x∈M∖A one has ⟨Nx∣f(x)⟩−ρ∑i=1k(⟨Nx∣gi(x)⟩)2<0, then   ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩<0 for all x∈M. Now, we prove that   ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩<0 for all x∈ℝn∖{0}. Let x∈ℝn∖{0}, there exists a unique (λ,y)∈(0,+∞)×M, such that x=δλ(y). A simple computation gives    ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩=⟨Nδλ(y)∣f(δλ(y))+∑i=1kui(δλ(y))gi(δλ(y))⟩=⟨CλAλ−1Ny∣f(δλ(y))+∑i=1kui(δλ(y))gi(δλ(y))⟩=⟨CλAλ−1Ny∣λpAλ(f(y)+∑i=1kui(y)gi(y))⟩=λpCλ⟨Ny∣f(y)+∑i=1kui(y)gi(y)⟩<0. By the fact that M is Φ-diffeomorphic to Sn−1, one can write   ℝn∖M=O1∪O2, where O1 and O2 are two connected open sets, we choose 0∈O1. We define Dλ=δλ(O¯1)=δλ(O1∪M). Using the fact that for all x∈ℝn∖{0}  one has  ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩<0, one can deduce that Xt(x0)∈Dλ° for all t>0 where Xt(x0) is the solution of the closed loop system (2) starting at x0∈Dλ¯∖D°λ=∂Dλ=δλ(M) and the proof follows from Theorem 3. 5. Application 5.1. Stabilization of a bilinear system in ℝ2 Let us consider the planar bilinear system described by   x˙=Ax+∑i=12uiBix,   x∈ℝ2 ,  u=(u1u2)∈ℝ2, (6) where A=(a11a12a21a22), B1=(λ100λ2) and B2=(01−10). We know that if λ1λ2>0, then system (6) can be stabilized by a feedback u=(u1u2)∈ℝ2 with u1=σ, u2=0 and |σ| is large enough to satisfy   eigenvalue(A+σB1)⊂ℝ−* and if λ1+λ2=0 and a11+a22>0 then system (6) cannot be stabilized by a feedback u=(u1u2)∈ℝ2 with u1=σ∈ℝ, u2=0. Moreover, if detA>0, then system (6) cannot be stabilized by a feedback u=(u1u2)∈ℝ2 with u1=0, u2∈ℝ. Now, we prove the stabilization of the system (6) by means of an homogeneous feedback u=(u1u2)∈ℝ2 with u1≠0, u2≠0. We introduce the function   θ:ℝ2→ℝx↦12||x−b||2, where b=(01). It is easy to verify that θ satisfies to conditions of Theorem 2. In fact: Let X(x)=(∇θ(x))⊤=(x1,x2−1)⊤, we have   ⟨X(x)|x⟩=x12+x22−x2=||x−b2||2−||b2||2≥|||x||−||b2|||2−||b2||2≥||x||(||x||−||b||). So, if we choose β>||b||=1 and δ=β(β−||b||)>0, we get for x satisfying ||x||≥β one has ⟨X(x)|x⟩≥β(β−||b||)=δ. Then according to Theorem 2, there exists k>0 such that M=θ−1{k} is a submanifold of ℝ2 of dimension 1. The constant k can be chosen such that k>1 and this to guarantee that for all x∈ℝ2, M∩Dx+ is a unique point. Now let x∈M, we verify that if ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0, then ⟨X(x)|Ax⟩<0. But ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0 is equivalent to   {λ1x12+λ2x2(x2−1)=0x1x2−x1(x2−1)=0 which gives   {λ2x2(x2−1)=0x1=0 This computation gives the set of points x=(x1x2)∈M satisfying ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0 is empty. By Theorem 6, we deduce that there exists a feedback u=(u1u2), such that ui is homogeneous of degree 0, for i=1,2, which stabilizes the system (6). 5.2. Stabilization of a bilinear system in ℝ3 Let us consider the multi-input bilinear system   x˙=Ax+∑i=12uiBix,   x∈ℝ3 ,  u=(u1u2)∈ℝ2, (7) where A=(a11a120a21a22000a), B1=(λ1000λ20000) and B2=(010−10000λ), λ1>0, λ2<0 and λ≠0.\ We define the function θ:ℝ3→ℝ, x=(x1x2x3)↦12||x−b||2, where b=(001). It is easy to verify that θ satisfies to conditions of Theorem 2. Indeed: Let X(x)=(∇θ(x))⊤=(x1,x2,x3−1)⊤, we have ⟨X(x)|x⟩=x12+x22+x32−x3≥||x||2−||x||≥||x||(||x||−1). So if we choose β>1 and δ=β(β−1)>0, we get for x satisfying ||x||≥β one has ⟨X(x)|x⟩≥β(β−1)=δ. Then according to Theorem 2, there exists k>0 such that M=θ−1{k} is a submanifold of ℝ3 of dimension 2. The constant k can be chosen such that k>1 and this to guarantee that for all x∈ℝ3, M∩Dx+ is a unique point. Now let x∈M, we verify that if ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0, then ⟨X(x)|Ax⟩<0. But ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0 is equivalent to   {λ1x12+λ2x22=0λx3(x3−1)=0 which gives {x22=−λ1λ2x12x3=0 or {x22=−λ1λ2x12x3=1 Using the fact that x∈M, we deduce {x22=−λ1λ2x12x3=0x12=λ2λ2−λ1(k−1) or {x22=−λ1λ2x12x3=1x12=λ2λ2−λ1k. So the set of points x=(x1x2x3)∈M satisfying ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0 is   B={x=(x1x2x3)∈M such that ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0}={x=(x1x2x3)∈ℝ3such that {x12=λ2λ2−λ1(k−1)x22=λ1λ1−λ2(k−1)x3=0 or{x12=λ2λ2−λ1kx22=λ1λ1−λ2kx3=1}. Let