DISC Systems and Control Theory of Nonlinear Systems 1 Lecture 1: Mathematical preliminaries and introduction to nonlinear controllability Nonlinear Dynamical Control Systems, Chapters 1, 2 + handout See www.math.rug.nl/˜arjan (under teaching) for info on course schedule and homework sets. DISC Systems and Control Theory of Nonlinear Systems 2 Very simple example of a nonlinear system: unicycle ẋ1 = u1 cos x3 ẋ1 = u1 sin x3 ẋ3 = u2 Example of a general nonlinear system ẋ = f (x, u), y = h(x, u) DISC Systems and Control Theory of Nonlinear Systems One approach to analysis and control of nonlinear systems: linearization. Let 0 = f (x̄, ū) Then linearized system is ż = Az + Bv where ∂f ∂f (x̄, ū), B = (x̄, ū) ∂x ∂u Approximation of the nonlinear system with z = x − x̄, v = u − ū. A= 3 DISC Systems and Control Theory of Nonlinear Systems Linearization of the unicycle at any point (x̄1 , x̄2 , x̄3 ) and (ū1 , ū2 ) = (0, 0) : ż1 cos x̄3 0 u ż1 = sin x̄3 0 1 u2 ż3 0 1 Is never controllable ! Contrary to intuition. How do we study nonlinear controllability ? 4 DISC Systems and Control Theory of Nonlinear Systems First observation: Nonlinearity shows up in • Nonlinearity of differential equations for the state evolution or nonlinear output map. • Nonlinearity also shows up in the structure of the state space, which is in general not anymore Rn . We will start by defining nonlinear state spaces; or in mathematical terminology, (smooth) manifolds. Analogy: Subspaces of Rn of dimension n − m are defined by m independent linear equations. Manifolds of dimension n − m are subsets of Rn , which are defined by m independent nonlinear equations. 5 DISC Systems and Control Theory of Nonlinear Systems 6 Definition 1 Let f1 , · · · , fm , m ≤ n, be smooth functions on an open part V of Rn . Define the set M = {x ∈ V |f1 (x) = · · · = fm (x) = 0} Suppose that the rank of the Jacobian matrix of f = (f1 , · · · , fm )T ∂f1 ∂x1 (x) ··· ∂fm ∂x1 (x) ··· .. . ∂f1 ∂xn (x) ∂fm ∂xn (x) ∂f =: (x) ∂x is m at each x ∈ M . Then M is a manifold of dimension n − m (if M is non-empty). DISC Systems and Control Theory of Nonlinear Systems Example 2 Every open subset V of Rn is a manifold of dimension n (Take m = 0). Example 3 The circle S 1 is a manifold of dimension 1, since S 1 = {(x1 , x2 ) ∈ R2 |x21 + x22 − 1 = 0}. Example 4 Consider the group O(n) of orthogonal (n, n)-matrices (i.e. A ∈ O(n) satisfies AT A = In ). n2 Consider the set g`(n) of all (n, n) matrices, identified with R . Define the map f from g`(n) to the space of symmetric (n, n) 1 matrices (identified with R 2 n(n+1) ) as f (A) = AT A Then O(n) = {A ∈ g`(n)|f (A) = In }. The rank of the Jacobian matrix 2 1 of f (seen as a map from Rn to R 2 n(n+1) ) equals 21 n(n + 1) at every point A ∈ O(n). Therefore O(n) is a smooth manifold of dimension 1 1 n − n(n + 1) = n(n − 1) 2 2 2 7 DISC Systems and Control Theory of Nonlinear Systems 8 The basic feature of a manifold M of dimension n − m is that it is locally Rn−m in the following sense. Let xo ∈ M . By permuting the coordinates x1 , · · · , xn for Rn we may assume that the (m, m) matrix ∂f1 ∂x1 ··· .. . ∂fm ∂x1 ··· ∂f1 ∂xm ∂fm ∂xm is non-singular at xo . By the implicit function theorem there now exists a neighborhood W1 ⊂ Rn of xo , a neighborhood W2 ⊂ Rn−m of (xom+1 , · · · , xon ), and a smooth map g : W2 → Rm such that M ∩ W1 equals {[g1 (xm+1 , · · · , xn ), · · · , gm (xm+1 , · · · , xn ), xm+1 , · · · , xn ] |(xm+1 , · · · , xn ) ∈ W2 } DISC Systems and Control Theory of Nonlinear Systems Then on U := M ∩ W1 we define coordinate functions ϕi , i = 1, · · · , n − m, by ϕi [g1 (xm+1 , · · · , xn ), · · · , gm (xm+1 , · · · , xn ), xm+1 , · · · , xn ] = xm+i U is called a coordinate neighborhood of xo . In this way the neighborhood U of xo becomes identified with an open part of Rn−m . 9 DISC Systems and Control Theory of Nonlinear Systems Example 5 Consider the circle S 1 = {(x1 , x2 )|x21 + x22 − 1 = 0}. Take any point xo = (xo1 , xo2 ) ∈ S 1 . If xo1 6= 0 we have that ∂ 2 2 o o ∂x1 (x1 + x2 − 1)|(x1 ,x2 ) 6= 0, and thus we can solve for x1 , i.e. p x1 = ± 1 − x22 (with sign depending on the sign of xo1 ). The x2 -coordinate may thus serve as coordinate function in both cases. p o Alternatively, if x2 6= 0 we solve for x2 , i.e. x2 = ± 1 − x21 , leading to neighborhoods Ũ1 and Ũ2 which are respectively in the upperand the lower half-plane. 10 DISC Systems and Control Theory of Nonlinear Systems 11 From now on we look at manifolds as objects on their own. Let h : M → R be a function on M . Let U be a coordinate neighborhood of xo ∈ M as above. Then h is smooth on U if the function h [g1 (xm+1 , · · · , xn ), · · · , gm (xm+1 , · · · , xn ), xm+1 , · · · , xn ] depends smoothly on its arguments xm+1 , · · · , xn . The function h is smooth on M if it is smooth on a covering set of coordinate neighborhoods of M . Let h1 , · · · , hk be smooth functions on M . Then h1 , · · · , hk are called independent on U if the functions hi [g1 (xm+1 , · · · , xn ), · · · , gm (xm+1 , · · · , xn ), xm+1 , · · · , xn ] , are independent as functions of xm+1 , · · · , xn . i = 1, · · · , k DISC Systems and Control Theory of Nonlinear Systems 12 With the aid of the above definition the notion of a coordinate neighborhood and of coordinate functions defined on it can be immediately generalized. Indeed, any open subset V of M with n (= dim M ) independent smooth functions (ϕ1 , · · · , ϕn ) defined on it defines a coordinate neighborhood and coordinate functions for M , or, briefly, a coordinate system (V, (ϕ1 , · · · , ϕn )) Definition 6 Let M now be a manifold of dimension n. A subset P ⊂ M is called a submanifold of dimension k < n if for each p ∈ P there exists a coordinate system (V, ϕ1 , · · · , ϕn ) for M about p such that P ∩ V = {q ∈ V |ϕi (q) = ϕi (p), i = k + 1, · · · , n} Notice that a submanifold P of a manifold M is a manifold in its own right, with coordinate system (P ∩ V, (ϕ1 , · · · , ϕk )). DISC Systems and Control Theory of Nonlinear Systems 13 Let M be an (n − m)-dimensional manifold. Let xo ∈ M , then the tangent space Tx0 M at x0 to the manifold M is given as the linear space ∂f ∂f Tx0 M = z ∈ Rn | (x0 )z = 0 = ker (x0 ) ∂x ∂x (Notice that because the rank of ∂f ∂x (x0 ) equals m the dimension of Tx0 M equals n − m, i.e. the dimension of the manifold M .) Furthermore the tangent bundle T M is defined as the manifold T M = {(x, z) ∈ V × Rn |f1 (x) = · · · = fm (x) = 0, and equals S x∈M Tx M . ∂f (x)z = 0} ∂x DISC Systems and Control Theory of Nonlinear Systems 14 Let (z1 , z2 , . . . , zn ) be a coordinate system (local on U ) on