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Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 0 These course-notes are following notation of EDS U versus U Φ(T,p,u) p.84 reachable sets exponential notation for nonlinear time varying flows, too

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  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 0

    These course-notes are

    following notation of EDSU versus UΦ(T, p, u)p.84 reachable setsexponential notation for nonlinear time varying flows, too

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 31

    3 Controllability

    3.1 General remarks

    The fundamental concept is very simple: A system is called controllable from a state p to astate q if there exists an admissible control u that steers the system from initial state p at timet0 to the final state q = Φ(t0 + T, p, t0, u). A system is called (globally) controllable if for everypair (p, q) of states it is controllable from p to q. There are many specializations of this concept,ranging from finite state machines (or automata) to systems governed by partial differentialequations. [References: S. Eilenberg, Automata, languages, and machines, and various recentpapers by J.-M. Coron, on controllability of various PDEs.] Here we shall focus on systemswith continuous states in Rn or on finite dimensional manifolds Mn, governed by controlleddifferential and difference equations.

    Exercise 3.1 Give an example of a control system with a finite state space, and define suitablenotions of controllability for this kind of system. (E.g. consider networks of routes served by anairline). Discuss whether the controllability notions are symmetric with respect to time reversal.

    A basic technical notion is that of a reachable set. For sake of clarity we shall assume herethat the system is time-invariant, allowing us to generally assume initialization at time t0 = 0.Define the reachable set from the state p at time T as

    RT (p) = {Φ(T, p, u) : u ∈ U admissible control} (1)

    It is convenient to also introduce

    R≤T =T⋃

    t=0

    Rt(p), and R≤∞ =⋃t≥0

    Rt(p) (2)

    The statement q ∈ Rp(T ) may be read as “q is reachable from p”, or as “the system is controllableto q from p”. The system is reachable from p if the entire state-space X ⊆ R≤∞(p) can be reachedfrom p. Analogously the system is q-controllable (“controllable to q”) if q ∈

    ⋂p∈X R≤∞(p).

    For many systems (but generally not for discrete-time systems) the reversal of time correspondsto interchanging the roles of reachabilty and controllability, and one commonly uses the termcontrollability when one technically analyzes reachability. In the sequel we shall investigate whenthe reachable set RT (p) contains an equilibrium point or a reference trajectory in its interior,and still use the term controllability. For typical applications one usually studies the systemobtained by replacing t by −t.Multidinuous special versions of the basic concept of controllability have been introduced. Im-portant for us is that in the linear setting many of them coincide with each other, whereas innonlinear settings one needs to very carefully distinguish between global and local controllabil-ity, and controllability in arbitrarily large or bounded, or arbitrarily small times, etc. A positiveway to read this is that there are plenty of open research problems, each corresponding to adifferent meaningful setting in which to analyze controllability.A central objective of the analysis is to obtain readily checkable conditions in terms of thesystems data that allow one to algorithmically decide whether a system is controllable or not.Such conditions generally are only existence results (∃u s.t. Φ(T, p, u) = q). Constructingcontrols that actually steer the system from a given state p to a given state q is generally adifferent problem which we shall revisit when discussing feedback stabilization.

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 32

    For finite dimensional systems the classical result is the Kalman rank condition which assertsthat the time-invariant system ẋ = Ax+Bu, x ∈ Rn, ∀t, u(t) ∈ U = Rm is globally controllableif and only if the rank of the compound matrix [B,AB,A2B, . . . , An−1B] is n. Remarkably,the same condition holds for discrete time systems with the corresponding data. Note thatthis condition applies to systems that impose no a-priori bound on the magnitude |u(·)| of thecontrol. The next subsection will study the Kalman rank condition in more detail.

    Exercise 3.2 Give an example of matrices A,B defining a linear system that is controllable forthe set of unbounded controls (U = Rm), but is not controllable with bounded controls takingvalues in U = [−1, 1]m.

    Different technical results are available for time-varying linear systems in which the matricesA = A(t) and B = B(t) are not necessarily constant. The next popular class of systems hasa bilinear structure ẋ = Ax +

    ∑mj=1 ujBjx. While such systems at first sight seem to much

    resemble linear systems, in many ways they display characteristics more reminiscent of generalnonlinear systems. Indeed, many nonlinear systems can be lifted to bilinear systems (on a muchhigher dimensional state-space) – and therefore, in many ways, bilinear systems must be at leastas complex as many nonlinear systems. However, what makes some bilinear systems particularlyattractive to study is that their underlying state-space may be taken to have the structure of a(matrix-)Lie group, thus affording a particularly rich set of tools for studying them. In the lasttwo decades many new results for controllability on Lie groups have been found, and this areacontinues to see much ongoing research.In our course, we shall jump ahead and next concentrate on nonlinear systems that are affine inthe control

    ẋ = f0(x) +m∑

    j=1

    ujfj(x) (3)

    defined by a finite number of smooth vector fields f0 and fi (which often are required to besmooth C∞, or analytic Cω). These are a special case of fully nonlinear systems ẋ = f(x, u) forwhich much fewer general results are available. In recent years, much research has focused onhybrid systems which mix continuous and discrete states as they occur in systems with hysteresisor in switched systems. Again, this area provides many opportunities for new research projects,but goes far beyond our course.In nonlinear systems, it is common practice to focus on the case of bounded controls. We shallusually assume that for all i and for all t, |ui(t)| ≤ 1, or

    ∑mi=1 u

    2i ≤ 1. More generally one may

    allow the controls u to take values in a convex, compact subset U ⊆ Rm. There may be manypossible explanations for this distinction of working with bounded and unbounded controls whenstudying nonlinear and linear systems. Clearly, in practical applications there are almost alwaysbounds on the admissible controls. However, the mathematical treatment may impose differentconditions to be feasible. This distinction will become clearer later when we study optimalcontrol problems which are closely related to controllability: Different system structures willsuggest different objective functions which in turn determine the underlying sets of admissiblecontrols.Back to the affine system (3), the best studied notion is that of small-time local controllability(STLC) about an equilibrium point of the drift vector field f0 for the control set U = [−1, 1]m,with STLC defined as ∀T > 0, p ∈ intRT (p). Algebraic conditions analogous to the Kalmanrank condition, however, are known only for the weaker notion of accessibility defined as ∀T > 0,

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 33

    ∅ 6= intRT (p). For practical purposes, more interesting is the notion of STLC about a notnecessarily stationary reference trajectory Φ(·, p, u∗), defined via ∀T > 0,Φ(T, p, u∗) ∈ intRT (p).

    Exercise 3.3 Calculate the reachable sets RT (0) from p = 0 for the system ẋ = y + u, ẏ = x,subject to the constraint ∀t, u(t) ∈ U = [0, 1]. Discuss various notions of controllability for thissystem (incl. global, local, and small-time local controllability).

    3.2 Linear systems

    Consider the finite dimensional linear systems with no bounds on the control

    ẋ = Ax+Bu, x ∈ Rn, u ∈ Rm. (4)

    Theorem 3.1 (Kalman Rank Condition) System (4) is globally controllable if and only if

    rank [B,AB,A2B, . . . An−1B] = n. (5)

    The first key observation regarding the reachable sets of system (4) is that they are always affinesubspaces on Rn. Indeed, consider the variation of parameters formula for the solution curvesof system (4) for any control u(·).

    Φ(T, p, u) = etAp+∫ T

    0e(t−s)ABu(s)ds (6)

    It is immediate that for any constants c1, c2 and any controls u1, u2

    Φ(T, p, c1u1 + c2u2)− etAp = c1(Φ(T, p, u1)− etAp) + c2(Φ(T, p, u2)− etAp) (7)

    and thus RT (0) ⊆ Rn is a subspace, and RT (p) = etAp+RT (0) is a translate of a subspace. Dueto the additive structure, it suffices to consider the reachable sets RT (0) for systems initializedat Φ(0) = 0.

    Exercise 3.4 What can you say in the case of a bounded control set U = [1, 1]m?

    Exercise 3.5 What can you say about the structure of the reachable sets for linear discrete timesystems x(t+ 1) = Ax(t) +Bu(t)?

    Recall the definition of the matrix exponential as an absolutely convergent power series. Next,using the Cayley Hamilton theorem, there exist analytic functions ck : R 7→ R such that

    etA =∞∑

    k=0

    (tA)k

    k!=

    n−1∑k=0

    ck(t)(tA)k

    k!(8)

    Exercise 3.6 Prove this assertion. Calculate the ci for the case of A =

    −1 1 0 00 −1 0 00 0 0 −10 0 1 0

    .