x∈B, we get ⟨X(x)|Ax⟩=a11x12+a22x22+(a12+a21)x1x2+ax3(x3−1)=(−a11λ2+a22λ1±(a12+a21)−λ1λ2)kλ1−λ2 or ⟨X(x)|Ax⟩=a11x12+a22x22+(a12+a21)x1x2+ax3(x3−1)=(−a11λ2+a22λ1±(a12+a21)−λ1λ2)k−1λ1−λ2. So, a necessary condition for the stabilization of the system (7) is to choose matrices A and B1 satisfying to the following (−a11λ2+a22λ1±(a12+a21)−λ1λ2)<0. By Theorem 6, we deduce that there exists a feedback u=(u1u2), such that ui is homogeneous of degree 0, for i=1,2, which stabilizes the system (7). Example 1 Let the system   x˙=Ax+∑i=12uiBix,   x∈ℝ3 ,  u=(u1u2)∈ℝ2, (8) where A=(−1−10100001), B1=(1000−10000) and B2=(010−100001). We define the function θ:ℝ3→ℝ, x=(x1x2x3)↦12||x−b||2, where b=(001). We can easily remark that system (8) cannot be stabilized by a feedback u=(u1u2)∈ℝ2 with u1=0 or u2=0. In the previous paragraph, we have proved that θ satisfies to conditions of Theorem 2. So for k=2, the set M=θ−1{k} is a submanifold of ℝ3 of dimension 2 and for all x∈M, Nx=(∇xθ)⊤||(∇xθ)⊤||=x−b||x−b||. Moreover, a simple computation gives for all x∈M, ⟨Nx|B1x⟩=0 and ⟨Nx)|B2x⟩=0 implies ⟨Nx|Ax⟩<0. Indeed,\ {x∈M⟨Nx|B1x⟩=0⟨Nx)|B2x⟩=0 This is equivalent to {x12+x22+(x3−1)2=4x12−x22=0x3(x3−1)=0 which implies ⟨Nx|Ax⟩=−x12+x3(x3−1)<0. So by Theorem 6, the feedback u=(u1u2), where ui(x)=−ρ⟨NΦ−1(β1,…,βn)(x)∣Bi(Φ−1(β1,…,βn)(x))⟩ for x∈ℝn∖{0} and ui(0)=0, stabilizes the system (8). The expression of the maps ui can be simplified as follow. For x∈ℝ3∖{0}, the exists a unique positive real α such that αx∈M. In addition αx=Φ−1(β1,…,βn)(x). A simple computation gives α=x3+x32+3||x||2||x||2. We conclude that   ui(x)=−ρα⟨αx−b||αx−b|||Bix⟩, where the positive real ρ will be computed in the follow. Let A={x∈M such that ⟨Nx∣Ax⟩≥0}, Let ω=infx∈A{(⟨Nx∣B1(x)⟩)2+(⟨Nx∣B2(x)⟩)2}>0. Now, we have to approximate the value of ω. Let x∈A, then x∈M and ⟨Nx∣Ax⟩≥0. So   {||x−b||2=4⟨Nx∣Ax⟩≥0⇔{||x−b||2=4⟨(x−b)∣Ax⟩≥0⇔{x12+x22+(x3−1)2=4−x12+x3(x3−1)≥0 On the other hand, we have   (⟨Nx∣B1x⟩)2+(⟨Nx∣B2x⟩)2=(⟨x−b∣B1x⟩)2+(⟨x−b∣B2x⟩)2=(x12−x22)2+(x3(x3−1))2. A numerical computation gives ω=infx∈A(⟨Nx∣B1x⟩)2+(⟨Nx∣B2x⟩)2≥0,15 Now, we calculate M=supx∈A⟨Nx∣Ax⟩. Let x∈A, one has   {||x−b||2=4⟨Nx∣Ax⟩≥0⇔{||x−b||2=4−x12+x3(x3−1)≥0⇔{x12+x22+(x3−1)2=4−x12+x3(x3−1)≥0 We have −x12+x3(x3−1)≤x3(x3−1)≤|x3||x3−1|≤6 by the fact that |x3−1|≤2. We deduce M≤6. Finally, we can choose ρ=60. In the next, we give an example for stabilization of driftless systems. Example 2 Let the system described by:   {x˙1=u2(x2+x3)x˙2=−u1x2−u2x13x˙3=−2u1x3−u2x13, (9) where u=(u1u2)∈ℝ2 and x=(x1,x2,x3)⊤∈ℝ3. Denote g1(x1,x2,x3)=(0,−x2,−2x3)⊤ and g2(x1,x2,x3)=(x2+x3,−x13,−x13)⊤. We can easily verify that g1 and g2 are homogeneous with respect to the dilation δr, r=(1,3,3), of degree 3. Let θ(x1,x2,x3)=14x14+12x22+(x3−1)2. θ is proper and satisfies to conditions of Theorem 2. Indeed, let X(x)=(∇θ(x))⊤=(x13,x2,2(x3−1))⊤, we have   ⟨X(x)|x⟩=x14+x22+2x32−2x3=x14+x22+x32+(x3−1)2−1=x14+x12−x12+x22+x32+(x3−1)2−1=||x||2+x12(x12−1)+(x3−1)2−1≥||x||2−3. It is clear that if ||x||≥2, then ⟨X(x)|x⟩≥1. Then according to Theorem 2, there exists k>0 such that M=θ−1{k} is a submanifold of ℝ3 of dimension 2. The constant k can be chosen such that k>1 and this to guarantee that for all x∈ℝ3, M∩Dx+ is a unique point. So for k=4, the set M=θ−1{k} is a submanifold of ℝ3 of dimension 2 and for all x∈M, Nx=(∇xθ)⊤||(∇xθ)⊤||=X(x)||X(x)||.\ Moreover a simple computation gives for all x∈M,   {⟨X(x)|g1(x)⟩=−x22−4x3(x3−1)⟨X(x)|g2(x)⟩=x13(x2+2x3)−x13x2−2x13(x3−1). We get   {x∈M⟨Nx|g1(x)⟩=0⟨Nx)|g2(x)⟩=0 (10) is equivalent to {14x14+12x22+(x3−1)2=4−x22−4x3(x3−1)=02x13=0 Which implies {12x22+(x3−1)2=4−x22−4x3(x3−1)=0x1=0 Thus {x22=−4x3(x3−1)−2x3(x3−1)+(x3−1)2=4x1=0 So {x22=−4x3(x3−1)−x32−3=0x1=0 It is easy to remark that the previous system has no solution in ℝ3, this implies M1∩M2=Ø. So by Theorem 5, if we choose p=4, the feedback u=(u1u2), where   ui(x)=−(β0(x))⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩, for x∈ℝn∖{0} and ui(0)=0, stabilizes the system (9). References Andriano V. ( 1993) Global feedback stabilization of the angular velocity of a symmetric rigid body. Systems Control Lett. , 20, 361– 364. Google Scholar CrossRef Search ADS   Artstein Z. ( 