The transformation from(13.1)to(13.3)is a typical example of feedback linearization, (13.1)is an underactuated model, i.e. when u(t)is restricted to a given subspace in p+ which uses a trong control authority to simplify system equations. For example, whe 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 coor- dinates q(t) and velocities i(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 How 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 i(t)=f(r(t))+g(r(t)u(t), (13.4) y()=h(x(t) (13.5) where a(t EU is the state vector ranging over a given open subset Xo of R", u(tER the control vector,y(t)∈R" is the output vector,f:Xo→R",h:Xo→R", and g: Xo HRXm 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 (t)=(x(t)+a(x(t)u(t) (13.6) and a state transformation z(t)=[1(1);0(t)=v(x(t) where v:X0→R",B:X0→R",a:X0→ R are continuously differentiable functions, such that the Jacobian of wb is not singular on Xo, and the relation between v(t), y(t) and z(t) subject to(13.6),(13.7)has the form (t)=Az(t)+ bu(t), y(t)=Cz(t), (13.8) (t)=ao(z1(t),20(t) (13.9) where A, B, C are constant matrices of dimensions k-by-k, k-by-m, and m-by-k respec tively, such that the pair (A, B)is controllable and the pair(C, A)is observable, and R" is a continuously differentiable function2 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 coor￾dinates 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 respec￾tively, 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