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 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 = 1 in (13.4))