1983) Stabilization with relaxed controls. Nonlinear Anal. , 7, 1163– 1173. Google Scholar CrossRef Search ADS   Hermes H. ( 1991) Nilpotent and high-order approximations of vector field systems. SIAM Rev. , 33, 238– 264. Google Scholar CrossRef Search ADS   Jerbi H. ( 2002) A manifold-like characterization of asymptotic stabilizability of homogeneous systems. Systems Control Lett. , 41, 173– 178. Google Scholar CrossRef Search ADS   Jerbi H., Kharrat T. ( 2005) Only a level set of a control Lyapunov function for homogeneous systems. Kybernetika , 41, 593– 600. Jerbi H., Kallel W., Kharrat T. ( 2008) On the stabilization of homogeneous perturbed systems. J. Dyn. Control Syst. , 14, 595– 606. Google Scholar CrossRef Search ADS   Kharrat T. ( 2017) Stability of homogeneous nonlinear systems. IMA J. Math. Control Inform. , 34, 451– 461. Google Scholar CrossRef Search ADS   Nakamura N., Nakamura H., Yamashita Y., Nishitani H. Homogeneous stabilization for input affine homogeneous systems. IEEE Trans. Automat. Control , 54, 2271– 2275. CrossRef Search ADS   Rosier L. ( 1992) Homogeneous Lyapunov function for homogeneous continuous vector field. Systems Control Lett. , 19, 467– 473. Google Scholar CrossRef Search ADS   Sepulchre R., Aeyels D. ( 1996) Homogeneous Lyapunov functions and necessary conditions for stabilization. Math. Control Signals Systems , 9, 34– 58. Google Scholar CrossRef Search ADS   Sontag E. D. ( 1989) A “Universal” construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett. , 13, 117– 123. Google Scholar CrossRef Search ADS   Tsinias J. ( 1988) Stabilization of affine in control nonlinear systems. Nonlinear Anal. , 12, 1283– 1296. Google Scholar CrossRef Search ADS   Tsinias J. ( 1989) Sufficient Lyapunov like conditions for stabilization. Math. Control Signals Systems , 2, 343– 357. Google Scholar CrossRef Search ADS   © Crown copyright 2017. This article contains public sector information licensed under the Open Government Licence v3.0 (http://www.nationalarchives.gov.uk/doc/open-government-licence/version/3/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

Stabilization of nonlinear homogeneous systems

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Oxford University Press
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0265-0754
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1471-6887
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10.1093/imamci/dnx040
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Abstract

Abstract In this article, we study the stabilization problem of a class of homogeneous multi-input systems. First we give a result for driftless homogeneous systems, then we study the stabilization problem of homogeneous affine in control systems. We introduce some hypothesis for the desired system on a submanifold of ℝn diffeomorphic to Sn−1 and we construct a stabilizing homogeneous feedback which preserve the homogeneity of the closed loop system. 1. Introduction For smooth multi-input systems that are affine in the control   x˙=f(x)+∑i=1kuigi(x), (1) where the state x∈ℝn, the input (u1,…,uk)⊤∈ℝk, f(0)=0 and f, g1, g2,…,gk are continuously differentiable vector fields, the basic stabilization Lyapunov condition provided in Artstein (1983), Tsinias (1988), Sontag (1989) and Tsinias (1989) can be expressed as follows: There exists a positive definite real function V: ℝn→ℝ (i.e., V(0)=0 and V(x)>0 for x≠0 near zero) such that for any x≠0 near zero with   {∇V.g1(x)=0⋮∇V.gk(x)=0} it holds ∇V.f(x)<0 In Artstein (1983), it was shown that if the above condition is fulfilled, then the system (1) is stabilizable at the origin by means of a nonlinear feedback law which is smooth for x≠0. The same result was proved independently in Tsinias (1988), Sontag (1989) and Tsinias (1989), where the corresponding stabilizing feedback laws are more explicitly identified. Further progress was provided in Sontag (1989) where an explicit proof of Artstein's theorem is presented. Among other things, Sontag proved that if the system (1) satisfies the above Lyapunov condition, then stabilization is possible by means of a feedback law that depends directly on the dynamics of the system. For homogeneous