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  • Nonlinear Control Systems 3. - Ordinary Differential Equations

    Dept. of Electrical Engineering

    Department of Electrical Engineering University of Notre Dame, USA

    EE67598-02

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 1 / 33

  • Dynamical Systems

    Consider a continuous-time dynamical system (X ,R, φ) where φ : R× X → X is a continuous map such that φ(0, p) = p and φ(s + t, p) = φ(s, φ(t, p)).

    Define the partial map Φt : X → X called the system’s flow where Φt(p) = φ(t; p).

    Define the partial map ξp : R→ X called the system’s trajectory from p that takes values ξp(t) = φ(t; p) for any p ∈ X and t ∈ R.

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 2 / 33

  • Uniqueness of Trajectories

    Consider the dynamical system (X ,R, φ) and assume there are two states q1, q2 ∈ X and p ∈ X such that

    φ(t, p) = q1 6= q2 = φ(t, p)

    Since we can go forward or backward in time, then

    φ(−t, q1) = p = φ(−t, q2)

    So consider a trajectory where we go from q1 to p to q2. This would mean by the group property of φ that

    q2 = φ(t, φ(−t, q1)) = φ(t − t, q1) = φ(0, q1) = q1

    This contradicts the assumption that q1 6= q2 which implies the uniqueness of ξp.

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 3 / 33

  • Orbits

    For any p ∈ X , we define the set

    Ωp = {y ∈ X : y = ξp(t) for any t ∈ R}

    is called the orbit of p.

    We introduce a binary relation ∼ such that for any p, q ∈ X that p ∼ q if and only if q ∈ Ωp. Clearly ∼ is reflexive, symmetric, and transitive, so that ∼ is an equivalence relation.

    The equivalence classes of ∼ are the orbits of the system

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 4 / 33

  • Smooth Dynamical Systems

    Consider the map f : X → Y from X ⊂ Rn into Y ⊂ Rm. This map f is (C k) differentiable if each of its component functions fi : X → R is C k for i = 1, 2 . . . , n.

    The map f : X → Y is a diffeomorphism if both f and f −1 : Y → X are differentiable mappings.

    The dynamical system (X ,R, φ) is (C k) smooth if φ is a (C k) differentiable mapping.

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 5 / 33

  • Smooth Dynamical Systems and IVP

    Given a smooth dynamical system φ, we define f : X → x of the flow Φt at a point p ∈ X as

    f (p) := d

    dt

    ∣∣∣∣ t=0

    Φt(p)

    We let x(t; x0) = ξx0(t) denote a trajectory of φ, then the components of f may be written as

    fi (x(t)) = dxi (t)

    dt := ẋi (t)

    where x(0) = x0 This means that the trajectory x(t; x0) satisfies the initial value problem (IVP)

    ẋ(t) = f (x(t)), x(0) = x0

    This shows that the trajectories of any smooth dynamical system may be implicitly represent as a solution to an ordinary differential equation. The converse, however, may not always be true.

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 6 / 33

  • IVP with no C 1 solution

    As an example of a differential equation for which a solution may not exist, let us consider an IVP of the following form,

    ẋ(t) =

    { 1 x(t) ≤ 0 −1 x(t) > 0 , x(0) = 0

    At the initial time, x(0) is zero and so ẋ(0) = 1. So an infinitesimal time after 0 we find x(�) > 0 which means that ẋ(�) = −1. This would immediately force x to go back to zero again, but as soon as it does, ẋ shifts back to being positive.

    In other words, this differential equation appears to force the system to chatter back and forth between being positive and zero; a solution that is definitely not smooth

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 7 / 33

  • ODE with finite escape

    consider the IVP

    ẋ(t) = −x2(t), x(0) = −1

    This is a reasonable ODE that may fit some physical process.

    This is a separable equation, so we can rewrite it as

    t = − ∫ x x0

    dx

    x2 =

    1

    x − 1

    x0 =

    1

    x + 1

    which implies that for t > 0 that

    x(t) = 1

    t − 1

    This trajectory only exists over the time interval [0, 1) and so it fails to generate a smooth dynamical system φ, since we define φ over all time.

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 8 / 33

  • IVP with nonunique solution

    The last example we’ll consider is the IVP,

    ẋ(t) = x1/3, x(0) = 0

    This IVP has two continuously differentiable solutions. The trivial

    trajectory x(t) = 0 satisfies the ODE and the function x(t) = (

    2t 3

    )3/2 also satisfies the ODE.