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 34

    To prove the only-if part of theorem 3.1 assume to the contrary that rank[B,AB, . . . An−1B] < n.Thus there exists a nonzero vector ξ ∈ Rn such that ξT · [B,AB,A2B, . . . An−1B] = 0 andconsequently also ξT · etA = 0 for all t. This implies that

    ξT · Φ(T, 0, u) = ξT ·∫ T

    0e(t−s)ABu(s)ds =

    ∫ T0

    (ξT · e(t−s)A)Bu(s)ds =∫ T

    00 · u(s)ds = 0 (9)

    and consequently R(T, 0) ⊆ {ξ}⊥ = {x ∈ Rn : x ⊥ ξ} is contained in a proper subspace.To prove the if-part of the theorem, one might try to compute the coefficients in a Taylor expan-sion of Φ(T, 0, u) about t = 0. But any straightforward such attempt will encounter derivativesof the control u, which generally is not even assumed to be continuous. One alternative is toapproximate any control u by a sequence of smooth controls, and then use analytic tools. Thisis more typical in the case of nonlinear systems. Here we follow [EDS p.89] and take advantageof the linear structure:Suppose the system is not controllable. By above observations, the reachable set is contained insome proper subspace and hence there exists a nonzero vector ξ ∈ Rn such that ξT ·Φ(T, 0, u) = 0for all T and all u. Now analyze the trajectory corresponding to the control u∗(t) = BT e(T−t)A

    0 = ξT · Φ(T, 0, u∗) = ξT ·∫ T

    0e(t−s)AB ·BT e(T−s)AT ξ ds =

    ∫ T0‖BT e(T−s)AT ξ‖2 ds (10)

    and hence BT e(T−s)ATξ ≡ 0 for almost all s ∈ [0, T ]. Now – with no control variables appearing

    anymore – the Taylor expansion about s = T

    0 ≡ ξT e(T−s)AB = ξT − ξTAB · (s− T ) + 12!ξTA2B · (s− T )2 − 1

    3!ξTA2B · (s− T )2 . . . (11)

    immediately yields the desired algebraic condition that for all k ≥ 0, ξT · AkB =0, and hencethe compound matrix is rank deficient.

    Exercise 3.7 Explore whether some variation of the above argument may be applied to nonlinearsystems. Explain what needs to be modified, and how. Or explain which arguments have littlehope of generalization.

    3.3 Nonlinear systems affine in the control

    Among nonlinear systems we concentrate on systems that are affine in the control

    ẋ = f0(x) +m∑

    j=1

    ujfj(x), ∀t, x(t) ∈ Rn, |uj(t)| ≤ 1 (12)

    defined by a finite number of analytic vector fields f0 and fj . The controls are generally as-sumed to be measurable and bounded, although for many subsequent arguments it will sufficeto consider controls that are piecewise constant. Meticulous analysis of the relation of the reach-able sets RT,pc(p) and RT,meas(p) under various hypotheses on the smoothness of the data (e.g.Cr, C∞, Cω and control bounds) may be found in numerous articles, especially [Grasse]. Wewill revisit this question in the context of optimal controls later in the course.For piecewise constant controls u : [0, T ] 7→ [−1, 1]m defined by the data 0 ≤ t1 ≤ t2 ≤ . . . tr = Tand control values u(t) = uk ∈ [−1, 1]m for tk−1 ≤ t < tk we consider the endpoint map

    (t1, t2, . . . tr, u1, . . . , ur) ∼ u 7→ Φ(T, p, u) ∈ Rn (13)

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 35

    While we do not a-priori limit the number r of pieces (or switchings), effectively we will alwaysconsider only a finite number of switchings. In some sense the rationale is that we are lookingfor some sort of rank condition, and the range of the map is finite-dimensional. Hence, we hopefor some finite derivative information to be sufficient to make decisions about controllability.Indeed, the questions of whether small-time local controllability even for analytic systems isfinitely determined is still an open question, see [Agrachev, open problems book vol. I].Consequently, we consider this as a map from a subset of the finite dimensional space R2r (ratherthan the infinite dimensional space U = {u : [0, T ] 7→ [−1, 1]m measurable } of all admissiblecontrols. The strategy, or game plan, is to use derivatives of this restriction of the endpoint mapto this finite dimensional space to obtain some algebraic criteria similar to the Kalman rankcondition of the linear setting. The objective is that the criteria be readily checkable in termsof algebraic computations involving the data f0 and fj , in particular, not involving the solvingof any differential equations, or calculations explicitly involving any controls.

    3.3.1 One basic variation, differential of the flow, and Lie derivatives

    As a start consider the most simple variations of a reference control u∗ : [0, T ] 7→ U ⊆ R thatchanges the value of the control only on some small subinterval [t0, t0 + ε] ⊆ [0, T ] by theamount v. Assuming that u∗ ± v still lies in the set U of admissible values of the control,the resulting endpoints may be considered as a curve in the state-space parameterized by theduration ε of the variation, or by the size v of the variation, or both. Using exponential notationalso for the flows of time-varying nonlinear vector fields, this curve is

    γ(ε, v) = e(T−t0−ε)(f+u∗g)eε(f+u

    ∗g+vg)et0(f+u∗g)(p) (14)

    As either one (or both) go to zero, the endpoints approach the endpoint of the reference trajec-tory. Differentiating with respect to either ε or v and evaluating at ε = 0 or v = 0 one obtains avariational vector at the endpoint. This provides infinitesimal information about the location ofreachable set R(T ) relative to γ(T ). Note that in order to differentiate with respect to the sizeof the control variation one generally may want to assume that u∗ ∈ intU , so that u∗ ± v ∈ Ufor all sufficiently small values of v. On the other hand, one may consider a fixed u∗ + v ∈ Uand only vary the times ε, even when u∗ ∈ ∂U is a boundary value of U .For various computational and conceptual reasons, it turns out to be more convenient to pullback all such endpoints and variational vectors to the starting point – e.g. in view of desirablealgebraic conditions in terms of (derivatives) of the data at the starting point. Thus we considerthe curve (parameterized by either v or ε)

    γ0(ε, v) = e−T (f+u∗g)γ(ε, v) = e−(t0+ε)(f+u

    ∗g)eε(f+u∗g+vg)et0(f+u

    ∗g)(p) (15)

    Note that even though can only move forward, not backward along drift field, in some expres-sions one still sees e−tf . To calculate these derivatives is a matter of using the chain-rule, anddifferentiating flows with respect to the initial conditions.

    ∂∂ε

    ∣∣ε=0

    γ0(ε, v) = ∂∂ε∣∣ε=0

    e−(t0+ε)(f+u∗g)eε(f+u

    ∗g+vg)et0(f+u∗g)(p), or (16)

    ∂∂v

    ∣∣v=0

    γ0(ε, v) = ∂∂v∣∣v=0

    e−(t0+ε)(f+u∗g)eε(f+u

    ∗g+vg)et0(f+u∗g)(p) (17)

    The basic form of this expression, and a convenient and suggestive notation for the derivative is

    ∂∂ε

    ∣∣ε=0

    e−tXeεY etX =(e−tX

    )∗ Y e

    tX (18)

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 36

    where Ψ∗ denotes the tangent map or differential of a map Ψ. In coordinates, Ψ∗ = DΨ is theJacobian matrix of partial derivatives of Ψ. In our case, Ψt(p) = Φ(t, p) = etX(p) is the flowof a vector field X evaluated at time t with initial point p, i.e. the solution of the initial valueproblem ẋ = X(x), x(0) = p. In coordinates, the differential Φ∗ = (etX)∗ is the solution of thetime-varying linear differential equation Ṁ(t) = (DX)(Φ(t, p)) ·M(t) with M(0) = I ∈ Rn×n(the identity matrix).

    Exercise 3.8 Observe that Φ∗(t, (1, 0)) = (cos(t), sin(t)) is one solution the uncontrolled dy-namical system ẋ = −y, ẏ = x(2− x2 − y2) with initial condition x(0) = 1, y(0) = 0.Write down the variational equation Ṁ(t) = (DF )(Φ∗(t)) ·M(t) on R2×2 obtained by lineariz-ing the system of differential equations about Φ∗. Solve the variational equation for the initialcondition M(0) = Id2×2 and use this result to estimate Φ(t, (1.1, 0.1)) and Φ(t, (0.9, 0.1)) (youmay have to resort to numerical integration).

    Exercise 3.9 For a fixed control u∗ and fixed initial condition, write down the variational equa-tion for the linear system ẋ = Ax+Bu∗. Discuss the usefulness of the variational equation forlinear differential equations, and speculate about what this means for the relation to local andglobal controllability of a linear system?

    In terms of the differential of the flow we now may define the Lie derivative of a vector field.For comparison, we also define the Lie derivative of a smooth function in the same language:

    (Xϕ)(p) = (LXϕ)(p) = limh→0

    (ϕ(Φ(h, p))− ϕ(p)) (19)

    (LXg)(p) = limh→0

    (g(p)− Φ∗(h, p)(g(Φ(−h, p))) (20)

    Exercise 3.10 Give an example – e.g. of two vector fields f and g in the plane – to show thatthe Lie derivative Lfg in general is NOT equal to the column vector (Lfg1, Lfg2, . . . Lfgn)T . ofdirectional derivatives of the components of g = (g1, g2, . . . , gn).

    3.3.2 Lie bracket of two vector fields, limit of flows

    As the exercise in the previous section illustrated, in general one does not expect to be able to findclosed-form expressions for the flow of a nonlinear system of differential equations. It is even lessdesirable to explicitly compute and explicitly solve the variational equation along a trajectorythat one may not get one’s hands on. The objective instead is to use infinitesimal (derivative)information at the starting point only in order to decide properties such as controllability (andlater, observability and optimality).This is achieved by considering Lie derivatives of the data. We first give a geometric defintion,then a purely algebraic condition, discuss why they define the same object, and also give a briefhand-on coordinate formula for actual calculations.The introductory discussion in the previous subsection showed that it is natural to go somedistance along the flow of a reference control, then take a variation, and then go back along theoriginal flow. The result should be compared to taking a corresponding variation at the startingpoint. Instead of the sequence “(f,g,-f)”, and comparison to g, one might as well compare“(f,g,-f,-g)” to zero. Using this basic building block of four pieces we differentiate the curve ofendpoints of the concatenation of flows γ(t) = e−tg e−tf etg etf p ∈ Rn at t = 0 and define theLie bracket of two vector fields f and g at the point p to be (compare EDS p. 152)

    [f, g](p) = limt→0

    12t2

    (e−tg e−tf etg etf p− p

    )(21)

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 37

    Exercise 3.11 Justify why this limit exists. Suggestion: First argue why γ(0)=p and γ′(0)=0.

