Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 13: Feedback linearization Using control authority to transform nonlinear models into linear ones is one of the most commonly used ideas of practical nonlinear control design. Generally, the trick helps one to recognize "simple"nonlinear feedback design tasks 13.1 Motivation and objectives In this section, we give a motivating example and state technical objectives of theory of feedback linearization 13.1.1 Example: fully actuated mechanical systems Equations of rather general mechanical systems can be written in the form M(a(t))q(t)+F(a(t), it))=u(t), (13.1) where q(t)E R is the position vector, u(t)is the vector of actuation forces and torques. F:R×R→ R is a given vector- valued function,andM:R←→ RXK is a given function taking positive definite symmetric matrix values(the inertia matrix). When u= u(t)is fixed(for example, when u(t)=uo cos(t)is a harmonic excitation (13. 1)is usually an extremely difficult task. However, when u(t) is an unrestricted control effort to be chosen, a simple change of control variable u(t)=M(q(t))(u(t)+F(q(t),i(t)) (13.2) transforms(13. 1)into a linear double integrator model Version of october 29. 2003
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 13: Feedback Linearization1 Using control authority to transform nonlinear models into linear ones is one of the most commonly used ideas of practical nonlinear control design. Generally, the trick helps one to recognize “simple” nonlinear feedback design tasks. 13.1 Motivation and objectives In this section, we give a motivating example and state technical objectives of theory of feedback linearization. 13.1.1 Example: fully actuated mechanical systems Equations of rather general mechanical systems can be written in the form M(q(t))¨q(t) + F(q(t), q˙(t)) = u(t), (13.1) where q(t) ∀ Rk is the position vector, u(t) is the vector of actuation forces and torques, F : Rk × Rk ≤� Rk is a given vector-valued function, and M : Rk ≤� Rk×k is a given function taking positive definite symmetric matrix values (the inertia matrix). When u = u(t) is fixed (for example, when u(t) = u0 cos(t) is a harmonic excitation), analysis of (13.1) is usually an extremely difficult task. However, when u(t) is an unrestricted control effort to be chosen, a simple change of control variable u(t) = M(q(t))(v(t) + F(q(t), q˙(t))) (13.2) transforms (13.1) into a linear double integrator model q¨(t) = v(t). (13.3) 1Version of October 29, 2003
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 function
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))
13.2.1 Relative degree and I/O feedback linearization Assume that functions h, f, g in(13. 4), (13.5 )are at least q+l times continuously differ entiable. We say that system(13. 4), (13.5) has relative degree q on Xo if Vh1()9(2)=0,,Vhq-1()9(2)=0,Vhq()9(2)≠0V∈X0, here hi: Xo b R are defined by h1=h,h+1=(Vh)f(i=1,…,q) By applying the definition to the LTI case f(r)=Ar, g(a)=B, h(r)=car one can see that an LTi system with a non-zero transfer function always has a relative degree which equals the difference between the degrees of numerator and denominator of its transfer function io. It turns out that systems with well defined relative degree are exactly those for which out/output feedback linearization is possible Theorem 13.1 Assuming that h, f, g are continuously differentiable n+ 1 times, the following conditions are equivalent (a)system(13. 4), (13.5) has relative degre (b)system(13. 4), (13.5) is input/ output feedback linearizable Moreover if conditions(a) is satisfied then (i) the gradients Vh(E)with k= l,..., q are linearly independent for every E Xo (which, in particalar, implies that q <n); i) vectors gk(i) defined by q1=9,9k+1=[f,9k](k= satis Vh4(z)9k()=Vh2+y-1()9(2)V∈Xo jori+j≤q+1; (iii) feedback linearization is possible with k=g, (2)=Vh()g(2),B()=Vh(z)∫(z) h1(z) =v() hg()
4 13.2.1 Relative degree and I/O feedback linearization Assume that functions h, f, g in (13.4),(13.5) are at least q + 1 times continuously differentiable. We say that system (13.4),(13.5) has relative degree q on X0 if ∈h1(¯x)g(¯x) = 0, . . . , ∈hq−1(¯x)g(¯x) = 0, ∈hq(¯x)g(¯ = x) → 0 � x¯ ∀ X0, where hi : X0 ≤� R are defined by h1 = h, hi+1 = (∈hi)f (i = 1, . . . , q). By applying the definition to the LTI case f(x) = Ax, g(x) = B, h(x) = Cx one can see that an LTI system with a non-zero transfer function always has a relative degree, which equals the difference between the degrees of numerator and denominator of its transfer function. It turns out that systems with well defined relative degree are exactly those for which input/output feedback linearization is possible. Theorem 13.1 Assuming that h, f, g are continuously differentiable n + 1 times, the following conditions are equivalent: (a) system (13.4),(13.5) has relative degree q; (b) system (13.4),(13.5) is input/output feedback linearizable. Moreover if conditions (a) is satisfied then (i) the gradients ∈hk(¯x) with k = 1, . . . , q are linearly independent for every x¯ ∀ X0 (which, in particular, implies that q � n); (ii) vectors gk(¯x) defined by g1 = g, gk+1 = [f, gk] (k = 1, . . . , q − 1) satisfy ∈hi(¯x)gk(¯x) = ∈hi+j−1(¯x)g(¯x) � x¯ ∀ X0 for i + j � q + 1; (iii) feedback linearization is possible with k = q, �(¯x) = ∈hq(¯x)g(¯x), �(¯x) = ∈hq(¯x)f(¯x), ⎤ � h1(¯x) � h ⎢ � ⎢ 2(¯x) z¯l = �l(¯x) = � . ⎢ . � . . ⎣ hq(¯x)
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