    This is problematic for us as well since we already know that smooth dynamical systems, φ, must generate unique trajectories by theorem ??.

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 9 / 33

  • Existence of Solutions to IVP

    Consider the initial value problem

    ẋ = f (t, x), x(t0) = x0 (1)

    where f ∈ C (U,Rn) and U is an open subset of Rn+1 with (t0, x0) ∈ U. We first note that integrating both sides of this equation with respect to t shows that the IVP is equivalent to the following integral equation

    x(t) = x0 +

    ∫ t 0 f (s, x(s))ds (2)

    Let x : R→ X be a C 1 solution to the IVP in the sense that it satisfies the above integral equation and note that

    x(h) = x0 + ẋ(0)h + o(h) = x0 + f (0, x0)h + o(h) := xh + o(h)

    where o(h)h → 0 as h→ 0. Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 10 / 33

  • Existence of IVP Solutions

    This suggests that an approximate solution to the IVP might be obtained by omitting the error term to construct the recursive procedure

    xh(tm+1) = xh(tm) + f (tm, xh(tm))h, tm = mh

    This procedure is known as Euler’s method.

    We expect xh(t) to converge to the actual trajectory as h ↓ 0.

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 11 / 33

  • Equicontinuous Functions an the Arzelá-Ascoli Theorem

    A set of functions {xm}m∈N is equicontinuous if for all � > 0 there is δ > 0 (independent of m) such that

    |t − s| ≤ δ ⇒ |xm(t)− xm(s)| ≤ �

    Key thing is independent of δ from m, so that equicontinuity may be viewed as the notion of uniform continuity for a family of functions (rather than a single one).

    The Arzelá-Ascoli theorem is the main tool we’ll need.

    Theorem 1

    (Arzelà-Ascoli) Suppose the set of functions {xm(t)}m∈N in C (I ,Rn) where I is a compact interval is equicontinuous. If the sequence {xm} is bounded, then there is a uniformly convergent subsequence.

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 12 / 33

  • Existence of IVP Solutions

    To prove the existence of IVP solutions, we consider a sequence {xh} of Euler solutions and show that this set is equicontinuous and bounded, thereby allowing us to establish the existence of a subsequential limit for {xhi}. We then show that this limit indeed satisfies the integral equation characterizing solutions to the IVP.

    Theorem 2

    (Peano) Suppose f is continuous on V = [t0, t0 + T ]× Nδ(x0) and denote the maximum of |f | on V as M. Then there exists at least one solution of the IVP for t ∈ [t0, t0 + T0] which remains in Nδ(x0) where T0 = min{T , δ/M}. The analogous assertion holds for the interval [t0 − T0, t0].

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 13 / 33

  • Proof of Peano Theorem

    First select δ,T > 0 so that

    V = [t0, t0 + T ]× Nδ(x0) is compact

    Define the family {xh} as

    xh(t) = x0 + m−1∑ j=0

    ∫ tj+1 tj

    f (tj ; xh(tj))χ(s)ds

    where {t0, t1, . . . , tm} is sequence of times such that tm = t and h = tj+1 − tj . Since f is continuous on a compact set, it attains its maximum

    M = max V |f (t, x)|

    and so f is uniformly bounded on V .

    Dept. of Electrical Engineering (ND) Nonlinear Control Systems 3. - Ordinary Differential Equations EE67598-02 14 / 33

  • Proof of Peano Theorem (2)

    From the definition of xh(t), we can show |xh(t)− xh(s)| ≤ M|t − s| and so {xh} is equicontinuous and bounded and there exists a subsequential limit x(t)

    What is that limit? Define ∆(h) as

    |f (t, y)− f (t, x)| ≤ ∆(h) when |y − x | < M and |s − t| < h

    Note that ∣∣∣∣xh(t)− x0 − ∫ t t0

    f (t0, xh)ds

    ∣∣∣∣ ≤ |t − t0|∆(h) and since ∆(h)→ 0 as h→ 0 This implies xh(t) is uniformly continuous and so

    x(t) = x0 + lim h→0

    ∫ t 0 f (s, xh(s))ds

    = x0 +

    ∫ t 0 f (s, x(s))