Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 12: Local controllability In this lecture, nonlinear Ode models with an input are considered. Partial answers to the general controllability question(which states can be reached in given time from a given state by selecting appropriate time-dependent control action are presented More precisely, we consider systems described by i(t=a(a(t), u(t), where a: R"XRHR is a given continuously differentiable function, and u=u(t is an m-dimensional time-varying input to be chosen to steer the solution a= r(t) in a desired direction. Let U be an open subset of R, To E R. The reachable set for a given T>0 the(U-locally) reachable set R(To, T)is defined as the set of all a(T) where 0,T+R", u: 0, T]H+Rm is a bounded solution of(12. 1)such that x(0) andx(t)∈ U for all t∈[0,m Our task is to find conditions under which R(To, T)is guaranteed to contain a neig borhood of some point in R, or, alternatively, conditions which guarante that RU(o, T) has an empty interior. In particular, when Io is a controlled equilibrium of (12.1),i.e a(io, io)=0 for some io E R", complete local controllability of(12.1)at To means that for every e>0 and T>0 there exists 8>0 such that R(, T)3 B6(Eo)for every TE B(io), where U= B(io) and B,(2)={正1∈R":|z1-≤r} denotes the ball of radius r centered at I Version of October 31. 2003
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 12: Local Controllability1 In this lecture, nonlinear ODE models with an input are considered. Partial answers to the general controllability question (which states can be reached in given time from a given state by selecting appropriate time-dependent control action) are presented. More precisely, we consider systems described by x˙ (t) = a(x(t), u(t)), (12.1) where a : Rn × Rm ⇒∀ R is a given continuously differentiable function, and u = u(t) n is an m-dimensional time-varying input to be chosen to steer the solution x = x(t) in a desired direction. Let U be an open subset of Rn, ¯x0 ∗ Rn. The reachable set for a given T > 0 the (U-locally) reachable set RU (¯x0, T) is defined as the set of all x(T) where x : [0, T] , ⇒∀ R u : [0, T] is a bounded solution of (12.1) such that x(0) = x¯0 n ⇒∀ Rm and x(t) ∗ U for all t ∗ [0, T]. Our task is to find conditions under which RU (¯x0, T) is guaranteed to contain a neigborhood of some point in Rn, or, alternatively, conditions which guarante that RU (¯x0, T) has an empty interior. In particular, when x¯0 is a controlled equilibrium of (12.1), i.e. x0, ¯ u0 ∗ Rm a(¯ u0) = 0 for some ¯ , complete local controllability of (12.1) at x¯0 means that for every φ > 0 and T > 0 there exists � > 0 such that RU (¯x, T) � B�(¯x0) for every x¯ ∗ B�(¯x0), where U = Bα(¯x0) and Br(¯x) = {x¯1 ∗ R x n : |x¯1 − ¯| √ r} denotes the ball of radius r centered at x¯. 1Version of October 31, 2003
12.1 Systems with controllable linearizations A relatively straightforward case of local controllability analysis is defined by systems with controllable linearizations 12.1.1 Controllability of linearized system Let To: 0, THR, uo: 0, T]H Rm be a bounded solution of (12. 1). The standard linearization of(12.1)around the solution(ro(), uo()) describes the dependency of small state increments 8(t)=r(t)-co(t)+o(5(t)) on small input increments ou(t)=u(t) 6n(t) 6x(t)=A(t)6(t)+B(t)6a(t) (122) where A(t) B(t)= =r0(t),u=a0(t) r=o(t) u=uo(t) are bounded measureable matrix-valued functions of time Let us call system(12.2)controllable on time interval [0, T] if for every 0,dTER there exists a bounded measureable function &u: 0, T]+Rm such that the solution of (12.2 )with 5(0)=80 satisfies S(T)=8.The following simple criterion of controllability is well known from the linear system theory Theorem 12.1 System(12.2) is controllable on interval 0, T] if and only if the matric M(t-B()(t(M(ty-dt is positive definite, where M= M(t) is the evolution matric of (12.2), defined b M(t)=A(t)M(t),M(O)=I Matrix Wc is frequently called the Grammian, or Gram matriz of (12. 