    Algebraically, we identify the vector field that maps the point p to column vector f(p) =(f1(p), f2(p), . . . fn(p))T with the first order partial differential operator

    f : p 7→ f1 ∂∂x1∣∣∣p+ f2 ∂∂x2

    ∣∣∣p+ . . .+ fn ∂∂xn

    ∣∣∣p

    (22)

    which now is interpreted as a function (operator) on the algebra C∞(Rn) of smooth functions.Clearly this identification is a linear one-to-one correspondence. A major advantage of thinkingof vector fields this way is that this notion makes complete sense on any smooth manifold,whereas column vectors and pictorial arrows are much more limited. Algebraically we maycharacterize a vector field as a derivation, that is a linear map f : C∞(Rn) 7→ C∞(Rn) that alsosatisfies the Leibniz (product) rule

    ∀ϕ,ψ ∈ C∞(Rn), f(ϕψ) = (fϕ)ψ + ϕ(fψ) (23)

    We say that a vector field f is smooth if for every smooth function ϕ ∈ C∞(Rn) the functionfϕ is smooth. It is not hard to show that this is equivalent to the, in local coordinates, thecomponent functions f i = fxi all being smooth.

    pages 37-40 in progress, numbering may change

    For two vector fields f, g : C∞(Rn) 7→ C∞(Rn) considered as first order differential operators,define their Lie bracket as the differential operator [f, g] : C∞(Rn) 7→ C∞(Rn) given for ϕ ∈C∞(Rn) by

    [f, g]ϕ = g(fϕ)− f(gϕ) (24)

    Exercise 3.12 Verify that unlike fg, the Lie bracket [f, g] is again a linear first order differentialoperator (or derivation): Check that for any two functions ϕ,ψ ∈ C∞(Rn) and c1, c2 ∈ R

    [f, g](c1ϕ+ c2ψ) = c1 · [f, g](ϕ) + c2 · [f, g](ψ), and (25)

    [f, g](ϕ · ψ)(p) = [f, g](ϕ)(p) · ψ(p) + ϕ(p) · [f, g](ψ)(p) (26)

    Exercise 3.13 Suppose that f and g are smooth vector fields. Show that f ◦ g : ϕ 7→ f(gϕ)) isgenerally not a vector field, but the commutator f ◦ g − g ◦ f : ϕ 7→ f(gϕ))− g(fϕ) always is asmooth vector field.

    In terms of this operator language, the preceding geometric definition of the Lie bracket of twovector fields becomes

    ∀ϕ ∈ C∞(Rn), [f, g](ϕ)(p) = limt→0

    12t2

    (ϕ(e−tg e−tf etg etf p)− ϕ(p)

    )(27)

    Exercise 3.14 Show that in this operator language, the Lie bracket of two smooth vector fields[f, g] as defined above is equal to the commutator g ◦ f − f ◦ g

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 38

    (XXXXXX check for possible sign error. Comment on BA-AB. XXXXXXX)

    Exercise 3.15 Show that the Lie bracket [f, g] of two smooth vector fields f and g equals theLie derivative Lfg of g in the direction f as defined earlier.

    The statement in the previous exercise is quite surprising as the definition of the Lie derivativeof a vector field in terms of the flow does not at all suggest that Lfg = −Lgf .

    Exercise 3.16 Verify that the Lie bracket satisfies the Jacobi identity.

    For all X,Y, Z ∈ Γ∞(Rn), [X, [Y, Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0 (28)

    Taken together, the anticommutativity and Jacobi identity, establishes that the set of all smoothvector fields Γ∞(Rn) forms a Lie algebra under the product (f, g) 7→ [f, g].

    limh→0

    1h (Φ(t, p+ hv)− Φ(t, p)) = Φ∗(Φ(t, p)) · v (29)

    ∂∂ε

    ∣∣ε=0

    e−tfeεgetf =(e−tf

    )∗g etf (30)

    commonly used notation

    adkf g = (adkf, g) = [f, [f, . . . [f︸ ︷︷ ︸

    k−times

    , g]. . .] (31)

    also Adset of all smooth vector fields Γ∞(Rn) : C∞(Rn) 7→ C∞(Rn)

    3.3.3 in coordinates, Jacobian matrices

    easy to calculate for any system

    Exercise 3.17 Suppose f0 and fj are smooth vector fields and f0(0) = 0. Let A be the Jacobianmatrix of partial derivatives of f0 at x = 0, and let B the matrix whose columns are the vectorsfj(0). Relate the matrix products AkB to the iterated Lie brackets of the vector fields f and fjevaluated at x = 0. Explain what corresponds in linear systems to iterated Lie brackets that haveseveral factors of controlled vector fields fj.

    Exercise 3.18 For the model of parallel parking a car (or rather a bicycle) defined by the vectorfields f0(x) = x2 tanx1 ∂∂x3 +x2 cosx3

    ∂∂x4

    +x2 sinx3 ∂∂x5 , and f1(x) =∂

    ∂x1, f2(x) = ∂∂x2 , calculate

    as many iterated Lie brackets as needed, and evaluate them at p = 0 ∈ R5 until this set of vectorsspans R5 (or technically, the tangent space T0R5).

    pages 37-40 in progress

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 41

    3.3.5 Involutive distributions and Frobenius theorem

    We saw that in the linear case the reachable set at any given time is always contained in anaffine (translated) subspace whose dimension is equal to the rank of the controllability matrix.The nonlinear smooth analogue is intimately related to the Frobenius theorem. This theoremhas applications in many different settings, and thus is presented in many different lookingforms. But at its core is the observation that mixed partial derivatives of a smooth function areequal. A most basic example (in informal notation) is the system of first order partial equationsux = y, uy = −x for a scalar function u = u(x, y). This does not have a solution since anysmooth solution u(x, y) must satisfy 1 = uxy

    != uyx = −1. In the language of differential forms,the same example reads: The differential form ω = dz − x dy + y dx is not closed (and hencenot exact) since dω = −2dy ∧ dx 6= 0. More suitable for our settings is the language of vectorfields (and control systems) in which this example may be written as f1(x, y, z) = ∂∂x − y

    ∂∂z , and

    f2(x, y, z) = ∂∂y +x∂∂z . Again, there does not exists a function ϕ : R

    3 7→ R that is constant alongtrajectories of the system ddt(x, y, z) = u1f1(x, y, z) + u2f2(x, y, z) because [f1, f2] = 2

    ∂∂z 6= 0.

    We start with some technical definitions.

    Definition 3.3 A distribution is a map ∆ which assigns to every point p ∈ Rn (or in a manifoldMn) a subspace ∆(p) of Rn (of the tangent space TpM of M at p).A distribution ∆ is regular if there exists an integer m such that for every p, dim ∆(p) = m.Conversely, the distribution ∆ is singular at the point p if in every neighborhood V of p there isa point q ∈ V such that dim ∆(q) 6= dim ∆(p)A distribution ∆ is smooth if for every point p there exists a neighborhood V of p and smoothvector fields f1, . . . fm defined on V such that for every q ∈ V , ∆(q) is the linear span of thevectors f1(q), . . . fm(q).

    In the case of a single (smooth) vector field f that does not vanish at a point p, f definesa one-dimensional distribution, or a line field in some open neighborhood V of p by ∆: q 7→{c1f1 : c1 ∈ R }. By the standard existence and uniqueness theorems for solutions of ordinaryequations, for every q ∈ V there exists a solution curve γ : (−ε, ε) 7→ Rn such that for everyt ∈ (−ε, ε), γ′(t) = f1(γ(t)) ∈ ∆(γ(t)).One might suspect that for two smooth vector fields f1 and f2 that are linearly independentin a neighborhood V of a point p, i.e. which define a two-dimensional smooth distribution∆: q 7→ {c1f1(q) + c2f2(q) : c1, c2 ∈ R } on V , it is possible to find a two-dimensional curve,i.e. a map γ : (−ε, ε)2 7→ Rn such that for every (s, t) ∈ (−ε, ε)2 both f1(γ(s, t)) and f2(γ(s, t))are tangent to the image of γ at the point γ(s, t). However, this expectation is completely false.One of the easiest counterexamples is the one discussed above where f1(x, y, z) = ∂∂x − y

    ∂∂z and

    f2(x, y, z) = ∂∂y + x∂∂z .

    Our main interest is in algebraic conditions that identify under exactly what circumstancesone can find such an integral surface. Colloquially speaking, for an m-dimensional smoothdistribution ∆, we define an integral surface through the point p to be a map (parameterizedsurface) Φ: (ε, ε)m 7→ Rn such that Φ(0) = p and such that for every t = (t1, . . . tm) ∈ (ε, ε)m,the image of the differential (DΦ)(t) is equal to ∆(Φ(t)). The analogous standard definition inthe language of manifolds is

    Definition 3.4 A submanifold manifold Sm ⊆ Mn is an integral manifold of a distribution ∆on Mn if at every point p ∈ S, ∆(p) = TpS ⊆ TpM .

