《非线性动力学》（英文版） Lecture 13 Feedback linearization.pdf_大学文库

2 The transformation from (13.1) to (13.3) is a typical example of feedback linearization, which uses a strong control authority to simplify system equations. For example, when (13.1) is an underactuated model, i.e. when u(t) is restricted to a given subspace in Rk, the transformation in (13.2) is not valid. Similarly, if u(t) must satisfy an a-priori bound, conversion from v to u according to (13.2) is not always possible. In addition, feedback linearization relies on access to accurate information, in the current example – precise knowledge of functions M, F and precise measurement of coordinates q(t) and velocities q˙(t). While in some cases (including the setup of (13.1)) one can extend the benefits of feedback linearization to approximately known and imperfectly observed models, information flow constraints remain a serious obstacle when applying feedback linearization. 13.1.2 Output feedback linearization Output feedback linearization can be viewed as a way of simplifying a nonlinear ODE control system model of the form x˙ (t) = f(x(t)) + g(x(t))u(t), (13.4) y(t) = h(x(t)), (13.5) where x(t) ∀ U is the state vector ranging over a given open subset X0 of Rn, u(t) ∀ Rm is the control vector, y(t) ∀ Rm is the output vector, f : X0 ≤� Rn, h : X0 ≤� Rm, and g : X0 ≤� Rn×m are given smooth functions. Note that in this setup y(t) has same dimension as u(t). The simplification is to be achieved by finding a feedback transformation v(t) = �(x(t)) + �(x(t))u(t), (13.6) and a state transformation z(t) = [zl(t); z0(t)] = �(x(t)), (13.7) where � : X0 ≤� Rn, � : X0 ≤� Rm, � : X0 ≤� Rm×m are continuously differentiable functions, such that the Jacobian of � is not singular on X0, and the relation between v(t), y(t) and z(t) subject to (13.6), (13.7) has the form z˙l(t) = Azl(t) + Bv(t), y(t) = Czl(t), (13.8) z˙0(t) = a0(zl(t), z0(t)), (13.9) where A, B, C are constant matrices of dimensions k-by-k, k-by-m, and m-by-k respectively, such that the pair (A, B) is controllable and the pair (C, A) is observable, and a0 : Rk × Rn−k ≤� Rn−k is a continuously differentiable function

More precisely, it is required that for every solution x: to, t1]b Xo, u: to, t1 Rm y: to, t1]+Rm of (13.4),(13.5)equalities(13.8),(13.9)must be satisfied for 2(t), v(t) defined by(13.6)and(13.7) As long as accurate measurements of the full state a(t) of the original system are available, Xo= R, and the behavior of y(t) and u(t) is the only issue of interest, the output feedback linearization reduces the control problem to a linear one. However, in a ddition to sensor limitations, Xo is rarely the whole R", and the state a (t) is typically required to remain bounded (or even to converge to a desired steady state value). Thus it is frequently impossible to ignore equation(13.9), which is usually refered to as the zero dynamics of (13.4), (13.5). In the best scenario(the so-called"minimum phase systems") the response of(13.9) to all expected initial conditions and reference signals y(t) can be proven to be bounded and generating a response r (t) confined to Xo. In general, the area Xo on which feedback linearization is possible does not cover of states of interest the zero dynamics is not as stable as desired and hence the benefits of output feedback linearization are limited 13.1.3 Full state feedback linearization Formally, full state feedback linearization applies to nonlinear ODE control system model of the form(13.4), without a need for a particular output y(t) to be specified tarAs in the previous subsection, the simplification is to be achieved by finding a feedback ansformation(13.6)and a state transformation a(t)=v(ar(t)) (13.10) with a non-singular Jacobian. It is required that for every solution a: [to, ti H+Xo a: to, t1 b Rm of (13.4)equality 2(t)=Az(t)+B(t) (13.11 must be satisfied for z(t), v(t) defined by(13.6)and(13.10) It appears that the benefits of having a full state linearization are substantially greater than those delivered by an output feedback linearization. Unfortunately, among systems of order higher than two the full state feedback linearizable ones form a set of "zero mea sure", in a certain sense. In other words, unlike in the case of output feedback lineariza- tion, which is possible, at least locally, "almost always", full state feedback linearizability requires certain equality constraints to be satisfied for the original system data, and hence does not take place in a generic setup 13.2 Feedback linearization with scalar control This section contains basic results on feedback linearization of single-input systems(the case when m=l in(13.4))

3 More precisely, it is required that for every solution x : [t0, t1] ≤� X0, u : [t0, t1] ≤� Rm, y : [t0, t1] ≤� Rm of (13.4), (13.5) equalities (13.8), (13.9) must be satisfied for z(t), v(t) defined by (13.6) and (13.7). As long as accurate measurements of the full state x(t) of the original system are available, X0 = Rn, and the behavior of y(t) and u(t) is the only issue of interest, the output feedback linearization reduces the control problem to a linear one. However, in a ddition to sensor limitations, X0 is rarely the whole Rn, and the state x(t) is typically required to remain bounded (or even to converge to a desired steady state value). Thus, it is frequently impossible to ignore equation (13.9), which is usually refered to as the zero dynamics of (13.4),(13.5). In the best scenario (the so-called “minimum phase systems”), the response of (13.9) to all expected initial conditions and reference signals y(t) can be proven to be bounded and generating a response x(t) confined to X0. In general, the area X0 on which feedback linearization is possible does not cover of states of interest, the zero dynamics is not as stable as desired, and hence the benefits of output feedback linearization are limited. 13.1.3 Full state feedback linearization Formally, full state feedback linearization applies to nonlinear ODE control system model of the form (13.4), without a need for a particular output y(t) to be specified. As in the previous subsection, the simplification is to be achieved by finding a feedback transformation (13.6) and a state transformation z(t) = �(x(t)) (13.10) with a non-singular Jacobian. It is required that for every solution x : [t0, t1] ≤� X0, u : [t0, t1] ≤� Rm of (13.4) equality z˙(t) = Az(t) + Bv(t) (13.11) must be satisfied for z(t), v(t) defined by (13.6) and (13.10). It appears that the benefits of having a full state linearization are substantially greater than those delivered by an output feedback linearization. Unfortunately, among systems of order higher than two, the full state feedback linearizable ones form a set of “zero measure”, in a certain sense. In other words, unlike in the case of output feedback linearization, which is possible, at least locally, “almost always”, full state feedback linearizability requires certain equality constraints to be satisfied for the original system data, and hence does not take place in a generic setup. 13.2 Feedback linearization with scalar control This section contains basic results on feedback linearization of single-input systems (the case when m = 1 in (13.4))