2)over [ 0, T It is easy to see why Theorem 12.1 is true: the variable change 8(t)=M(t)z(t)reduces (122)to 2(t)=M()-B(t)6n(t) Moreover. since x(T)=M(t)-B(t)ou(t)dt is a linear integral dependence, function Ou can be chosen to belong to any subclass which is dense in L(0, T). For example, du(t) can be selected from the class of polynomials class of piecewise constant functions, etc Note that controllability over an interval A implies controllability over every interval A+ containing A, but in general does not imply controllability over all intervals A contained in A. Also, system(12. 2)in which A(t)= Ao and B(t)= Bo are constant is equivalent to controllability of the pair(A, B)
2 12.1 Systems with controllable linearizations A relatively straightforward case of local controllability analysis is defined by systems with controllable linearizations. 12.1.1 Controllability of linearized system Let x0 : [0, T] , ⇒∀ R u0 : [0, T] be a bounded solution of (12.1). The standard n ⇒∀ Rm linearization of (12.1) around the solution (x0(·), u0(·)) describes the dependency of small state increments �x(t) = x(t) − x0(t) + o(�x(t)) on small input increments �u(t) = u(t) − �u(t): � ˙ x(t) = A(t)�x(t) + B(t)�u(t), (12.2) where ⎛ ⎛ da⎛ da⎛ A(t) = ⎛ , B(t) = ⎛ (12.3) ⎛ dx⎛ du x=x0(t),u=u0(t) x=x0(t),u=u0(t) are bounded measureable matrix-valued functions of time. Let us call system (12.2) controllable on time interval [0, T] if for every � ¯0 � ¯T ∗ Rn x, x there exists a bounded measureable function �u : [0, T] ⇒∀ R such that the solution of m ¯ (12.2) with �x(0) = � ¯0 satisfies �x(T) = �T x x . The following simple criterion of controllability is well known from the linear system theory. Theorem 12.1 System (12.2) is controllable on interval [0, T] if and only if the matrix � T Wc = M(t) −1 B(t)B(t) � (M(t) � ) −1 dt 0 is positive definite, where M = M(t) is the evolution matrix of (12.2), defined by M˙ (t) = A(t)M(t), M(0) = I. Matrix Wc is frequently called the Grammian, or Gram matrix of (12.2) over [0, T]. It is easy to see why Theorem 12.1 is true: the variable change �x(t) = M(t)z(t) reduces (12.2) to z˙(t) = M(t) −1 B(t)�u(t). Moreover, since � T z(T) = M(t) −1 B(t)�u(t)dt 0 is a linear integral dependence, function �u can be chosen to belong to any subclass which is dense in L1(0, T). For example, �u(t) can be selected from the class of polynomials, class of piecewise constant functions, etc. Note that controllability over an interval � implies controllability over every interval �+ containing �, but in general does not imply controllability over all intervals �− contained in �. Also, system (12.2) in which A(t) = A0 and B(t) = B0 are constant is equivalent to controllability of the pair (A, B)
12.1.2 Consequences of linearized controllability Controllability of linearization implies local controllability. The converse is not true: a nonlinear system with an uncontrollable linearization can easily be controllable Theorem12.2Leta:R"×R→ r be continuously differentiable. Let To:[0,→ R, uo: 0,THRm be a bounded solation of (12.1). Assume that system(12. 2 ), defined by(12.3), is controllable over 0, T]. Then for every e>0 there erists 8>0 such that for all To, ir satisfying o-x00)<6,|zr-xo(T)|<6 there earist functions z: 0, T+", u:[ 0, THRm satisfying the ODE in(12.2) and conditions x(0)=0,a(T)=Zr,x(t)-x0(t)<e,|()-0(1)<∈t∈0, In other words, if linearization around a trajectory(ao, uo) is controllable then from every point in a sufficiently small neigborhood of o() the solution of (12. 