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 42

    Definition 3.5 A smooth distribution ∆ is involutive if for every pair of vector fields f, g whichare such that for every q, f(q), g(q) ∈ ∆(q) also [f, g](q) ∈ ∆(q).

    Theorem 3.4 (Frobenius) Suppose that ∆ is a smooth m-dimensional regular distributiondefined on a neighborhood V of p. Then there exists an m-dimensional integral manifolds of ∆through every point q ∈ V if and only if ∆ is involutive on V .

    We sketch the key ideas of a standard proof – for the detailed geometric version on manifolds see[Spivak vol.I]. The only if part is a direct consequence of the equality of mixed partial derivativesof smooth functions. Thus suppose that ∆ is a smooth m-dimensional regular distributiondefined on a neighborhood V of p ∈ Rn. After possibly reordering the coordinates and shrinkingV , we may, without loss of generality, assume that there exist smooth vector fields f1, . . . fmdefined on V that span ∆ and which have the form fk = ∂∂xk +

    ∑nj=m+1 +βkj(x)

    ∂∂xj

    for suitablesmooth functions βkj defined on V .

    Exercise 3.27 Justify why it is possible to assume that ∆ is spanned by such fields near p.

    One immediately calculates that for any pair i, j ≤ m, the Lie bracket of the vector fields fi andfj is of the form

    [fi, fj ] =n∑

    `=m+1

    α`ij(x)∂

    ∂x`(32)

    for suitable smooth functions αij : V 7→ R. On the other hand, by the assumption of involutivity,there exist smooth functions c`ij : V 7→ R such that

    [fi, fj ] =m∑

    `=1

    c`ij(x)f` =m∑

    `=1

    c`ij(x)∂

    ∂x`+

    n∑j=m+1

    α̃`ij(x)∂

    ∂x`(33)

    for suitable smooth functions α̃ij : V 7→ R. Comparison of (32) with (33) establishes that for alli, j, ` ≤ m, c`ij ≡ 0. Consequently the vector fields fi, i ≤ m commute pairwise and an integralmanifold of ∆ may be locally parameterized by the map γ : (−ε, ε)m 7→ Rn (for sufficiently smallε > 0) defined by

    γ(t1, . . . tm) = etmfm ◦ . . . ◦ et2f2 ◦ et1f1(p) (34)

    For such map it is always true that

    ∂∂tn

    γ(t) = fm(etmfm ◦ . . . ◦ et2f2 ◦ et1f1(p)) = fm(γ(t)) (35)

    but in general

    ∂∂tjγ(t) =

    (etmfm

    )∗◦ . . . ◦

    (etj+1fj+1

    )∗

    (fj(etjfj ◦ . . . ◦ et2f2 ◦ et1f1(p))

    )6= fj(γ(t)) (36)

    for j < m. However, as argued above the vector fields fi and fj commute pairwise, and hence sodo their local flows. Consequently, the last inequality in (36) actually is an equality, and hencethe tangent space of the parameterized surface γ at γ(t) ∈ V is indeed equal to ∆(γ(t)), thusestablishing the theorem.

    Exercise 3.28 Consider the smooth distribution ∆: R3 7→ TR3 spanned by the two vector fieldsX = z ∂∂y − y

    ∂∂z and Y = y

    ∂∂z − z

    ∂∂y . Find the involutive closure of ∆, that is, the distribution of

    smallest dimension that contains ∆ and which is involutive. Suggestion: Calculate Z = [X,Y ]and analyze [X,Z] and [Y, Z]. Then find all integral manifolds of this involutive closure.

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 43

    3.3.6 Chow’s theorem and the Lie algebra rank condition

    The previous subsection established one direction of a nonlinear analogue of the Kalman rankcondition – that the reachable sets at any time are always contained in a subspace or submanifold(integral manifold) of dimension equal to the rank of some compound matrix. For the otherdirection, we very much relied on linearity, using a neat argument related to the controllabilityGramian. There is little hope for any nonlinear analogue for this. Still we expect that at anytime T > 0 the reachable sets to have relatively open interior in the integral manifolds describedin the previous section. This is indeed true, but the straightforward argument quickly becomesunappealingly messy.Intuitively, one expects that since (by definition!) one can approximate the flow et[f,g] of theLie bracket of two vector fields f and g by a concatenation of flows e±tf and e±tg in a suitableorder, one also can approximate the flows of higher order iterated Lie brackets of f and g by,albeit more complicated, concatenations of flows e±tf and e±tg. Assume that certain iterated Liebrackets fπj of the system vector fields span the tangent space at p, and hence in a neighborhoodof p. It is easy to show that then the map γ(−ε, ε)n 7→ Rn defined by

    γ(t1, . . . tm) = etmfπm ◦ . . . ◦ et2fπ2 ◦ et1fπ1 (p) (37)

    will be a diffeomorphism (and in particular, a bijection) onto its image for sufficiently smallε > 0. Combining this with some scheme of approximating the flows etjf

    πj by concatenationsof the flows of the original vector fields should yield the desired proof – but it is far from anappealing strategy.Our textbook [EDS] gives an elegant (but less constructive) argument, with key intermediateresults summarized in lemma 4.28. The first idea is to define k as the largest integer for whichthere exists some k-tuple of vector fields gi =

    ∑mj=1 uijfj (linear combinations of the systems

    vector fields, or elements of the distribution spanned by the fields fj) for which the (differentialof the) map

    σ(t1, . . . tk) = etkgk ◦ . . . ◦ et2g2 ◦ et1g1(p) (38)

    has maximal rank k. Arguing indirectly and using the involutivity one argues that each of thevector fields gi must be tangent to the image of σ. On the other hand, the tangent space of theimage is an involutive distribution that must contain all the vector fields fi, and hence mustalso contain the Lie algebra generated by the fi. Consequently, the (relative) interiors of thesets reachable by concatenating flows of vector fields in the original distribution, and reachableby concatenating flows of vector fields in its involutive closure agree. For details please see theoriginal argument in [EDS, pp.152-154].In conclusion we have a (first order) nonlinear analogue of the Kalman Rank Condition forcontrollability. Recall that the system (3) is called accessible from p if for all possible timesT > 0 the reachable set RT (p) has nonempty (n-dimensional) interior.

    Theorem 3.5 The system (3) with analytic vector fields f0, f1, . . . fm is accessible from p if andonly if the Lie algebra rank condition (LARC) dimL(F0, f1, . . . , fm)(p) = n is satisfied.

    There are many fine points about this theorem. Clearly the necessity is only true under theassumption of analytic vector fields. (Just recall the Taylor expansion of the analytic functionϕ : R 7→ R defined by ϕ(0) = 0 and ϕ(x) = exp(−x−2).) Moreover, it makes a difference whetherone assumes that the LARC is satisfied at one point only, or in an open neighborhood of the

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 44

    initial point. Theorems covering the different settings are known under the names of Chow’stheorem and the Nagano/Sussmann theorem. See the bibliography in EDS for details.

    In general, accessibility is weaker than even local controllability as is obvious in the simpleexample {

    ẋ1 = u x(0) = 0ẋ2 = xk1 |u(t)| ≤ 1

    (39)

    Here, f0(x) = xk1∂

    ∂x2, f1 ≡ ∂∂x1 , and up to the obvious relations in any Lie algebra, the only

    iterated Lie bracket that does not vanish at x(0) = p = 0 is (adkf1, f0) ≡ (−)k k! ∂∂x2 . Thus, bythe LARC, for any integer k ≥ 1, this system is accessible from x(0) = 0. It is also clear that ifk is an even integer, then for any control u and any T one always has that x2(Φ(T, u, (0, 0))) ≥ 0and hence x(0) 6∈ intRT (p) for any T ≥ 0, i.e. the system is not locally controllable fromx(0) = 0.On the other hand, if k is an odd integer, then one readily sees that for all T ≥ 0, for alladmissible controls u (and p = x(0) = (0, 0))

    Φ(T, p,−u) = −Φ(T, p, u) (40)

    In this case, a simple calculation shows that for k an odd integer the system is indeed small-timelocally controllable (STLC). Yet the observation made here is leads much further. The basicidea is that if a system is accessible, and for every control u : [0, t] 7→ U there exists a controlu− : [0, t] 7→ U such that Φ(T, p, u−) = −Φ(T, p, u) then the system is STLC. In particular,if the system (3) has no drift term, i.e. f0 ≡ 0 and the set U of admissible control valuesis symmetric about the origin (i.e. if u ∈ U then also −u ∈ U), then accessibility impliessmall-time local controllability. The key technical arguments rely on the convexity of high-orderapproximating cones (of high-order variational vectors) of the reachable set / endpoint map.This class may address some of these issues later in the context of optimal control.

    We just mention one positive result which is very useful in practice, and which seems to bevery suggestive. To facilitate the statement define the following subspaces of the Lie algebragenerated by the system vector fields. S1 = spanR{(adkf0, f1) : k ∈ Z+0 }, and inductively fork ≥ 1, Sk+1 = [S1,Sk] = spanR{[v, w] : v ∈ S1, w ∈ Sk }. With this terminology, we state thefollowing sufficient condition for STLC of analytic single-input systems (m = 1)

    Theorem 3.6 (Hermes condition) An analytic single-input systems of form (3) is STLCabout p ∈ Rn if it is accessible from p ∈ Rn and if for every k ≥ 0, S2k(p) ⊆ S2k−1(p).

