Geometric control and applications Ludovic Rifford Universit e - PowerPoint PPT Presentation

Geometric control and applications Ludovic Rifford Universit´ e Nice Sophia Antipolis Fields Institute - November 6-7, 2014 Ludovic Rifford Fields Institute minischool

Outline Lecture 1: A controllability result: The Chow-Rashevsky Theorem Lecture 2: An optimal control study: Sub-Riemannian geodesics Ludovic Rifford Fields Institute minischool

Lecture 1 A controllability result: The Chow-Rashevsky Theorem Ludovic Rifford Fields Institute minischool

Control of an inverted pendulum Ludovic Rifford Fields Institute minischool

Control systems A general control system has the form x = f ( x , u ) ˙ where x is the state in M u is the control in U Proposition Under classical assumptions on the datas, for every x ∈ M and every measurable control u : [0 , T ] → U the Cauchy problem � ˙ � � x ( t ) = f x ( t ) , u ( t ) a.e. t ∈ [0 , T ] , x (0) = x admits a unique solution x ( · ) = x ( · ; x , u ) : [0 , T ] − → M . Ludovic Rifford Fields Institute minischool

b b Controllability issues Given two points x 1 , x 2 in the state space M and T > 0, can we find a control u such that the solution of � ˙ � � x ( t ) = f x ( t ) , u ( t ) a.e. t ∈ [0 , T ] x (0) = x 1 satisfies x ( T ) = x 2 ? x 2 x 1 Ludovic Rifford Fields Institute minischool

Controllability of linear control systems in R n An autonomous linear control system in R n has the form ˙ ξ = A ξ + B u , with ξ ∈ R n , u ∈ R m , A ∈ M n ( R ) , B ∈ M n , m ( R ). Theorem The following assertions are equivalent: (i) For any T > 0 and any ξ 1 , ξ 2 ∈ R n , there is u ∈ L 1 ([0 , T ]; R m ) such that � � ξ T ; ξ 1 , u = ξ 2 . (ii) The Kalman rank condition is satisfied: B , AB , A 2 B , · · · , A n − 1 B � � rk = n . Ludovic Rifford Fields Institute minischool

Proof of the theorem Duhamel’s formula � T = e TA ξ + e TA e − tA B u ( t ) dt . � � ξ T ; ξ, u 0 Then the controllability property (i) is equivalent to the surjectivity of the mappings � T F T : u ∈ L 1 ([0 , T ]; R m ) �− e − tA B u ( t ) dt . → 0 Ludovic Rifford Fields Institute minischool

Proof of (ii) ⇒ (i) If F T is not onto (for some T > 0), there is p � = 0 n such that � T � � e − tA B u ( t ) dt ∀ u ∈ L 1 ([0 , T ]; R m ) . p , = 0 0 Using the linearity of �· , ·� and taking u ( t ) = B ∗ e − tA ∗ p , we infer that p ∗ e − tA B = 0 ∀ t ∈ [0 , T ] . Derivating n times at t = 0 yields p ∗ B = p ∗ A B = p ∗ A 2 B = · · · = p ∗ A n − 1 B = 0 . Which means that p is orthogonal to the image of the n × mn matrix � B , AB , A 2 B , · · · , A n − 1 B � . Contradiction !!! Ludovic Rifford Fields Institute minischool

Proof of (i) ⇒ (ii) If B , AB , A 2 B , · · · , A n − 1 B � � rk < n , there is a nonzero vector p such that p ∗ B = p ∗ A B = p ∗ A 2 B = · · · = p ∗ A n − 1 B = 0 . By the Cayley-Hamilton Theorem, we deduce that p ∗ A k B = 0 ∀ k ≥ 1 , and in turn p ∗ e − tA B = 0 ∀ t ≥ 0 . We infer that � T � � e − tA B u ( t ) dt ∀ u ∈ L 1 ([0 , T ]; R m ) , ∀ T > 0 . p , = 0 0 Contradiction !!! Ludovic Rifford Fields Institute minischool

Application to local controllability Let ˙ x = f ( x , u ) be a nonlinear control system with x ∈ R n , u ∈ R m and f : R n × R m → R n of class C 1 . Theorem Assume that f ( x 0 , 0) = 0 and that the pair A = ∂ f B = ∂ f ∂ x ( x 0 , 0) , ∂ u ( x 0 , 0) , satisfies the Kalman rank condition. Then for there is δ > 0 such that for any x 1 , x 2 with | x 1 − x 0 | , | x 2 − x 0 | < δ , there is u : [0 , 1] → R m smooth satisfying � � x 1; x 1 , u = x 2 . Ludovic Rifford Fields Institute minischool

b b b Local controllability around x 0 x 2 x 0 x 1 Ludovic Rifford Fields Institute minischool

Proof of the Theorem Define G : R n × L 1 ([0 , 1]; R m ) → R n × R n by � � � � G x , u := x , x (1; x , u ) . The mapping G is a C 1 submersion at (0 , 0). Thus there are n controls u 1 , · · · , u n in L 1 ([0 , 1]; R m ) such that G : R n × R n R n × R n ˜ − → x , � n � i = k λ k u k � ( x , λ ) �− → G is a C 1 diffeomorphism at (0 , 0). Since the set of smooth controls is dense in L 1 ([0 , 1]; R m ), we can take u 1 , . . . , u n to be smooth. We apply the Inverse Function Theorem. Ludovic Rifford Fields Institute minischool

