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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

  • 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 a state q if there exists an admissible control u that steers the system from initial state p at time t0 to the final state q = Φ(t0 + T, p, t0, u). A system is called (globally) controllable if for every pair (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 differential equations. [References: S. Eilenberg, Automata, languages, and machines, and various recent papers by J.-M. Coron, on controllability of various PDEs.] Here we shall focus on systems with continuous states in Rn or on finite dimensional manifolds Mn, governed by controlled differential and difference equations.

    Exercise 3.1 Give an example of a control system with a finite state space, and define suitable notions of controllability for this kind of system. (E.g. consider networks of routes served by an airline). 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 here that 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 controllable to q from p”. The system is reachable from p if the entire state-space X ⊆ R≤∞(p) can be reached from 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 corresponds to interchanging the roles of reachabilty and controllability, and one commonly uses the term controllability when one technically analyzes reachability. In the sequel we shall investigate when the 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 system obtained 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 in nonlinear 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 positive way to read this is that there are plenty of open research problems, each corresponding to a different meaningful setting in which to analyze controllability. A central objective of the analysis is to obtain readily checkable conditions in terms of the systems 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). Constructing controls that actually steer the system from a given state p to a given state q is generally a different 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 asserts that the time-invariant system ẋ = Ax+Bu, x ∈ Rn, ∀t, u(t) ∈ U = Rm is globally controllable if 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 that this condition applies to systems that impose no a-priori bound on the magnitude |u(·)| of the control. 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 for the set of unbounded controls (U = Rm), but is not controllable with bounded controls taking values in U = [−1, 1]m.

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

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

    resemble linear systems, in many ways they display characteristics more reminiscent of general nonlinear systems. Indeed, many nonlinear systems can be lifted to bilinear systems (on a much higher dimensional state-space) – and therefore, in many ways, bilinear systems must be at least as complex as many nonlinear systems. However, what makes some bilinear systems particularly attractive 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 last two decades many new results for controllability on Lie groups have been found, and this area continues to see much ongoing research. In our course, we shall jump ahead and next concentrate on nonlinear systems that are affine in the 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 be smooth C∞, or analytic Cω). These are a special case of fully nonlinear systems ẋ = f(x, u) for which much fewer general results are available. In recent years, much research has focused on hybrid systems which mix continuous and discrete states as they occur in systems with hysteresis or 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 shall usually assume that for all i and for all t, |ui(t)| ≤ 1, or

    ∑m i=1 u

    2 i ≤ 1. More generally one may

    allow the controls u to take values in a convex, compact subset U ⊆ Rm. There may be many possible explanations for this distinction of working with bounded and unbounded controls when studying nonlinear and linear systems. Clearly, in practical applications there are almost always bounds on the admissible controls. However, the mathematical treatment may impose different conditions to be feasible. This distinction will become clearer later when we study optimal control problems which are closely related to controllability: Different system structures will suggest different objective functions which in turn determine the underlying sets of admissible controls. 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 Kalman rank 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 not necessarily 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 this system (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 affine subspaces on Rn. Indeed, consider the variation of parameters formula for the solution curves of system (4) for any control u(·).

    Φ(T, p, u) = etAp+ ∫ T

    0 e(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. Due to the additive structure, it suffices to consider the reachable sets RT (0) for systems initialized at 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 time systems 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 0 0 −