systems, in Nakamura et al. (2009), the authors design homogeneous controllers using homogeneous control Lyapunov functions. However, there is no a constructive method for a control Lyapunov function. For this, in Jerbi (2002) and Jerbi et al. (2008), the authors give some hypothesis to construct a submanifold of ℝn diffeomorphic to Sn−1 and under some hypothesis on this submanifold, they study the stabilization problem of homogeneous systems. In this article, first we study the stabilization problem of driftless homogeneous systems. We construct a homogeneous stabilizing feedback that preserve the homogeneity of the closed loop system. Second, we study the stabilization problem for homogeneous nonlinear affine systems in the form (1). We assume that f, gi, for i∈{1,…,k}, are continuously differentiable and homogeneous with respect to the same dilation. We use results given in Jerbi (2002), Jerbi & Kharrat (2005) and Jerbi et al. (2008) to prove under some conditions the stabilization of affine homogeneous systems (1) by means of a homogeneous feedback law. We conclude by applying the proven results to a class of bilinear systems in ℝ2 and ℝ3. 2. Preliminaries We start by recalling the following definitions and results. Definition 1 (Sepulchre & Aeyels 1996) Let {r1,r2,…,rn} a family of fixed positive reals and r=(r1,r2,…,rn). Let δε(x)=(εr1x1,…,εrnxn), x∈ℝn, ε>0, a dilation on ℝn. (i) We say that a function h:ℝn→ℝ is homogenous of degree k with respect to δ, if   h(δε(x))=εkh(x), ∀x∈ℝn, ∀ε>0. (ii) We say that f:ℝn→ℝn is homogenous of degree k, if each fi,  i∈{1,…,n}, is homogeneous of degree k+ri. Definition 2 Let M be a submanifold of ℝn of dimension n−1. We say that M is Φ−diffeomorphic to Sn−1, if the map   Φ: M→Sn−1  x↦x||x|| is a diffeomorphism of C∞. Notations For x∈ℝn∖{0}, we denote Dx+={λx,  λ>0} and Dx−={λx,  λ<0}. ⟨..∣..⟩ denotes the Euclidean inner product and ||x||=⟨x∣x⟩ for x∈ℝn. For x∈ℝn we define ⟨x⟩={λx ; λ∈ℝ}. M⊤ denotes the transpose matrix of M. Let θ: ℝn→ℝ be a smooth map, we denote ∇θ(x)=(∂θ∂x1(x),….,∂θ∂xn(x)). Now, we recall the following definitions and results. Theorem 1 (Jerbi, 2002) Let M be a submanifold of ℝn of dimension n−1. M is Φ−diffeomorphic to Sn−1 if and only if the following holds (1) ∀x∈M one has ℝn=TxM⊕⟨x⟩, where TxM is the tangent space at the point x to the submanifold M, (2) for all x∈ℝn∖{0}, M∩Dx+ is a unique point. The main results of this article are based on the existence of a submanifold of ℝn Φ-diffeomorphic to Sn−1. The following theorem is useful for the construction of such submanifold. Theorem 2 (Jerbi & Kharrat, 2005) Let θ:ℝn→ℝ a map of class C1 and denote X(x)=(∇θ(x))⊤. If θ is proper and the vector field X satisfies the following condition (P) there exists R>0 and ρ>0 such that for ∥x∥>R one has ⟨X(x)∣x⟩≥ρ, then there exists k∈ℝ such that θ−1{k}=M is a sub manifold of ℝn, Φ-diffeomorphic to Sn−1. Moreover Nx=X(x)||X(x)||. Lemma 1 (Rosier, 1992) (1) The map   α:(0,+∞)×Sn−1→ℝn∖{0} (t,(y1,…,yn))↦(tr1y1,…,trnyn) is a bijection, and its inverse function α−1, which we write β=(β0,β1,…,βn), is of class C∞. (2) The function β0 satisfies   limx→0 x≠0β0(x)=0 and lim||x||→+∞β0(x)=+∞ Proposition 1 (Jerbi et al., 2008) Let M be a sub manifold of ℝn, Φ−diffeomorphic to Sn−1. The map   σ:(0,+∞)×M→ℝn∖{0}     (t,z)↦(tr1z1||z||,…,trnzn||z||) is a bijection of class C∞. Its inverse function σ−1:=γ=(γ0,γ1,…,γn) satisfies γ0(x)=β0(x). Remark 1 If M is a submanifold of ℝn, Φ−homeomorphic to Sn−1, then α is an homeomorphism. We recall the following theorem which is fundamental to conclude the stability results given in this article. Theorem 3 (Andriano, 1993) Consider the system   x˙=h(x),   x∈ℝn, where h is a continuously differentiable function satisfying h(0)=0. Suppose that there exist compact subsets {Dλ}λ∈ℝ+ such that (i) ∩λ∈ℝ+Dλ={0}, (ii) for all λ1<λ2, Dλ1⊂Dλ2°, (iii) for all x∈ℝn, there exists λ>0 such that x∈∂Dλ, (iv) for all λ>0 and x0∈∂Dλ one has Xt(x0)∈Dλ°, ∀t>0; where Xt(x0) is the solution of the system x˙=h(x) starting at x0∈∂Dλ, then the system x˙=h(x) is G.A.S. Theorem 4 (Jerbi et al., 2008) Let M be a submanifold of ℝn of dimension n−1, Φ−diffeomorphic to Sn−1. For λ>0, we denote δλr the map defined by    δλ:M→δλ(M)(x,…,xn)→(λr1x1,…,λrnxn), where ri,  i∈{1,…,n} are fixed positive reals. δλ is a diffeomorphism and for all y∈δλ(M), there exists a unique x∈M such that y=δλ(x) and Nδλ(x)=CλAλ−1Nx, where Cλ=∥∇xφn∥∥Aλ−1∇xφn∥, Aλ=diag(λr1,…,λrn) and φn is the n-th component of φ; ((Ux,φ) is a chart of M at the point x; ( φ:Ux→φ(Ux)⊂ℝn−1×{0}) is a diffeomorphism)). 