Back to the inverted pendulum The equations of motion are given by θ 2 sin θ x + m ℓ ¨ θ cos θ − m ℓ ˙ ( M + m ) ¨ = u m ℓ 2 ¨ θ − mg ℓ sin θ + m ℓ ¨ x cos θ = 0 . Ludovic Rifford Fields Institute minischool

Back to the inverted pendulum x = θ = ˙ The linearized control system at x = ˙ θ = 0 is given by x + m ℓ ¨ ( M + m ) ¨ θ = u m ℓ 2 ¨ θ − mg ℓ θ + m ℓ ¨ x = 0 . It can be written as a control system ˙ ξ = A ξ + B u , x , θ, ˙ with ξ = ( x , ˙ θ ),     0 1 0 0 0 − mg 1 0 0 0     M A = and B = M  .     0 0 0 1 0    ( M + m ) g − 1 0 0 0 M ℓ M ℓ Ludovic Rifford Fields Institute minischool

Back to the inverted pendulum The Kalman matrix ( B , AB , A 2 , A 3 B ) equals 1 mg 0 0   M 2 ℓ M 1 mg 0 0 M 2 ℓ  M   . − ( M + m ) g  − 1  0 0  M 2 ℓ 2 M ℓ − ( M + m ) g − 1 0 0 M 2 ℓ 2 M ℓ Its determinant equals − g 2 M 4 ℓ 4 < 0 In conclusion, the inverted pendulum is locally controllable around (0 , 0 , 0 , 0) ∗ . Ludovic Rifford Fields Institute minischool

Movie Ludovic Rifford Fields Institute minischool

The Chow-Rashevsky Theorem Theorem (Chow 1939, Rashevsky 1938) Let M be a smooth manifold and X 1 , · · · , X m be m smooth vector fields on M. Assume that X 1 , . . . , X m � � Lie ( x ) = T x M ∀ x ∈ M . Then the control system m � u i X i ( x ) x = ˙ i =1 is locally controllable in any time at every point of M. Ludovic Rifford Fields Institute minischool

b b b Comment I The local controllability in any time at every point means that for every x 0 ∈ M , every T > 0 and every neighborhood U of x 0 , there is a neighborhood V ⊂ U of x 0 such that for any x 1 , x 2 ∈ V , there is a control u ∈ L 1 ([0 , T ]; R m ) such that the trajectory x ( · ; x 1 , u ) : [0 , T ] → M remains in U and steers x 1 to x 2 , i.e. x ( T ; x 1 , u ) = x 2 . x 2 x 0 x 1 V U Local controllability in time T > 0 ⇒ Local controllability in time T ′ > 0 , ∀ T ′ > 0 Ludovic Rifford Fields Institute minischool

Comment II If M is connected then Local controllability ⇒ Global controllability Let x ∈ M be fixed. Denote by A ( x ) the accessible set from x , that is � � � | T ≥ 0 , u ∈ L 1 � A ( x ) := x T ; x , u | u ∈ L 1 � � � � = x 1; x , u . By local controllability, A ( x ) is open. Let y be in the closure of A ( x ). The set A ( y ) contains a small ball centered at y and there are points of A ( x ) in that ball. Then A ( x ) is closed. By connectedness of M , we infer that A ( x ) = M for every x ∈ M , and in turn that the control system is globally controllable in any time. Ludovic Rifford Fields Institute minischool

The Chow-Rashevsky Theorem Theorem (Chow 1939, Rashevsky 1938) Let M be a smooth manifold and X 1 , · · · , X m be m smooth vector fields on M. Assume that X 1 , . . . , X m � � Lie ( x ) = T x M ∀ x ∈ M . x = � m i =1 u i X i ( x ) is locally Then the control system ˙ controllable in any time at every point of M. The condition in red is called H¨ ormander’s condition or bracket generating condition. Families of vector fields satisfying that condition are called nonholonomic, completely nonholonomic, or totally nonholonomic. Ludovic Rifford Fields Institute minischool

Comment III Definition Given two smooth vector fields X , Y on R n , the Lie bracket [ X , Y ] at x ∈ R n is defined by [ X , Y ]( x ) = DY ( x ) X ( x ) − DX ( x ) Y ( x ) . The Lie brackets of two smooth vector fields on M can be defined in charts with the above formula. Given a family F of smooth vector fields on M , we denote by Lie {F} the Lie algebra generated by F . It is the smallest vector subspace S of smooth vector fields containing F that also satisfies [ X , Y ] ∈ S ∀ X ∈ F , ∀ Y ∈ S . Ludovic Rifford Fields Institute minischool

b Comment III e tX ( x ) b x Ludovic Rifford Fields Institute minischool

b Comment III b e tY ◦ e tX ( x ) e tX ( x ) b x Ludovic Rifford Fields Institute minischool

b b Comment III b e tY ◦ e tX ( x ) e − tX ◦ e tY ◦ e tX ( x ) e tX ( x ) b x Ludovic Rifford Fields Institute minischool

b b Comment III b e tY ◦ e tX ( x ) e − tX ◦ e tY ◦ e tX ( x ) b e − tY ◦ e − tX ◦ e tY ◦ e tX ( x ) e tX ( x ) b x Ludovic Rifford Fields Institute minischool

b b Comment III b e tY ◦ e tX ( x ) e − tX ◦ e tY ◦ e tX ( x ) b e − tY ◦ e − tX ◦ e tY ◦ e tX ( x ) e tX ( x ) b x Ludovic Rifford Fields Institute minischool

b b b b b Comment III Exercise We have e − tY ◦ e − tX ◦ e tY ◦ e tX � � ( x ) − x [ X , Y ]( x ) = lim . t 2 t ↓ 0 x Ludovic Rifford Fields Institute minischool