1)can be steered to every point in a sufficiently small neigborhood of o(T) by applying a small perturbation u= u(t) of the nominal control uo(t). In particular, this applies when to(t)=To, uo(t)= io is a conditional equilibrium, in which case A, B are constant hence controllability of(12.2) is easy to verify se When system(12. 2)is not controllable, system(12. 1) could still be: for example, the econd order ODE model has an uncontrollable linearization around the equilibrium solution 1=0, 12=0, but is nevertheless locally controllable The proof of Theorem 12.2 is based on the implicit mapping theorem. Let e1,..., 6, be the standard basis in r", Let o,= sk be the controls which cause the solution of (12.2) wit 8(0)=0 to reach 8(T)=ek. For e>0 let B={∈R:|<e} The function S:B×B4→R", which maps={o;u2;…;n]∈ Be and v∈Bto S(, v)=r(T), where a =a(t) is the solution of (12.1)with a(0)=ro(0)+v and l k=1 is well defined and continuously differentiable when e >0 is sufficiently small. The derivative of S with respect to w at w=U=0 is identity. Hence, by the implicit mapping theorem, equation S(, v)=i has a solution w a 0 whenever l and I-To(T)I are mall enough
� 3 12.1.2 Consequences of linearized controllability Controllability of linearization implies local controllability. The converse is not true: a nonlinear system with an uncontrollable linearization can easily be controllable. Theorem 12.2 Let a : Rn × Rm be continuously differentiable. Let x0 : [0, T] Rn, u0 : ⇒∀ Rn ⇒∀ [0, T] be a bounded solution of (12.1). Assume that system (12.2), defined by (12.3), is controllable over [0, T]. Then for every φ > 0 there exists � > 0 such that for ⇒∀ Rm all x¯0, x¯T satisfying |x¯0 − x0(0)| | < �, x¯T − x0(T)| < � there exist functions x : [0, T] ⇒∀ R , u : [0, T] satisfying the ODE in (12.2) and n ⇒∀ Rm conditions x(0) = x¯0, x(T) = ¯xT , x| | | (t) − x0(t) < φ, u(t) − u0(t) < φ � t ∗ [0, T]. In other words, if linearization around a trajectory (x0, u0) is controllable then from every point in a sufficiently small neigborhood of x0(0) the solution of (12.1) can be steered to every point in a sufficiently small neigborhood of x0(T) by applying a small perturbation u = u(t) of the nominal control u0(t). In particular, this applies when x0(t) ≥ x¯0, u0(t) ≥ u¯0 is a conditional equilibrium, in which case A, B are constant, and hence controllability of (12.2) is easy to verify. When system (12.2) is not controllable, system (12.1) could still be: for example, the second order ODE model 3 x˙ 1 = x2, x˙ 2 = u has an uncontrollable linearization around the equilibrium solution x1 ≥ 0, x2 ≥ 0, but is nevertheless locally controllable. The proof of Theorem 12.2 is based on the implicit mapping theorem. Let e1, . . . , φn be the standard basis in Rn. Let �u = �k be the controls which cause the solution of u (12.2) wit �x(0) = 0 to reach �x(T) = ek. For φ > 0 let x ∗ Rn Bα = {¯ ¯ : | | x < φ}. The function S : Bα × Bα ⇒∀ R , which maps w = [w1; w2; . . . ; wn] ∗ Bα and v ∗ Bα to n S(w, v) = x(T), where x = x(t) is the solution of (12.1) with x(0) = x0(0) + v and n u(t) = u0 + wk�k u(t), k=1 is well defined and continuously differentiable when φ > 0 is sufficiently small. The derivative of S with respect to w at w = v = 0 is identity. Hence, by the implicit mapping theorem, equation S(w, v) = x¯ has a solution w � 0 whenever | | | | v and x¯ − x0(T) are small enough