3. Stabilization of driftless homogeneous systems Let the system described by   x˙=∑i=1kuigi(x),   x∈ℝn,  u=(u1⋮uk)∈ℝk, (2) where each gi is continuously differentiable and homogeneous of degree qi for all i∈{1,…,k} with respect to the same dilation δ, where δλ(x)=(λr1x1,…,λrnxn), λ>0 and ri are fixed positive reals. We establish the following lemma which is useful for the proofs of the theorems. Lemma 2 Let the system   x˙=h(x), (3) where h is continuous locally Lipshitz and homogeneous of degree d with respect of the dilation δ. If there exists a submanifold M of ℝn Φ-diffeomorphic to Sn−1 such that ⟨Ny|h(y)⟩<0 for all y∈M, then ⟨Nx|h(x)⟩<0 for all x∈ℝn∖{0}. Proof Let x∈ℝn∖{0}, there exists a unique (λ,y)∈(0,+∞)×M, such that x=δλ(y). A simple computation gives     ⟨Nx∣h(x)⟩=⟨Nδλ(y)∣h(δλ(y))⟩=⟨CλAλ−1Ny∣h(δλ(y))⟩=⟨CλAλ−1Ny∣λdAλh(y)⟩=λdCλ⟨Ny∣h(y)⟩<0.    □ Lemma 3 Let p∈ℝ, the maps ui, i∈{1,…,k}, defined by   {ui(x)=−(β0(x))p−qi⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩u(0)=0 (4) are well defined, continuous over ℝn\{0} and homogeneous of degree p−qi. If in addition p>qi, for all i then ui is continuous at the origin. Proof Let i∈{1,…,k}, we define ui(x)=−(β0(x))p−qi⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩ and ui(0)=0. The map ui is well defined and continuous over ℝn\{0}. The continuity of ui is a trivial consequence of the continuity of the maps Φ−1,β,N,gi and the inner product. The maps   θ:(0,+∞)×M→(0,+∞)×Sn−1    (t,y)↦(t,y||y||) and   α:(0,+∞)×Sn−1→ℝn∖{0}       (t,z)↦(tr1z1,…,trnzn) are bijections of class C∞. Then for all x∈ℝn∖{0}, there exists a unique (ε,y)∈(0,+∞)×M, such that x=α∘θ(ε,y). We have α∘θ(ε,y)=α(ε,Φ(y))=x. So (ε,Φ(y))=β(x)=(β0(x),β1(x),…,βn(x)). Then ε=β0(x) and y=Φ−1(β1(x),…,βn(x)). Thus, we can write ui(x)=−εp−qi⟨Ny∣gi(y)⟩. Let λ>0, we have   δλ(x)=(λr1x1,…,λrnxn)=(λr1tr1y1∥y∥,…,λrntrnyn∥y∥)=((λt)r1y1∥y∥,…,(λε)rnyn∥y∥)=α(λε,y∥y∥)=α(θ(λε,y)) So β0(δλ(x))=λε. Now, we prove that the map ui is homogeneous of degree p−qi. Let x∈ℝn\{0}, there exist ε>0 and y∈M such that y=δε(x). So for λ>0, one has   ui(δλ(x))=−(β0(δλ(x))p−q⟨Ny∣gi(y))⟩=−(λε)p−qi⟨Ny∣gi(y)⟩=−λp−qi(β0(x))p−qi⟨Ny∣gi(y)⟩=λp−qiui(x). If in addition p>qi, then ui is continuous at 0, indeed for x∈ℝn\{0}, there exist ε>0 and y∈M such that y=δε(x). Moreover, by Lemma 1, ε tends to 0 as x tends to zero. By the continuity of ui and the fact that y is in a compact set, one has ui is in a compact set. So limx→0ui(x)=limε→0εp−qiui(y)=0=ui(0).    □ Notation Let M be a submanifold of ℝn, Φ-diffeomorphic to Sn−1. We denote   Mi={x∈M such that ⟨Nx|gi(x)⟩=0}. Now, we establish the following result Theorem 5 Let p>sup{qi, 1≤i≤k}. If ∩i∈{1,…,k}Mi=Ø, then the system (2) is globally asymptotically stabilizable at the origin by a continuous feedback u=(u1,…,uk) satisfying each ui is homogeneous of degree p−qi. Proof We define for i∈{1,…,k},   {ui(x)=−(β0(x))p−qi⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩ for x∈ℝn∖{0}ui(0)=0, where β0,β1,…,βn are the components of the function β given in Lemma 1. It was proven in Lemma 3 that ui is homogeneous of degree p−qi for all i and continuous on ℝn. Let i∈{1,…,k}, y∈M, one has   ui(y)=−⟨Ny∣gi(y)⟩. So   ⟨Ny|∑i=1kui(y)gi(y)⟩=−∑i=1k(⟨Ny|gi(y)⟩)2. By the hypothesis ∩i∈{1,…,k}Mi=Ø, there exists i∈{1,…,k} such that ⟨Ny|gi(y)⟩≠0. We deduce that ⟨Ny|∑i=1kui(y)gi(y)⟩<0. Let x∈ℝn∖{0}, by the hypothesis M is Φ-diffeomorphic to Sn−1, there exists λ>0 and y∈M such that x=δλ(x). So     ⟨Nx|∑i=1kui(x)gi(x)⟩=−∑i=1kui(x)⟨Nx|gi(x)⟩=−∑i=1kλp−qi⟨Ny|gi(y)⟩⟨CλAλ−1Ny|λqiAλgi(y)⟩=−∑i=1kλpCλ(⟨Ny|gi(y)⟩)2<0. By the fact that M is Φ-diffeomorphic to Sn−1, one can write   ℝn∖M=O1∪O2, where O1 and O2 are two connected open sets, we choose 0∈O1. We define Dλ=δλ(O¯1)=δλ(O1∪M). Using the fact that for all x∈ℝn∖{0}  one has  ⟨Nx∣∑i=1kui(x)gi(x)⟩<0, one can deduce that Xt(x0)∈D°λ for all t>0, where Xt(x0) is the solution of the closed loop system (2) starting at x0∈Dλ; that is,   (d/dt)Xt(x0)=∑i=1kui(Xt(x0))gi(Xt(x0)) and X0(x0)=x0. Then, we deduce that Xt(x0)∈Dλ° for all t>0 where Xt(x0) is the solution of the closed loop system (2) starting at x0∈Dλ¯∖D°λ=∂Dλ=δλ(M) and the proof follows from Theorem 3.     □ 4. Stabilization of homogeneous affine in control systems Let us consider the multi-input systems that are affine in control:   x˙=f(x)+∑i=1kuigi(x), (5) where the state x∈ℝn, the input (u1,…,uk)⊤∈ℝk, f(0)=0, f is continuously differentiable homogeneous of degree p and each  gi, i∈{1,…,k}, is continuously differentiable and homogeneous of degree qi. Using the same notations of Section 2, one has the following result. Theorem 6 If there exists a submanifold M of degree n−1 of ℝn, Φ−diffeomorphic to Sn−1 and satisfying   for all x∈∩i∈{1,…,k}Mi,one has ⟨Nx∣f(x)⟩<0, then there exists a feedback u=(u1⋮uk) such that ui is homogeneous of of degree p−qi, for i=1,…,k, which stabilizes the system (5). Proof Suppose that there exists a submanifold M of degree n−1 of ℝn, Φ−diffeomorphic to Sn−1 and satisfying for all x∈∩i∈{1,…,k}Mi one has ⟨Nx∣f(x)⟩<0. We define the set A={x∈M such that ⟨Nx∣f(x)⟩≥0}. We have two cases If A=Ø, then the system x˙=f(x) is globally asymptotically stable. If A≠Ø, we can easily deduce that A is a compact set. Moreover   A⊂{x∈M such that ∑i=1k⟨Nx∣gi(x)⟩2≠0 }. We define ω=infx∈A{∑i=1k⟨Nx∣gi(x)⟩2}>0 and M=supx∈A⟨Nx∣f(x)⟩≥0. Let ρ>Mω>0, we define for i∈{1,…,k},   {ui(x)=−ρ(β0(x))p−qi⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩ if x∈ℝn∖{0}ui(0)=0, where β0,β1,…,βn are the components of the function β given in Lemma 1. According to Lemma 3, the maps ui are well defined, continuous on ℝn∖{0} and homogeneous of degree p−qi. Moreover, the maps ui satisfy ui(x)=−ρ(⟨Nx∣gi(x)⟩) for all x∈M. Under the choice of the constant ρ, we can write for x∈A  ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩=⟨Nx∣f(x)⟩−ρ∑i=1k(⟨Nx∣gi(x)⟩)2≤M−ρω<0, and since for all x∈M∖A one has ⟨Nx∣f(x)⟩−ρ∑i=1k(⟨Nx∣gi(x)⟩)2<0, then   ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩<0 for all x∈M. Now, we prove that   ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩<0 for all x∈ℝn∖{0}. Let x∈ℝn∖{0}, there exists a unique (λ,y)∈(0,+∞)×M, such that x=δλ(y). A simple computation gives    ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩=⟨Nδλ(y)∣f(δλ(y))+∑i=1kui(δλ(y))gi(δλ(y))⟩=⟨CλAλ−1Ny∣f(δλ(y))+∑i=1kui(δλ(y))gi(δλ(y))⟩=⟨CλAλ−1Ny∣λpAλ(f(y)+∑i=1kui(y)gi(y))⟩=λpCλ⟨Ny∣f(y)+∑i=1kui(y)gi(y)⟩<0. By the fact that M is Φ-diffeomorphic to Sn−1, one can write   ℝn∖M=O1∪O2, where O1 and O2 are two connected open sets, we choose 0∈O1. We define Dλ=δλ(O¯1)=δλ(O1∪M). Using the fact that for all x∈ℝn∖{0}  one has  ⟨Nx∣f(x)+∑i=1kui(x)gi(x)⟩<0, one can deduce that Xt(x0)∈Dλ° for all t>0 where Xt(x0) is the solution of the closed loop system (2) starting at x0∈Dλ¯∖D°λ=∂Dλ=δλ(M) and the proof follows from Theorem 3. 5. Application 5.1. Stabilization of a bilinear system in ℝ2 Let us consider the planar bilinear system described by   x˙=Ax+∑i=12uiBix,   x∈ℝ2 ,  u=(u1u2)∈ℝ2, (6) where A=(a11a12a21a22), B1=(λ100λ2) and B2=(01−10). We know that if λ1λ2>0, then system (6) can be stabilized by a feedback u=(u1u2)∈ℝ2 with u1=σ, u2=0 and |σ| is large enough to satisfy   eigenvalue(A+σB1)⊂ℝ−* and if λ1+λ2=0 and a11+a22>0 then system (6) cannot be stabilized by a feedback u=(u1u2)∈ℝ2 with u1=σ∈ℝ, u2=0. Moreover, if detA>0, then system (6) cannot be stabilized by a feedback u=(u1u2)∈ℝ2 with u1=0, u2∈ℝ. Now, we prove the stabilization of the system (6) by means of an homogeneous feedback u=(u1u2)∈ℝ2 with u1≠0, u2≠0. We introduce the function   θ:ℝ2→ℝx↦12||x−b||2, where b=(01). It is easy to verify that θ satisfies to conditions of Theorem 2. In fact: Let X(x)=(∇θ(x))⊤=(x1,x2−1)⊤, we have   ⟨X(x)|x⟩=x12+x22−x2=||x−b2||2−||b2||2≥|||x||−||b2|||2−||b2||2≥||x||(||x||−||b||). So, if we choose β>||b||=1 and δ=β(β−||b||)>0, we get for x satisfying ||x||≥β one has ⟨X(x)|x⟩≥β(β−||b||)=δ. Then according to Theorem 2, there exists k>0 such that M=θ−1{k} is a submanifold of ℝ2 of dimension 1. The constant k can be chosen such that k>1 and this to guarantee that for all x∈ℝ2, M∩Dx+ is a unique point. Now let x∈M, we verify that if ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0, then ⟨X(x)|Ax⟩<0. But ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0 is equivalent to   {λ1x12+λ2x2(x2−1)=0x1x2−x1(x2−1)=0 which gives   {λ2x2(x2−1)=0x1=0 This computation gives the set of points x=(x1x2)∈M satisfying ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0 is empty. By Theorem 6, we deduce that there exists a feedback u=(u1u2), such that ui is homogeneous of degree 0, for i=1,2, which stabilizes the system (6). 