the n-dimensional manifold M . This defines on every tangent space Tp M , with p ∈ U , a basis for this linear space, denoted as ∂ ∂z1 ,··· , p ∂ ∂zn p Indeed, every tangent vector Xp ∈ Tp M, p ∈ M can be associated with a derivation. Define c : (−, ) −→ M, > 0, c(0) = p such that c0 (0) = dc dt (0) = Xp . For any function h : M → R define the derivative of h in the direction Xp at the point p ∈ M as Xp (h) := d h(c(t))|t=0 dt DISC Systems and Control Theory of Nonlinear Systems The derivation corresponding to 15 ∂ ∂zi |p is defined as ∂ ∂h |p h = (p) ∂zi ∂zi We have defined what we mean by a smooth function on a manifold M . Similarly we define what we mean by a smooth mapping F : M1 → M2 with M1 and M2 manifolds. Indeed, let M1 and M2 be manifolds of dimension n1 and n2 , respectively. Then for any p ∈ M1 there exist local coordinate systems (U, (ϕ1 , · · · , ϕn1 )) for p and (V, (ψ1 , · · · , ψn2 )) for F (p) ∈ M2 . We now require that the maps F̂ := ψ ◦ F ◦ ϕ−1 : ϕ(U ) ⊂ Rn1 → ψ(V ) ⊂ Rn2 where ϕ = (ϕ1 , · · · , ϕn1 )T , ψ = (ψ1 , · · · , ψn2 )T , are smooth maps. DISC Systems and Control Theory of Nonlinear Systems 16 F̂ is nothing else than the local coordinate expression of the map F : M → N . Similarly we may rephrase the definition of a smooth function h : M → R by requiring that the functions ĥ := h ◦ ϕ−1 : ϕ(U ) ⊂ Rn → R are smooth, where (U, (ϕ1 , · · · , ϕn )) is a local coordinate system for M. Let now F : M → N be a smooth map. Define a linear map (called the tangent map of F at p ∈ M ) F∗p : Tp M → TF (p) N as follows. Let Xp ∈ Tp M . For any f ∈ C ∞ (F (p)) set F∗p Xp (f ) = Xp (f ◦ F ) where Xp ∈ Tp M is identified with the corresponding derivation at p ∈ M. DISC Systems and Control Theory of Nonlinear Systems 17 Definition 7 A (smooth) vector field X on a manifold M is defined as a smooth mapping X : M −→ T M satisfying π(X(p)) = p, ∀p ∈ M , where π : T M → M is the canonical projection mapping (p, Xp ) ∈ T M to p ∈ M . Thus a vector field X on M assigns to every point p ∈ M an element of Tp M : X(p) ∈ Tp M Let now (U, ϕ1 , · · · , ϕn ) = (U, x1 , · · · , xn )be a coordinate system for ∂ ∂ M . For every p ∈ U this yields a basis ∂x , · · · , for Tp M . p ∂xn 1 p DISC Systems and Control Theory of Nonlinear Systems 18 It thus follows that locally on U the vector field X can be expressed by a column-vector X1 (x1 , · · · , xn ) .. X(x) = . Xn (x1 , · · · , xn ) It follows that in local coordinates x1 , · · · , xn a vector field X corresponds to the n-dimensional set of first-order differential equations ẋ1 = X1 (x1 , · · · , xn ) .. . ẋn = Xn (x1 , · · · , xn ) DISC Systems and Control Theory of Nonlinear Systems 19 This implies that a vector field f transforms in a special way under any coordinate transformation z = S(x). Indeed, if x satisfies the differential equation ẋ = f (x) then z = S(x) should satisfy ż = where S. ∂S ∂x (x) ∂S −1 (S (z))f (S −1 (z)) ∂x denotes the Jacobian of the coordinate transformation It follows that f (x) transforms under z = S(x) to −1 f˜(z) := ∂S (z))f (S −1 (z)). Here, f˜ denotes the same vector field, ∂x (S but now expressed in the new coordinates. As an example, any linear set of differential equations ẋ = Ax transforms under a linear coordinate transformation z = Sx (with S an invertible matrix) to ż = SAS −1 z DISC Systems and Control Theory of Nonlinear Systems 20 Using this machinery we are now able to give a coordinate-free definition of a nonlinear state space system ẋ = f (x) + g(x)u y = h(x) , u ∈ Rm , x ∈ X, , y ∈ Rp , living on a state space X that is a manifold. Indeed, f (x) is the local coordinate expression of a vector field on X (called the drift vector field), and also the columns of g(x) are local coordinate expressions of vector fields on X (the input-vector fields), while h is a smooth mapping from X to Rp . DISC Systems and Control Theory of Nonlinear Systems Lie brackets of vector fields For X and Y any two vectorfields on M we define the Lie bracket of X and Y , denoted [X, Y ], by setting [X, Y ]p (f ) = Xp (Y (f )) − Yp (X(f )) for every function f : M → R. It can be checked that [X, Y ]p 21 DISC Systems and Control Theory of Nonlinear Systems 22 belongs to the space of derivations at p. Indeed [X, Y ]p (f g) = Xp (Y (f g)) − Yp (X(f g)) = = Xp {Y (f ) · g + f · Y (g)} − Yp {X(f ) · g + f · X(g)} = = Xp [Y (f )]g(p) + Yp (f )Xp (g) + Xp (f )Yp (g) + f (p)Xp (Y (g)) − Yp (X(f ))g(p) − Xp (f )Yp (g) − Yp (f )Xp (g) − f (p)Yp (X(g)) = [X, Y ]p (f ) · g(p) + f (p) · [X, Y ]p (g) Thus [X, Y ]p can be uniquely identified with an element in the tangent space Tp M , and [X, Y ] defines a new vectorfield on M . (1) DISC Systems and Control Theory of Nonlinear Systems In local coordinates the Lie bracket takes the following form: Proposition 8 Let X and Y be vectorfields on M , given in local coordinates (x1 , · · · , xn ) as X(x) = (X1 (x), · · · , Xn (x))T and Y (x) = (Y1 (x), · · · , Yn (x))T . Then the local coordinate expression of [X, Y ] is given as ∂Y ∂X [X, Y ](x) = (x)X(x) − (x)Y (x) ∂x ∂x with ∂Y ∂x , ∂X ∂x denoting the Jacobian matrices. 23 DISC Systems and Control Theory of Nonlinear Systems 24 Proof Compute for any j = 1, · · · , n [X, Y ]p (xj ) = Xp (Y (xj )) − Yp (X(xj )) = = Xp (Yj ) − Yp (Xj ) = n h P ∂Yj i=1 ∂xi Xi − ∂Xj ∂xi Yi i (x(p)) Since [X, Y ]p (xj ) is the j -th component of [X, Y ]p in these coordinates the result follows. DISC Systems and Control Theory of Nonlinear Systems 25 It readily follows that the Lie bracket satisfies the following properties (a) [f X, gY ] = f g[X, Y ] + f · LX g · Y − g · LY f · X (b) [X, Y ] = −[Y, X] (c) [X, Y1 + Y2 ] = [X, Y1 ] + [X, Y2 ] Furthermore, the following property can be checked [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0 f, g ∈ C ∞ (M ) DISC Systems and Control Theory of Nonlinear Systems Towards nonlinear controllability Consider the unicycle example 0 cos x3 ẋ = sin x3 u1 + 0 u2 0 1 The Lie bracket of the two input vector fields is given as 0 0 − sin x3 0 sin x3 − 0 0 cos x3 0 = − cos x3 0 0 0 1 0 which is a vector field that is independent from the two input vector fields. Claim: This new independent direction guarantees controllability of the unicycle system. 26 DISC Systems and Control Theory of Nonlinear Systems 27 Interpretation of the Lie bracket Proposition 9 Let X, Y be two vector fields such that [X, Y ] = 0 Then the solution flows of the vector fields are commuting. In fact, we may find local coordinates x1 , . . . , xn such that ∂ X= , ∂x1 ∂ Y = ∂x2 Thus, the Lie bracket [X, Y ] characterizes the amount of non-commutativity of the vector fields X, Y .