    For systems in the plane, i.e. n = 2, this theorem was first proved by Hermes in the early1980s, and by Sussmann for general n ≥ 2 a few years later. Several more general conditions(both necessary ones, and sufficient ones) in a similar spirit have been established (includinggeneralizations of the Hermes condition to more than one input). However, conditions for STLCwhich are both necessary and sufficient, and at the same time are as algorithmic, algebraicallycomputable have been elusive. Indeed, a modern formulation of the big question is whether(small-time) local controllability is finitely determined [Agrachev, open problems book, vol.I].Roughly speaking, suppose that one establishes accessibility of a system of form (3) on an n-dimensional manifold by computing iterated Lie brackets of the vector fields fi of length at mostN . The question is, whether there exist a global bound M , depending on n and N , such thatSTLC is determined by the values of all iterated Lie brackets of the vector fields fi of length atmost M .

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 45

    This questions naturally leads into the next topic, nilpotent approximating systems, which insome sense are systems which preserve iterated Lie brackets (at a point) up to a certain order,and typically set all higher order terms equal to zero.

    3.4 Nilpotent approximating systems

    It is routine to calculate solutions of linear dynamical systems using simple explicit formulas.On the other hand, even if only some amazingly minor nonlinear terms appear in a system ofdifferential equations, there is generally no hope at all for any kind of a closed-form, explicitsolution formula. This is nicely illustrated in the simplicity of the classical Lorenz system whichexhibits chaotic dynamics.

    ẋ = −10x+ 10yẏ = 28x− y − xzż = −83z + xy

    (41)

    Thus it is natural to search for approximating systems which reasonably well locally approximatethe system dynamics and preserve key properties of interest, while permitting one to get somesort of solutions without excessive efforts.Common basic choices in analysis are linearizations, or quadratic or higher order Taylor approx-imations of functions. In dynamical systems the first basic choices are local linearizations aboutfixed point or about a reference trajectory (as in the discussions preceding the introduction ofthe Lie derivative). In the case of affine control systems of the form ẋ = f0(x) +

    ∑mi=1 uifi(x)

    with the origin an equilibrium point of the drift, i. e. f0(0) = 0, a natural choice is the Jacobianlinearization (in column vector notation)

    ẋ = Ax+Bu where A = ∂f∂x (0), and B = [f1(0), . . . fm(0)] (42)

    Exercise 3.29 Calculate the Jacobian linearization of the parallel parking example (43), andshow that this linearization is not controllable.

    φ̇ = u1v̇ = u2θ̇ = v tanφẋ = v cos θẏ = v sin θ

    or f0(z) =

    00

    z2 tan z1z2 cos z3z2 sin z3

    , f1(z) =

    10000

    , f2(z) =

    01000

    (43)This example nicely illustrates that linearizations may simply be too crude, even locally, as theymay preserve too few properties of interest. The next most simple approximation which comesto mind is to use bilinear systems. These are most useful when working in a neighborhood ofa common equilibrium point of the drift f and of the controlled vector fields fj . There existssome nice literature available about this approach, which closely interfaces with systems on Liegroups. We shall not further pursue this approach here. The next step after this might bepolynomial cascade systems of the form:

    ẋ1 = uẋ2 = p2(x1)ẋ3 = p3(x1, x2)

    ...ẋs = ps(x1, x2, . . . xs−1)

    with u, x1 ∈ Rn1 , x2 ∈ Rn2 , . . . xs ∈ Rns . (44)

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 46

    Systems of this form are attractive as for any given control u(t), i.e. a function of time, everyresulting trajectory Φ(t, x0, u) can be computed by successive quadratures only – that is, by inte-grations of functions of time – integrating ẏ = f(t), no need to solve any (nonlinear) differentialequations ẏ = ft(y). Clearly there are functions of time which may not have nice formulas fortheir antiderivatives – but this is still much simpler than integrating general nonlinear differentialequations such as (41).Thus polynomial cascade systems satisfy at least one of the objectives – being amenable toanalysis and to obtaining formulas for solution curves. In the sequel we will argue that theyalso preserve properties of interest when employed as suitably constructed approximating sys-tems. They “often” preserve (small-time local) controllability. They preserve accessibility, andoften have the same optimal controls. Moreover, stabilizing feedback laws of the approximatingsystems (when properly constructed) will also locally stabilize the original system.However, from a geometric perspective this characterization (44) is unappealing as it dependson the choice of coordinates – even a simple linear coordinate change such as a rotation maycompletely mess up this structure. On the other end, it might be possible to bring a complicated-looking system into such nice form by a simple change of coordinates. A beautiful geometriccriterion that characterizes systems of this form, up to a change of coordinates are the following(the definition of nilpotent comes after the theorem).

    Theorem 3.7 If a control system ẋ = f0(x) +∑m

    i=1 uifi(x) is in polynomial cascade form, thethe Lie algebra L(f0, f1, . . . fm) is nilpotent.

    Theorem 3.8 (MK 1986) If a control system ẋ = f0(x) +∑m

    i=1 uifi(x) with analytic vectorfields f0 and fi is such that the Lie algebra L(f0, f1, . . . fm) is nilpotent, and it satisfies the Liealgebra rank condition (3.5) for accessibility at a point p, then there exist local coordinates aboutp such that in these coordinates the system takes the form (44).

    One actually can make stronger statements about the degrees of the polynomials in terms of thedimensions of certain subalgebras of L(f, f1, . . . fm).The proof of the first is an immediate consequence of some of the later discussions in this section.The proof of the second theorem is a fairly straightforward construction that obtains the newcoordinates in terms of solutions of n differential equations. In some sense the argument is similarto proving that every analytic system of differential equations – including chaotic systems suchas (41) – near every nonsingular point is equivalent – upon choosing suitable coordinates – to thesystem ẋ1 = 1, ẋ2 = 0, . . . ẋn = 0. Depending on one’s point of view, this may be consideredan explicit construction, or useless in practice.Nilpotency in the Lie algebra setting is closely related to the corresponding notions in linearalgebra and groups. Recall that a matrix A is nilpotent if there exists an integer N such thatAN = 0. In the Lie algebra setting we define two sequences of subalgebras

    L(1) = L, ∀k ≥ 1, L(k+1) = [L,L(k)] = spanR{[v, w] : v ∈ L, w ∈ L(k)} (45)

    L〈0〉 = L, ∀k ≥ 0, L〈k+1〉 = [L〈k〉, L〈k〉] = spanR{[v, w] : v, w ∈ L〈k〉} (46)The sequence {L(k)}∞k=1 is called the central descending series of L, and the sequence {L〈k〉}∞k=1is called the derived series of L. A Lie algebra L is called nilpotent, or respectively solvable, ifthere exists an integer N such that L(N) = {0}, or L〈N〉 = {0}, respectively.Exercise 3.30 Show that every nilpotent Lie algebra is solvable, and that if L is a solvable Liealgebra, then the first derived ideal L〈1〉 is a nilpotent Lie algebra.

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 47

    In the following we shall exhibit an algorithmic procedure on how to construct nilpotent approx-imating systems. This procedure was independently presented by Hermes and Belläıche in themid 1980s and early 1990s, but also known and used by Stefani, Bressan, Sussmann and others.A nice summary appeared in the 1991 SIAM Review [Hermes].Suppose we are given analytic vector fields f0 and fi on Rn which satisfy the Lie algebra rankcondition (3.5) at the point p ∈ Rn, i. e. dimL(f0, f1, . . . fm)(p) = n. We shall construct a newset of local coordinates about p (that does not require any solution of differential equations),and new vector fields f̃0, f̃1, . . . f̃m such that in the new coordinates p is the origin and the newsystem is of polynomial cascade form (44).

    We first need to make a short technical digression related to notion such as the number of factorsin a product or the first factor of a product. For illustration consider that 3·4 = 12 = 2·6 = 2·3·2.Since all of these are equal, it does not make sense to talk about either the number of factorsin 3 · 4 or the first factor of 3 · 4. On the other hand, one may consider formal products, that isfinite sequences such as (3, 4), (2, 6), or (2, 3, 2). For each of these there is a well defined lengthof the sequence and a well defined first term etc. In the Lie algebra setting, consider the vectorfields f(x, y) = ∂∂x and g(x, y) = e

    x ∂∂x . Then [f, g] = g = [f, [f, g]] = [f, [f, [f, g]]] and thus it

    does not make sense to talk about the number of factors in the iterated Lie bracket [f, [f, g]].Instead we shall start with the set of binary labelled trees or the set parenthesized words (alsocalled the free magma in [Bourbaki] generated by a set of pairwise distinct indeterminatesX0, X1, . . .m . For each formal expression such as (X0, (X1, X0)), (X0, X1)) we do have a welldefined length, and a well-defines left (first) factor etc.For any set of analytic vector fields f0, f1, . . . fm there exists a natural evaluation map ψ whichsends each formal bracket (parenthesized word or labelled tree) Xπ into the corresponding iteratedLie bracket fπ, by substituting fj for each Xj and mapping parenthesized pairs to Lie bracketsof vector fields. All of this can me made utterly precise, but that is not the thrust of our studies.