5.2. Stabilization of a bilinear system in ℝ3 Let us consider the multi-input bilinear system   x˙=Ax+∑i=12uiBix,   x∈ℝ3 ,  u=(u1u2)∈ℝ2, (7) where A=(a11a120a21a22000a), B1=(λ1000λ20000) and B2=(010−10000λ), λ1>0, λ2<0 and λ≠0.\ We define the function θ:ℝ3→ℝ, x=(x1x2x3)↦12||x−b||2, where b=(001). It is easy to verify that θ satisfies to conditions of Theorem 2. Indeed: Let X(x)=(∇θ(x))⊤=(x1,x2,x3−1)⊤, we have ⟨X(x)|x⟩=x12+x22+x32−x3≥||x||2−||x||≥||x||(||x||−1). So if we choose β>1 and δ=β(β−1)>0, we get for x satisfying ||x||≥β one has ⟨X(x)|x⟩≥β(β−1)=δ. Then according to Theorem 2, there exists k>0 such that M=θ−1{k} is a submanifold of ℝ3 of dimension 2. The constant k can be chosen such that k>1 and this to guarantee that for all x∈ℝ3, M∩Dx+ is a unique point. Now let x∈M, we verify that if ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0, then ⟨X(x)|Ax⟩<0. But ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0 is equivalent to   {λ1x12+λ2x22=0λx3(x3−1)=0 which gives {x22=−λ1λ2x12x3=0 or {x22=−λ1λ2x12x3=1 Using the fact that x∈M, we deduce {x22=−λ1λ2x12x3=0x12=λ2λ2−λ1(k−1) or {x22=−λ1λ2x12x3=1x12=λ2λ2−λ1k. So the set of points x=(x1x2x3)∈M satisfying ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0 is   B={x=(x1x2x3)∈M such that ⟨X(x)|B1x⟩=0 and ⟨X(x)|B2x⟩=0}={x=(x1x2x3)∈ℝ3such that {x12=λ2λ2−λ1(k−1)x22=λ1λ1−λ2(k−1)x3=0 or{x12=λ2λ2−λ1kx22=λ1λ1−λ2kx3=1}. Let x∈B, we get ⟨X(x)|Ax⟩=a11x12+a22x22+(a12+a21)x1x2+ax3(x3−1)=(−a11λ2+a22λ1±(a12+a21)−λ1λ2)kλ1−λ2 or ⟨X(x)|Ax⟩=a11x12+a22x22+(a12+a21)x1x2+ax3(x3−1)=(−a11λ2+a22λ1±(a12+a21)−λ1λ2)k−1λ1−λ2. So, a necessary condition for the stabilization of the system (7) is to choose matrices A and B1 satisfying to the following (−a11λ2+a22λ1±(a12+a21)−λ1λ2)<0. By Theorem 6, we deduce that there exists a feedback u=(u1u2), such that ui is homogeneous of degree 0, for i=1,2, which stabilizes the system (7). Example 1 Let the system   x˙=Ax+∑i=12uiBix,   x∈ℝ3 ,  u=(u1u2)∈ℝ2, (8) where A=(−1−10100001), B1=(1000−10000) and B2=(010−100001). We define the function θ:ℝ3→ℝ, x=(x1x2x3)↦12||x−b||2, where b=(001). We can easily remark that system (8) cannot be stabilized by a feedback u=(u1u2)∈ℝ2 with u1=0 or u2=0. In the previous paragraph, we have proved that θ satisfies to conditions of Theorem 2. So for k=2, the set M=θ−1{k} is a submanifold of ℝ3 of dimension 2 and for all x∈M, Nx=(∇xθ)⊤||(∇xθ)⊤||=x−b||x−b||. Moreover, a simple computation gives for all x∈M, ⟨Nx|B1x⟩=0 and ⟨Nx)|B2x⟩=0 implies ⟨Nx|Ax⟩<0. Indeed,\ {x∈M⟨Nx|B1x⟩=0⟨Nx)|B2x⟩=0 This is equivalent to {x12+x22+(x3−1)2=4x12−x22=0x3(x3−1)=0 which implies ⟨Nx|Ax⟩=−x12+x3(x3−1)<0. So by Theorem 6, the feedback u=(u1u2), where ui(x)=−ρ⟨NΦ−1(β1,…,βn)(x)∣Bi(Φ−1(β1,…,βn)(x))⟩ for x∈ℝn∖{0} and ui(0)=0, stabilizes the system (8). The expression of the maps ui can be simplified as follow. For x∈ℝ3∖{0}, the exists a unique positive real α such that αx∈M. In addition αx=Φ−1(β1,…,βn)(x). A simple computation gives α=x3+x32+3||x||2||x||2. We conclude that   ui(x)=−ρα⟨αx−b||αx−b|||Bix⟩, where the positive real ρ will be computed in the follow. Let A={x∈M such that ⟨Nx∣Ax⟩≥0}, Let ω=infx∈A{(⟨Nx∣B1(x)⟩)2+(⟨Nx∣B2(x)⟩)2}>0. Now, we have to approximate the value of ω. Let x∈A, then x∈M and ⟨Nx∣Ax⟩≥0. So   {||x−b||2=4⟨Nx∣Ax⟩≥0⇔{||x−b||2=4⟨(x−b)∣Ax⟩≥0⇔{x12+x22+(x3−1)2=4−x12+x3(x3−1)≥0 On the other hand, we have   (⟨Nx∣B1x⟩)2+(⟨Nx∣B2x⟩)2=(⟨x−b∣B1x⟩)2+(⟨x−b∣B2x⟩)2=(x12−x22)2+(x3(x3−1))2. A numerical computation gives ω=infx∈A(⟨Nx∣B1x⟩)2+(⟨Nx∣B2x⟩)2≥0,15 Now, we calculate M=supx∈A⟨Nx∣Ax⟩. Let x∈A, one has   {||x−b||2=4⟨Nx∣Ax⟩≥0⇔{||x−b||2=4−x12+x3(x3−1)≥0⇔{x12+x22+(x3−1)2=4−x12+x3(x3−1)≥0 