Note that, unlike the Frobenius theorem, Theorem 13. 1 is not local: it provides feed back linearization on every open set Xo on which the relative degree is well defined. Also in the case of linear models, where f(r)= Ar and g(ar)=B, it is always possible to get the zero dynamics depending on y only, i.e. to ensure that a0(2,x0)=a0(C2,x0) This, however, is not always possible in the nonlinear case. For example, for system x1+ there exists no function p: Xo H R defined on a non-empty open subset of R' such that Vp(a)f(a)=b(a1, p(a)), Vp(a)g(a)=0, Vp(a)t0 E Xo Indeed, otherwise the system with new output ynew= p(a) would have relative degree 3, which by Theorem 13.1 implies that(Vp)g1=(Vp)92=0, and hence by the Frobenius theorem the vector fields 0 gn(x)=1|,g(x) would define an involutive distribution, which they do not 13.2.2 Involutivity and full state feedback linearization It follows from Theorem 13. 1 that system(13.4),(13.5) which has maximal possible relative degree n is full state feedback linearizable. The theorem also states that, given smooth functions f, g, existence of h defining a system with relative degree n implies linear independence of vectors 91(0),., n(i)for all I E Xo, and involutivity of the regular distribution defined by vector fields 91, .. gn-1. The converse is also true, which allows one to state the following theorem Theorem13.2Letf:X0→→ R" and g:X0→ r be n+1 times continuously differentiable functions defined on an open subset Xo ofR". Let gk with k=1,., n be defined as in Theorem 13.1 (a)If system(13. 4) is full state feedback linearizable on Xo then vectors g1(i),.,n(i) form a basis in r for all T E Xo, and the distribution defined by vector fields 91, .., 9n-1 in involutive on Xo (b)If for some io E Xo vectors g1(I),., n(i)form a basis in R", and the distribution defined by vector fields g1, .. 9n-1 in involutive in a neigborhood of To, there erists an open subset Xo of Xo such that to E Xo and system(13. 4) is full state feedback linearizable on x

5 Note that, unlike the Frobenius theorem, Theorem 13.1 is not local: it provides feedback linearization on every open set X0 on which the relative degree is well defined. Also, in the case of linear models, where f(x) = Ax and g(x) = B, it is always possible to get the zero dynamics depending on y only, i.e. to ensure that a0(zl, z0) = a¯0(Czl, z0). This, however, is not always possible in the nonlinear case. For example, for system ⎤ � ⎤ � x1 x2 d � x2 ⎣ = � u ⎣ , y = x1 dt 2 x3 x1 + x2 there exists no function p : X0 ≤� R defined on a non-empty open subset of R3 such that ∈p(x)f(x) = b(x1, p(x)), ∈p(x)g(x) = 0, ∈p(x) →= 0 � x ∀ X0. Indeed, otherwise the system with new output ynew = p(x) would have relative degree 3, which by Theorem 13.1 implies that (∈p)g1 = (∈p)g2 = 0, and hence by the Frobenius theorem the vector fields ⎤ � ⎤ � 0 1 g1(x) = � 1 ⎣ , g2(x) = � 0 ⎣ 0 2x2 would define an involutive distribution, which they do not. 13.2.2 Involutivity and full state feedback linearization It follows from Theorem 13.1 that system (13.4), (13.5) which has maximal possible relative degree n is full state feedback linearizable. The theorem also states that, given smooth functions f, g, existence of h defining a system with relative degree n implies linear independence of vectors g1(¯x), . . . , αn(¯ ¯ x) for all x ∀ X0, and involutivity of the regular distribution defined by vector fields g1, . . . , gn−1. The converse is also true, which allows one to state the following theorem. Theorem 13.2 Let f : X0 ≤� Rn and g : X0 ≤� Rn be n + 1 times continuously differentiable functions defined on an open subset X0 of Rn. Let gk with k = 1, . . . , n be defined as in Theorem 13.1. (a) If system (13.4) is full state feedback linearizable on X0 then vectors g1(¯x), . . . , αn(¯x) form a basis in R ¯ n for all x ∀ X0, and the distribution defined by vector fields g1, . . . , gn−1 in involutive on X0. (b) If for some x¯0 ∀ X0 vectors g1(¯x), . . . , αn(¯x) form a basis in Rn, and the distribution defined by vector fields g1, . . . , gn−1 in involutive in a neigborhood of x¯0, there exists ¯ ¯ an open subset X0 of X0 such that x¯0 ∀ X0 and system (13.4) is full state feedback ¯ linearizable on X0