    The first choice in the construction is to select weights w(X0) = θ ∈ [0, 1] and, for simplicity herefor all i ≥ 1, w(Xi) = 1. Different choices of weights may yield different approximating systemswhich may suit different purposes. In many applications one considers only weights θ ∈ {0, 1}.However, the endpoint value θ = 0 may not yield nilpotent, only solvable approximating systems.Alternatively, one may consider arbitrarily small positive weights θ > 0 that yield almost thesame approximating systems, but with nilpotent Lie algebra. Extend the weight assignmentto all formal brackets by recursively defining w((Xσ, Xπ) = w(Xσ) + w(Xπ). For example, ifθ = 12 , then w((X1, X0), ((X1, X0), X1))) = 4.Since the system satisfies the Lie algebra rank condition(3.5), there exist formal brackets Xπ1 ,Xπ2 , . . . Xπn such that fπ1(p), fπ2(p), . . . fπn(p) such that are linearly independent. Among allpossible such formal brackets, choose those such that the sequence of weights

    r = (r1, r2, . . . , rn) = (w(Xπ1), w(Xπ2), . . . w(Xπn)) (47)

    is the smallest possible in lexicographical order. This step amounts to no more than calcu-lating Lie brackets of vector fields, i.e. in coordinates computing Jacobian matrices of partialderivatives, matrix multiplication, and function evaluation.Next perform a constant linear (affine) coordinate change such that x(p) = 0 and in the newcoordinates for 1 ≤ i ≤ n, fπi(0) = ∂∂xi

    ∣∣∣p. This step only involves finding the inverse of a

    constant nonsingular matrix and some matrix multiplication.

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 48

    To facilitate the arguments introduce the notion of a group of dilations, that is a parameterizedfamily of maps ∆: R+ × Rn 7→ Rn defined by

    ∆ε(x) = ∆ε(x1, x2, . . . xn) = (εr1x1, εr2x2, . . . εrnxn, ) (48)

    where ri ∈ R+ are fixed exponents. In many applications one considers ri ∈ Z+ and customarilylabels that the coordinates and exponents such that 1 = r1 ≤ r2 ≤ . . . ≤ rn, but we shall allowri ∈ Q+ in the following. In the construction of the approximating system these exponents arethe weights ri = w(Xπi). For a coordinate-free geometric version see [Kawski 1995].A family of dilation induces graded structures on the vector spaces of polynomials and polynomialvector fields on Rn. In the spaces P of all polynomial functions on Rn and V of all polynomialvector fields on Rn define for m, k ∈ R define the subspaces of ∆-homogeneous polynomialfunctions and polynomial vector fields by

    Hm = {p ∈ P : ∀ε > 0, p ◦∆ε = εm · p }nk = {f ∈ V : ∀m ∈ Q, ∀p ∈ Hm, fp ∈ Hm+k }

    (49)

    It is easy to see that e.g. the coordinate functions xi and partial derivatives ∂∂xi are homogeneousof degrees ri and −ri, respectively.

    Lemma 3.9 Let ∆ be a family of dilations as above with exponents 1 = r1 ≤ . . . ≤ rn. Then apolynomial vector field f = p1 ∂∂x1 + . . . pn

    ∂∂xn

    ∈ nm if and only if for all 1 ≤ i ≤ n, pi ∈ Hm+ri.Moreover, for all k < −rn, nk = {0}, and for all m, k ∈ Q,

    • If m 6= k then Hm ∩Hk = {0} and nm ∩ nk = {0}• Hm ·Hk ⊆ Hm+k• nm · nk ⊆ nm+k• Hm · nk ⊆ nm+k

    Exercise 3.31 Show that each Hm and each nk is indeed a subspace of P or V, respectively.Prove the statements of lemma 3.9

    Returning to the construction of the nilpotent approximating system, expand each componentfij(x) = (fi(x))(xj) = 〈dxj , fi〉 (of the original vector fields in the new adapted coordinates) ina Taylor series about x = 0. Truncate the series at terms of order at most rj for f0 and at termsof order at most rj − 1 for fi, i ≥ 1. Denote the resulting vector fields by f̂0, f̂1, . . . f̂m.

    Exercise 3.32 Show that these vector fields f̂0, f̂1, . . . f̂m generate a nilpotent Lie algebra, andthat they approximate the original systems in the sense that they preserve accessibility.

    In general, the resulting vector fields f̂i are not homogeneous with respect to the dilation ∆,but they already have a desirable cascade polynomial structure. In general one has:

    f̃0 ∈⊕

    k≤−θnk and for i ≥ 1, f̃1 ∈

    ⊕k≤−1

    nk. (50)

    Technically, similar to the graded structures of P and V defined by Hm and nk, one considersthe filtrations of P and V defined by the subspaces Pm =

    ⊕k

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 49

    then Pm∩Pk = Pk and Pm∪Pk = Pm, etc. However, for later arguments, especially for feedbackstabilization it is convenient to work with truly homogeneous vector fields, and it is well worth tomake the extra effort to fully adapt the coordinates. This may be achieved via a simple furthercoordinate change of triangular polynomial form which has a triangular polynomial inverse. Thisstep is completely algorithmic, involving only differentiation, function evaluation at a point andmatrix multiplication. The final result will be homogeneous vector fields f̃j that do not exhibitgeometrically irrelevant ghost terms. Not only are the resulting formulas simpler and nicer, thisform also opens the door to using powerful results that relate the stability of systems nonlinearsystems to that of homogeneous approximating systems.

    For illustration, first consider the “desired” homogeneous system and two triangular polynomialcoordinate changes y = Φ(z) and x = Ψ(y)

    ż1 = = uż1 = = z1ż1 = = z101ż1 = = z1001

    y1 = z1y2 = z2y3 = z3y4 = z4 + z53

    and

    x1 = y1x2 = y2x3 = y3 + y32x4 = y4 + y42

    (51)

    Exercise 3.33 Transform the given clean system in the z-coordinates to the equivalent systemin the mildly mixed-up x-coordinates. Explicitly write out the differentials Φ∗ and Ψ∗, and theinverses z = Φ−1(y) and y = Φ−1(x). Demonstrate that when evaluating the iterated Lie bracketsof the system vector fields at the origin, the added off-diagonal terms do not show up at all.

    It turns out to be relatively straightforward to eliminate such ghost-terms in the vector fieldswhich have no geometric role for controllability etc, yet they potentially destroy the homogeneityproperties of the vector fields, thereby preventing one from a battery of nice theorems for e.g.stability and stabilization.

    Exercise 3.34 (Continuation of exercise 3.33) Starting with the system in the x-coordinates,reconstruct the coordinate changes by successively performing transformations of triangular formx(k+1) = Φ(k)(x(k)) that may also be defined geometrically by expressions of the form:

    x(i+1)k = x

    (i)k −

    ∑j

    (x

    (i)i

    )jj!

    · Ljfπi∣∣∣x=0

    (x(i)k ) xxx Check this again xxx (52)

    where the sum ranges over all j such that jri < rk−1 or jri < rk−θ. Use the illustrating exampleabove to decide in which order to make the corrections (i.e. for i increasing or decreasing).Demonstrate that in the new “adapted” coordinates the truncated vector fields are homogeneousw.r.t. ∆.

    We summarize the result in the following theorem, where we again use x = (x1, . . . , xn) to nowdenote the final adapted coordinates obtained after at most n successive steps as outlined inexercise 3.34, and we denote the resulting vector fields in the new coordinates by f̃0, f̃1, . . . f̃m.

    Theorem 3.10 The vector fields f̃0, f̃1, . . . f̃m constructed above are ∆-homogeneous vectorfields (with the exponents ri and adapted coordinates) such that f̃0 ∈ n−θ and for i ≥ 1, f̃1 ∈ n−1.

    Theorem 3.11 If θ > 0 then the Lie algebra L(f̃0, f̃1, . . . f̃m) by the newly constructed vectorfields is nilpotent. If θ = 0, the the Lie algebra L(S1(f̃0, f̃1, . . . f̃m)) is nilpotent where

    S1(f̃0, f̃1, . . . f̃m)) = spanR {(adj f̃0, f̃k) : j ≥ 0, and 1 ≤ k ≤ n}. (53)

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 50

    Exercise 3.35 Prove theorem 3.11.

    Exercise 3.36 Apply the algorithm described above to construct nilpotent approximating sys-tems for different choices of θ, e.g. θ ∈ {0, , 12 , 1} for the model (43) for parallel parking a car.Verify that the approximating systems do indeed still satisfy the Lie algebra rank condition foraccessibility.

    Exercise 3.37 Show that if one chooses the weight θ = w(X0) = 0, then the approximatingvector fields f̃0, . . . f̃m satisfy the Hermes condition [?] for small-time local controllability if andonly if the original vector fields f0, . . . fm do.