12.2 Controllability of driftless models In this section we consider OdE models in which the right side is linear with respect to the control variable, i.e. when(12.1) has the special form i(t)=g(x()()=∑9(x()(),(0)= 12. k=1 functions deli."are given Coo(i.e. having continuous derivatives of arbitrary order) where gk functions defined on an open subset Xo of R, and u(t)=u(t);.; um(t)) is the vector control input. Note that linearization(12.2)of(12.4)around every equilibrium solution co(t)=io= const, uo(t)=0 yields A=0 and B= g(io), which means that the linearization is never controllable unless m n. Nevertheless. it turns out that. for a generic"function g, system(12. 4)is expected to be completely controllable, as long as m>1 12.2.1 Local controllability and Lie brackets Let us say that system(12. 4)is locally controllable at a point o E Xo if for every e>0 T>0, and I E Xo such that Ii-iol e there exists a bounded measureable function u:0, T]H+Rm defining a solution of(12.4)with (0)=Io such that a(r)=T and r()-<∈yt∈[0, The local controlability conditions to be presented in this section are based on the notion of a Lie bracket. Let us write h=[h1, h2(which reads as"h3 is the Lie bracket of h1 and h2")when hk: XoHR are continuous functions defined on an open subset Xo of R", functions h1, h2 are continuously differentiable on Xo, and ha3(z)=h1(E)h2()-h2()h1() for all I E Xo, where hx(i) denotes the Jacobian of hk at D The reasoning behind the definition, as well as a more detailed study of the properties of Lie brackets, will be postponed until the proof of the controllability results of this subsection Let us call a set of functions hk: Xo HR",(k=1,., g complete at a point T E Xo if either the vectors hi(i)with i= l,..., m span the whole r or there exist functions hk:X0→R",(k=q+1,,N), such that for every h> q we have hk=[h,h。」for some i, s< h, and the vectors hi(a)with i=1,., N span the whole R system(12.4) is locally controllable at io.k R"form a complete set at Lo E Xo then Theorem 12. 3 If Co functions gk
� 4 12.2 Controllability of driftless models In this section we consider ODE models in which the right side is linear with respect to the control variable, i.e. when (12.1) has the special form m x˙ (t) = g(x(t))u(t) = gk(x(t))u(t), x(0) = x¯0, (12.4) k=1 where gk : are given C� X0 ⇒∀ R (i.e. having continuous derivatives of arbitrary order) n functions defined on an open subset X0 of Rn, and u(t) = [u1(t); . . . ; um(t)] is the vector control input. Note that linearization (12.2) of (12.4) around every equilibrium solution x0(t) ≥ x¯0 = const, u0(t) = 0 yields A = 0 and B = g(¯x0), which means that the linearization is never controllable unless m = n. Nevertheless, it turns out that, for a “generic” function g, system (12.4) is expected to be completely controllable, as long as m > 1. 12.2.1 Local controllability and Lie brackets Let us say that system (12.4) is locally controllable at a point x¯ if 0 ∗ X0 for every φ > 0, T > 0, and x¯ ∗ X0 such that x¯ − x¯0| < φ there exists a bounded measureable function u : [0, T] defining a solution of (12.4) with x(0) = ¯ x and | ⇒∀ R x0 such that x(T) = ¯ m | | x(t) − x¯0 < φ � t ∗ [0, T]. The local controlability conditions to be presented in this section are based on the notion of a Lie bracket. Let us write h3 = [h1, h2] (which reads as “h3 is the Lie bracket of h1 and h2”) when hk : X0 ⇒∀ R are continuous functions defined on an open subset n X0 of Rn, functions h1, h2 are continuously differentiable on X0, and h3(¯x) = h x)h2(¯ x)h1(¯ ˙ 1(¯ x) − h˙ 2(¯ x) x ∗ X0, where h˙ for all ¯ k(¯x) denotes the Jacobian of hk at x¯. The reasoning behind the definition, as well as a more detailed study of the properties of Lie brackets, will be postponed until the proof of the controllability results of this subsection. Let us call a set of functions hk : X0 ⇒∀ R , (k = 1, . . . , q) complete at a point x¯ ∗ X0 n if either the vectors hi(¯ ⇒∀ Rn x) with i = 1, . . . , m span the whole Rn or there exist functions hk : X0 , (k = q + 1, . . . , N), such that for every k > q we have hk = [hi, hs] for some i, s < k, and the vectors hi(¯x) with i = 1, . . . , N span the whole Rn. Theorem 12.3 If C� functions gk : X0 ⇒∀ Rn form a complete set at x¯0 ∗ X0 then system (12.4) is locally controllable at x¯0