We have −x12+x3(x3−1)≤x3(x3−1)≤|x3||x3−1|≤6 by the fact that |x3−1|≤2. We deduce M≤6. Finally, we can choose ρ=60. In the next, we give an example for stabilization of driftless systems. Example 2 Let the system described by:   {x˙1=u2(x2+x3)x˙2=−u1x2−u2x13x˙3=−2u1x3−u2x13, (9) where u=(u1u2)∈ℝ2 and x=(x1,x2,x3)⊤∈ℝ3. Denote g1(x1,x2,x3)=(0,−x2,−2x3)⊤ and g2(x1,x2,x3)=(x2+x3,−x13,−x13)⊤. We can easily verify that g1 and g2 are homogeneous with respect to the dilation δr, r=(1,3,3), of degree 3. Let θ(x1,x2,x3)=14x14+12x22+(x3−1)2. θ is proper and satisfies to conditions of Theorem 2. Indeed, let X(x)=(∇θ(x))⊤=(x13,x2,2(x3−1))⊤, we have   ⟨X(x)|x⟩=x14+x22+2x32−2x3=x14+x22+x32+(x3−1)2−1=x14+x12−x12+x22+x32+(x3−1)2−1=||x||2+x12(x12−1)+(x3−1)2−1≥||x||2−3. It is clear that if ||x||≥2, then ⟨X(x)|x⟩≥1. Then according to Theorem 2, there exists k>0 such that M=θ−1{k} is a submanifold of ℝ3 of dimension 2. The constant k can be chosen such that k>1 and this to guarantee that for all x∈ℝ3, M∩Dx+ is a unique point. So for k=4, the set M=θ−1{k} is a submanifold of ℝ3 of dimension 2 and for all x∈M, Nx=(∇xθ)⊤||(∇xθ)⊤||=X(x)||X(x)||.\ Moreover a simple computation gives for all x∈M,   {⟨X(x)|g1(x)⟩=−x22−4x3(x3−1)⟨X(x)|g2(x)⟩=x13(x2+2x3)−x13x2−2x13(x3−1). We get   {x∈M⟨Nx|g1(x)⟩=0⟨Nx)|g2(x)⟩=0 (10) is equivalent to {14x14+12x22+(x3−1)2=4−x22−4x3(x3−1)=02x13=0 Which implies {12x22+(x3−1)2=4−x22−4x3(x3−1)=0x1=0 Thus {x22=−4x3(x3−1)−2x3(x3−1)+(x3−1)2=4x1=0 So {x22=−4x3(x3−1)−x32−3=0x1=0 It is easy to remark that the previous system has no solution in ℝ3, this implies M1∩M2=Ø. So by Theorem 5, if we choose p=4, the feedback u=(u1u2), where   ui(x)=−(β0(x))⟨NΦ−1(β1,…,βn)(x)∣gi(Φ−1(β1,…,βn)(x))⟩, for x∈ℝn∖{0} and ui(0)=0, stabilizes the system (9). References Andriano V. ( 1993) Global feedback stabilization of the angular velocity of a symmetric rigid body. Systems Control Lett. , 20, 361– 364. Google Scholar CrossRef Search ADS   Artstein Z. ( 1983) Stabilization with relaxed controls. Nonlinear Anal. , 7, 1163– 1173. Google Scholar CrossRef Search ADS   Hermes H. ( 1991) Nilpotent and high-order approximations of vector field systems. SIAM Rev. , 33, 238– 264. Google Scholar CrossRef Search ADS   Jerbi H. ( 2002) A manifold-like characterization of asymptotic stabilizability of homogeneous systems. Systems Control Lett. , 41, 173– 178. Google Scholar CrossRef Search ADS   Jerbi H., Kharrat T. ( 2005) Only a level set of a control Lyapunov function for homogeneous systems. Kybernetika , 41, 593– 600. Jerbi H., Kallel W., Kharrat T. ( 2008) On the stabilization of homogeneous perturbed systems. J. Dyn. Control Syst. , 14, 595– 606. Google Scholar CrossRef Search ADS   Kharrat T. ( 2017) Stability of homogeneous nonlinear systems. IMA J. Math. Control Inform. , 34, 451– 461. Google Scholar CrossRef Search ADS   Nakamura N., Nakamura H., Yamashita Y., Nishitani H. Homogeneous stabilization for input affine homogeneous systems. IEEE Trans. Automat. Control , 54, 2271– 2275. CrossRef Search ADS   Rosier L. ( 1992) Homogeneous Lyapunov function for homogeneous continuous vector field. Systems Control Lett. , 19, 467– 473. Google Scholar CrossRef Search ADS   Sepulchre R., Aeyels D. ( 1996) Homogeneous Lyapunov functions and necessary conditions for stabilization. Math. Control Signals Systems , 9, 34– 58. Google Scholar CrossRef Search ADS   Sontag E. D. ( 1989) A “Universal” construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett. , 13, 117– 123. Google Scholar CrossRef Search ADS   Tsinias J. ( 1988) Stabilization of affine in control nonlinear systems. Nonlinear Anal. , 12, 1283– 1296. Google Scholar CrossRef Search ADS   Tsinias J. ( 1989) Sufficient Lyapunov like conditions for stabilization. Math. Control Signals Systems , 2, 343– 357. Google Scholar CrossRef Search ADS   © Crown copyright 2017. This article contains public sector information licensed under the Open Government Licence v3.0 (http://www.nationalarchives.gov.uk/doc/open-government-licence/version/3/).

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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Sep 19, 2017

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