    4 Feedback and stabilization

    4.1 Feedback and equivalence: General remarks

    Stenographic summary.Want to characterize when two systems are basically the same. Think broadly and compare withour areas of mathematics. Here, typical candidates are: Change of time-scale (constant rescalingt′ = ct, or any order preserving smooth reparameterization t′ = ϕ(t). Change of coordinates ofstate space x′ = Φ(x) – often want to preserve (or improve) general structure such as linearity.State-dependent change of coordinates in the input space u = α(x) + β(x)u′ (usually written inthis reverse order). In more abstract terms, one considers one-to-one correspondences betweenthe sets of all possible trajectories of two systems, preserving time parameterization or allowingfor changes of the time parameterization. There are lots of different special versions of thisendeavor, with many details to be carefully analyzed, e.g. the change of the set U of admissiblecontrols, smoothness, local versus global etc.Generally, one considers sets of transformations form a group under composition, and studiesthe orbits under the group action – which partition the set of all systems into equivalence classesthat are pairwise equivalent.To identify the orbits, one looks for (complete sets) of invariants, that is functions from the set ofsystems taking values wherever useful which are constant on the orbits of the equivalence underconsideration (i.e. on any two equivalent systems, the functions take the same values). Such setof invariants is complete if for any two nonequivalent systems there is at least one function inthis set which distinguishes (i.e. has different values) on these.One generally would like to identify one normal form for each orbit, that is one (usually nice)system in each equivalence class that represents all systems in that set. (Strictly speaking, onemay consider such normal forms as a special kind of invariants.)In the sequel we shall consider very simple, special equivalence transformations, often with aneye on applications to feedback stabilization. But it is worthwhile to keep in mind this biggerscheme, which is played in many areas of mathematics.The kind of transformation that is special to control systems is to allow for feedback transfor-mations.

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 51

    4.2 Feedback and equivalence: Pole placement for linear systems

    Following closely the textbook – refer to EDS pp. 183-196

    • Consider linear coordinate changes in state space for linear system. Analogous to similaritytransformations in linear algebra, but with the twist that now we work on pairs (A,B) ∈Rn×n × Rn×m.

    • Complete set of invariants on the set of controllable systems is the characteristic polyno-mial. This is an if and only if statement. One direction is clear from linear algebra. Otherdirection just takes a little thought.

    • In single input case, use reachability matrix R = [b, Ab, . . . An−1b] for the similarity trans-formation to get system into normal form (A†, b†).

    Exercise 4.1 Work out how to do the analogue for multi-input systems m > 1 by usingselect columns reachability matrix R = [B,AB, . . . An−1B].

    • The controller form (companion matrix) has same characteristic polynomial, and thus isin same equivalence class.

    Exercise 4.2 Can you find a formula for the similarity transformation that brings thepair (A†, b†) into the controller form (A[, b[)?

    • Pole-placement. Using full state feedback (later we will also consider output feedback anddynamic output feedback).This is a big result: In the linear case, controllability implies stabilizability by static statefeedback u = Fx.

    Exercise 4.3 Carefully study pages 185 to 188 and work the exercises in these pages.

    4.3 Exact feedback linearization

    Following closely the textbook – refer to EDS pp. 197-207

    • A simple question: When does the orbit of an nonlinear system that is affine in the controllike (3) contain a linear system of the form (4) when considering feedback transformationsof the form u = α(x) + β(x)u′ together with local coordinate changes x′ = Φ(x) (α and βare assumed to be smooth, and β also invertible)?The answer is very algorithmic and geometric, and was a major discovery in the early1980s, usually attributed to Hunt, u, and Meyer, and independently to Jakubczyk andRespondek, but apparently the scheme was already partially known by others before this.Real examples – helicopters?Caveats – how much do β and Φ interfere with the control objectives of the linear design?

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 52

    4.3.1 Coordinate changes leave Lie bracket relations unchanged

    The following is a coordinate-free, reference calculation, that explains why local coordinatechanges alone are not expected to much simply a truly nonlinear control system. In geometriclanguage, this calculation involves the pushforward of a smooth vector field X : TM 7→M by adiffeomorphism Φ: M 7→ N . We also use here p ∈M and ϕ ∈ C∞(N).

    • Spivak I, p. 89 Φ∗p• Spivak I, p. 109 (Φ∗Xp)ϕ = Xp(ϕ ◦ Φ)• Spivak I, p. 186 (Φ∗X)q = Φ∗(XΦ−1(q)) = Φ∗Φ−1(q)(XΦ−1(q))• Spivak I, p. 257 Φ-related vector fields, Φ not necessarily a diffeo or even one-to-one

    The symbol Φ∗ is often abused to denote two different maps e.g. the shorthand (Φ∗X) iscommonly used for a vector field on N which correctly should be denoted (Φ∗X) ◦ Φ−1.One finds, for example: The map Φ∗ has all maps [for all q ∈M ] Φ∗q and Φ−1 built in.”At a point – as long as one evaluates at a point, most notation will make sense from the context.

    (Φ∗X)Φ(p)ϕ = (Φ∗pXp)ϕ = Xp(ϕ ◦ Φ) (54)

    or without using subscripts, getting closer to function composition notation

    ((Φ∗X)ϕ)(Φ(p)) = (X(ϕ ◦ Φ))(p) (55)

    But when taking second derivatives, the first derivative is not evaluated at one fixed point, i.e.one needs to work with the function!

    ((Φ∗X)ϕ) ◦ Φ = X(ϕ ◦ Φ) or equivalently ((Φ∗X)ϕ) = X(ϕ ◦ Φ) ◦ Φ−1 (56)

    Now using either one to show that the Lie bracket commutes with the tangent map (differential).

    [Φ∗X,Φ∗Y ]ϕ = (Φ∗X)((Φ∗Y )ϕ)− (Φ∗Y )((Φ∗X)ϕ)= (Φ∗X)((Y (ϕ ◦ Φ)) ◦ Φ−1)− . . .= (X((Y (ϕ ◦ Φ)) ◦ Φ−1 ◦ Φ)) ◦ Φ−1 − . . .= (X(Y (ϕ ◦ Φ))) ◦ Φ−1 − (Y (X(ϕ ◦ Φ))) ◦ Φ−1

    = ((X(Y (ϕ ◦ Φ)))− (Y (X(ϕ ◦ Φ)))) ◦ Φ−1

    = ([X,Y ](ϕ ◦ Φ)) ◦ Φ−1

    = (Φ∗[X,Y ])ϕ

    (57)

    And once more the same, even nicer, with the Φ on the other side of the equation

    ([Φ∗X,Φ∗Y ]ϕ)) ◦ Φ = ((Φ∗X)((Φ∗Y )ϕ)− (Φ∗Y )((Φ∗X)ϕ) ◦ Φ= ((Φ∗X)((Φ∗Y )ϕ)) ◦ Φ− ((Φ∗Y )((Φ∗X)ϕ)) ◦ Φ= X(((Φ∗Y )ϕ) ◦ Φ)− Y (((Φ∗X)ϕ) ◦ Φ)= X(Y (ϕ ◦ Φ))− Y (X(ϕ ◦ Φ))= [X,Y ](ϕ ◦ Φ)= ((Φ∗[X,Y ])ϕ) ◦ Φ

    (58)

    Exercise 4.4 Use coordinates and Jacobian matrices of partial derivatives to repeat the abovecalculation in multi-variable calculus notation. Clearly, it is all in the chain-rule – and the keypoint is to evaluate each matrix at the appropriate point.

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 53

    4.3.2 Exact linearization via coordinate change only

    Since Lie brackets commute with tangent maps, i.e. for the differential (or tangent map) Φ∗ of adiffeomorphism Φ and two smooth vector fields X and Y , [Φ∗X,Φ∗Y ] = Φ∗[X,Y ], no structuralchange is possible by changing coordinates. If f̃0, f̃1, . . . , f̃m denote the transformed systemvector fields f0, f1, . . . , fm under a smooth coordinate change (diffeomorphism), then the Liealgebra L(f̃0, f̃1, . . . , f̃m) is isomorphic (as a Lie algebra) to L(f0, f1, . . . , fm). Consequently anyLie algebra based arguments are blind to anything that is affected by coordinate changes. Thisshould not at all come as a surprise, given e.g. the geometric definition of the Lie bracket of twovector fields as the limit of a composition of their flows.For example, if L(f0, f1, . . . , fm) is not already isomorphic to the Lie algebra generated bythe vector fields of a linear system, then there is no way that the nonlinear system could betransformed into a linear system by a coordinate change only.

    Exercise 4.5 Describe the Lie algebra of linear systems of the form (4) in both the single-input(m = 1) and the multi-input (m > 1) case. Suggestion: Find identities that are always satisfied,especially which fields will always commute, which sets will always span involutive distributions.

    4.3.3 Exact linearization via feedback and coordinate change

    In view of this negative answer, it may come as a surprise that using a simple feedback trans-formation (together with possibly a smooth coordinate change) it is often possible to transformnonlinear systems into linear ones.To clarify the exposition we consider the single-input case only, but the following considerationsgeneralize to multi-input settings. For easier readability denote the uncontrolled and controlledvector fields of the original system by f and g, and those of the transformed system by f̃ and g̃.Assume f(p) = 0. We consider feedback of the form u = α(x) + β(x)v for smooth functions αand β, with β invertible – which preserves the affine structure of the system – together with alocal coordinate change z = Φ(x) about 0 = Φ(p). Since the coordinate change does not effectthe Lie algebra structure we first consider the feedback transformation alone.

    ẋ = f(x) + ug(x) = f(x) + (α(x) + β(x)v)g(x) = (f(x) + α(x)g(x))︸ ︷︷ ︸f̃(x)

    +v · (β(x)g(x))︸ ︷︷ ︸g̃(x)

    (59)

    Exercise 4.6 Suppose that f̃ and g̃ define a linear system, i.e. there exist aij , bi ∈ R such that

    f̃(z) =n∑

    i,j=1

    aijzj∂

    ∂ziand g̃(z) =

    n∑i1

    bi∂

    ∂zi. (60)

    Use homogeneity of these vector fields w.r.t. the standard dilation r = (1, 1, . . . 1) to verify thatfor all i, j ≥ 0, [(adi f̃ , g̃), (adj f̃ , g̃)] ≡ 0.