Theorem 12.3 provides a sufficient criterion of local controllability in terms of the span of all vector fields which can be generated by applying repeatedly the Lie bracket operation to gk. This condition is not necessary, as can be seen from the following example: the order system o(x1)u2, where function o: R H R is infinitely many times continuously differentiable and such that 0(0)=0,d(y)>0fory≠0,()(0)=0Vk is locally controllable at every point io E r despite the fact that the corresponding set of vector fields 9()=/1 0 o(x1) is not complete at i=0. On the other hand, the example of the system L hich is not locally controlable at I=0, but is defined by a(single element) set of vector fields which is complete at every point except i=0, shows that there is little room for relaxing the sufficient conditions of Theorem 12.3 12.2.2 Proof of heorem 12.3 Let s denote the set of all continuous functions s: Qs H+ Xo, where Qs is an open subset of R x Xo containing 0x Xo(Qs is allowed to depend on s). Let Sk E S be the elements of s defined b Sk(r,)=x(7):i(t)=9k(x(t),x(0)= Let Sg be subset of s which consists of all functions which can be obtained by recursion sk+1(,T)=Sa()(sk(,T),k(7),0(z,7)= where a(k)E (1, 2,..., m) and Ok: R-Rare continuous functions such that Ok(0)=0 One can view elements of Sg as admissible state transitions in system(12. 2)with piecewise constant control depending on parameter T in such a way that T=0 corresponds to the identity transition. Note that for every s∈ S, there exists an“ Inverse”s'∈ S such that s(S(,T),7)=V(z,)∈9, defined by applying inverses Sa()(, -ok(T)of the basic transformations Sa()(, %(T) n the reverse order
� � � � 5 Theorem 12.3 provides a sufficient criterion of local controllability in terms of the span of all vector fields which can be generated by applying repeatedly the Lie bracket operation to gk. This condition is not necessary, as can be seen from the following example: the second order system x˙ 1 = u1, x˙ 2 = α(x1)u2, where function α : R ⇒∀ R is infinitely many times continuously differentiable and such that α(0) = 0, α(y) > 0 for y = 0, α(k) ∈ (0) = 0 � k, is locally controllable at every point x¯0 ∗ Rn despite the fact that the corresponding set of vector fields 1 0 g1(x) = , g2(x) = 0 α(x1) is not complete at x¯ = 0. On the other hand, the example of the system x˙ = xu, which is not locally controlable at x¯ = 0, but is defined by a (single element) set of vector fields which is complete at every point except x¯ = 0, shows that there is little room for relaxing the sufficient conditions of Theorem 12.3. 12.2.2 Proof of Theorem 12.3 Let S denote the set of all continuous functions s : �s ⇒∀ X0, where �s is an open subset of R×X0 containing {0}×X0 (�s is allowed to depend on s). Let Sk ∗ S be the elements of S defined by Sk(δ, x¯) = x(δ ) : x˙ (t) = gk(x(t)), x(0) = x. ¯ Let Sg be subset of S which consists of all functions which can be obtained by recursion sk+1(¯x, δ ) = S�(k)(sk(¯x, δ ), αk(δ )), β0(¯x, δ ) = ¯x, where �(k) ∗ {1, 2, . . . , m} and αk : R ⇒∀ R are continuous functions such that αk(0) = 0. One can view elements of Sg as admissible state transitions in system (12.2) with piecewise constant control depending on parameter δ in such a way that δ = 0 corresponds to the identity transition. Note that for every s ∗ Sg there exists an “inverse” s� ∗ Sg such that s(s� (¯ x x, δ ) ∗ �s x, δ ), δ ) = ¯ � (¯ �, defined by applying inverses S�(k)(·, −αk(δ )) of the basic transformations S�(k)(·, αk(δ )) in the reverse order