    Proposition 4.1 With the notation as above, if there exist α and β such that the system afterfeedback is linear in the form of (4), then for all i, j ≥ 0, [(adi f, g), (adj f, g)] ≡ 0.

    Exercise 4.7 Prove proposition 4.1. Suggestions: Note that the feedback transformation (59) isinvertible, i.e. for suitable α̃, β̃ one has f = f̃ + α̃g̃ and g = β̃g̃, use exercise 4.6, and use thatfor smooth vector fields X and Y and a scalar function ϕ, [X,ϕY ] = (LXϕ) · Y + ϕ · [X,Y ].

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 54

    For notational ease define, for a fixed single-input system of form (3), the distribution

    ∆k(f, g) = spanR{g, [f, g], (ad2f, g), . . . . . . (adkf, g)}. (61)

    Thus a necessary condition for the system with fields f and g to be exactly feedback linearizableis that for each k ∈ Z+0 the distribution ∆k(f, g) is involutive. Moreover, if the resultingsystem is controllable, then it is also necessary that at the equilibrium point p of the drift f ,dim∆n−1(f, g)(p) = n.

    Exercise 4.8 Suppose f(p) = 0, ∆n−2(f, g) is involutive, and dim∆n−1(f, g)(p) = n.Show that this implies that for each k ∈ Z+0 , ∆k(f, g) is involutive. Compare also exercise 4.11.

    It turns out that these necessary conditions are also sufficient:

    Theorem 4.2 (Jakubczyk and Respondek, also: Hunt, Su and Meyer, early 1980s)The system ẋ = f(x) + ug(x) with smooth vector fields f and g is locally equivalent to a con-trollable linear system ż = Az+ bu via a feedback transformation u = α(x) + β(x)v with smoothfunctions α and β, β invertible, together with a local coordinate change z = Φ(x), 0 = Φ(p), ifand only if ∆n−2(f, g) is involutive, and dim∆n−1(f, g)(p) = n.

    The key to unravelling the coordinate change together with constructing the necessary feedbackis to consider the desired linear system in controller canonical form. Clearly, composing thefeedback and coordinate change that bring a system into controllable linear form with anotherlinear feedback and coordinate change that bring that system into controller canonical formyields just another transformation of the first kind.

    f̃(z) =n−1∑i=1

    zi+1∂

    ∂ziand g̃(z) = ∂∂zn . (62)

    It is clear that the coordinate function z1 together with its first (n− 1) time derivatives defineall n desired coordinates. Moreover, the function z1 is constant along trajectories of any of thevector fields (adk f̃ , g̃) = ∂∂zk+1 for k = 1, . . . n − 2. Reversing this observation, assuming that∆n−2(f, g) is involutive, and dim∆n−1(f, g)(p) = n, by Frobenius theorem there exists a smoothfunction ϕ such that ϕ(p) = 0, dϕ(p) 6= 0, and near p the integral manifolds of ∆n−1(f, g) aregiven by ϕ = const., i.e. near p

    for k ≤ n− 2, L(adkf,g)ϕ ≡ 0, and L(adn−1f,g)ϕ 6= 0 (63)

    For 1 ≤ k ≤ n define the functions zk = Lk−1f ϕ and calculate using the chain rule for differenti-ating of a scalar function ψ along a solution curve of ẋ = f + ug, i.e. ddtψ = (Lf + uLg)ψ

    ddtz1 = Lf+ugϕ = (Lf + uLg)ϕ = (Lfϕ) + uLgϕ = (Lfϕ) = z2 since Lgϕ ≡ 0 (64)

    Continue

    ddtz2 = (Lf + uLg)(Lfϕ) = (L

    2fϕ) + uLg(Lfϕ)

    = (L2fϕ) + u(Lf (Lgϕ)− L[f,g]ϕ

    )= (L2fϕ) = z3 since Lgϕ ≡ 0 and L[f,g]ϕ ≡ 0.

    (65)

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 55

    Exercise 4.9 Use induction to show that if for 0 ≤ k ≤ n − 2, L(adkf,g)ϕ ≡ 0 then for each0 ≤ k ≤ n− 2 also LgLkfϕ ≡ 0. Suggestion: Write

    L(adk+1f,g)ϕ = L[f,(adkf,g)]ϕ = Lf (L(adkf,g))ϕ− L(adkf,g)(Lfϕ) (66)

    Consequently, the first (n − 1) equations of the transformed system have the desired formż1 = z2, ż2 = z3, . . . żn−1 = zn. However, the last equation needs separate attention - itdetermines the feedback u = α(x) + β(x)v from

    ddtzn−1 = (Lf + uLg)(L

    n−2f

    ϕ) = (Ln−1f ϕ) + (α+ βv)Lg(Ln−2f

    ϕ) != v. (67)

    Exercise 4.10 Continuing the previous exercise, show that under the assumptions of the theo-rem Lg(Ln−2f ϕ)(p) 6= 0.

    By the exercise we have Lg(Ln−2f ϕ)(p) 6= 0 and thus in some open set about p, Lg(Ln−2f

    ϕ) 6= 0.Consequently, in that open set the functions

    α = −Ln−1f ϕ

    Lg(Ln−2f ϕ)and α =

    1Lg(Ln−2f ϕ)

    (68)

    are well-defined, and they clearly achieve the objective.

    Exercise 4.11 Consider the system given by the vector fields f = x2 ∂∂x1 +x3∂

    ∂x2+(x4+x21)

    ∂∂x3

    +x5

    ∂∂x4

    + x6 ∂∂x5 and g =∂

    ∂x6about p = 0 ∈ R6. Verify that ∆1(f, g) is not involutive, but

    [g, [f, g]] ∈ ∆4(f, g). Is this system exactly feedback linearizable? What exactly goes wrong?Compare also exercise 4.8.

    The critical step in this construction is to find the function ϕ. While computing Lie brackets andLie derivatives are purely algebraic operations (assuming that the data, the vector fields f and gare given by nice algebraic formulas, not by numeric approximations). However, the function ϕis defined as a solution of the system of (n− 1) first order partial differential equations.

    {Lgϕ = 0, L[f,g]ϕ = 0, L(adf2,g)ϕ = 0, . . . L(adfn−1,g)ϕ = 0, } (69)

    In general, there is little hope to find closed-form solutions for such a system of coupled nonlinearpartial differential equations. However, as numerous examples have shown, in practice the systemmay often have a special structure that allows one to to find one solution quite easily. (Note,that if ϕ is a solution and ψ is a smooth scalar function with ψ(0) = 0, ψ′(0) 6= 0 then (ψ ◦ ϕ)is also a solution.)

    If it impossible to exactly transform a system into a linear one, one might be tempted to askwhen is it possible to transform it into the next most simple system. Systems which generate anilpotent Lie algebra come to mind. But except for a very useful, special class of such systems– termed “in chained form, or “differentially flat” which possess a similar “flat output” ϕ – verylittle seems to be known.

    Exercise 4.12 Similar to proposition 4.1 and the subsequent exercise explore how a feedbacktransformation u = α+βv could possibly change the structure of the Lie algebra L(f, g). Assumethat L(f̃ , g̃) is nilpotent and explore what this requires for the iterated Lie brackets of the originalvector fields f = f̃ − αβ g̃ and g =

    1β g̃.

  • Classnotes for Mathematical Control Theory. Matthias Kawski. October 10, 2005 56

    4.4 Disturbance rejection and invariant subspaces / manifolds

    Following closely the textbook – refer to EDS pp. 207-210. Very brief intro only.

    • Objective: Use feedback to ensure that an output is not affected by a disturbance• Simple example• Geometric formulation: A-invariant subspace• Algorithm to compute maximal (A,B) invariant subspace.• Outlook: nonlinear (smooth) analogue, distributions and integral manifolds.

    in progress

    4.5 Feedback stabilization

    • Definitions: (Lyapunov-)stable, attractive, asystable (asymptotically) stable. Omit defi-nitions for time-varying systems (uniformly asystable etc.). Local versus global, basin ofattraction. Inlinear case local same as global. (Asymptotically) feedback stabilizable.

    • Linear systems – recall controller normal form and pole placement. Relation between sta-bilizability, controllability (at equilibrium), and null-controllability (e.g. bilinear systems).Uncontrollable modes must be stable.

    • If (Jacobian) linearization is asystable, then original system is locally asystable. feedbackasystabilizes (Jacobian) linearization is asystable, same feedback asystabilizes original sys-tem. Proof?Introduce Lyapunov functions – motivate via energy, passivity (make passive by feedback).Stability argument does not require Lyapunov function to be differentiable, only derivativesalong solution curves are needed, but perturbation arguments rely on Lyapunov functionto be differentiable.

    • LaSalle’s Invariance Principle• Generalize to ∆-homogeneous Lyapunov functions for ∆-homogeneous approximating sys-

    tem• Maybe: Homogeneous systems – eigenray placement ??• control-Lyapunov functions. Universal formula p.248.

    5 Output, realization, observation, and observers

    • observability: motivation, definition, and linear criterion.Combine with 4 canonical forms: controller controllable observer observable

    • Survey Volterra and Chen-Fliess series here.• Realization: Jakubcayk: output space?? YuanWang EDS• Observers. Dynamic feedback stabilization.

    in progress