5. Recall that for a nonlinear system of the form = f(x) + g(x)u, y =...

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5. Recall that for a nonlinear system of the form = f(x) + g(x)u, y = h(x), the normal form with relative degree r (<n) is given by where ₁ y, 2 = Ý,..., and μ² = [μ₁ μ₂ Anh nhỉ = [/1 /2 Note that the internal dynamics is given by = w(, ). Zero dynamics: We will consider the problem of finding the initial states ao and the input functions uo(t) such that the output y(t) = 0, Vt. In the normal form coordinates, the con- straint that y(t) = 0 means that j = 0,..., 0 or u(t)= 0, Vt. Note that the initial condition for the internal states given by (0) can be chosen arbitrarily, and the evolution of (t) is given by μ1 = μ2 μ2 = 13 ⠀ jbr-1 = fr jr = a(,) + b(u, v)u y = w(μ, y) (t) =w(0,4), (0) = 40. This is referred to as the "zero dynamics." The input that keeps the output at zero is given by: G(s) Un-r]. = 0= a(0, 2) + b(0, )uo(t) uo(t) where (t) evolves according to the zero dynamics. Problem: Consider the linear system with the following transfer function a(0,4) b(0,4) sn-r+Bn-r-18n-r-1 + +B18+ Bo sn + an-18-1+.. + α18 + a0 where r <n. Consider the minimal state space realization of G(s): * = Ax + Bu y = Cx where x = [1, 2,..., n] 0 1 0 0 0 1 : ⠀ DH B 0 0 1 -XX0 -α1 -02 -an-1 Show that the normal form of this linear system is given by A || Q 0 0 ⠀ || μ₁ = μ2 μ^₂ = μ3 ⠀ jr_1 = flr jr = Rp + S + u & = = Pµ+Q where R and S are row vectors of dimension r and n-r, respectively, P is an (nr) x r matrix and Q is a rx (n-r) matrix given by 1 0 0 1 ⠀ ⠀ 0 0 0 -Bo -B1 B2 C = [Bo Bi Note that the zero dynamics is given by 0 0 Bn-r-1 1 0.. : 1 -Bn-r-1. 0]. $ = Q4, whose eigenvalues are the zeros of the transfer function G(s). Hint: Differentiate the output y to obtain the normal form states ₁. Then, consider the other nr states ; to be i, i = 1,...,nr. Note that with this choice of states, z = =(x) is a diffeomorphism where z = []. 5. Recall that for a nonlinear system of the form = f(x) + g(x)u, y = h(x), the normal form with relative degree r (<n) is given by where ₁ y, 2 = Ý,..., and μ² = [μ₁ μ₂ Anh nhỉ = [/1 /2 Note that the internal dynamics is given by = w(, ). Zero dynamics: We will consider the problem of finding the initial states ao and the input functions uo(t) such that the output y(t) = 0, Vt. In the normal form coordinates, the con- straint that y(t) = 0 means that j = 0,..., 0 or u(t)= 0, Vt. Note that the initial condition for the internal states given by (0) can be chosen arbitrarily, and the evolution of (t) is given by μ1 = μ2 μ2 = 13 ⠀ jbr-1 = fr jr = a(,) + b(u, v)u y = w(μ, y) (t) =w(0,4), (0) = 40. This is referred to as the "zero dynamics." The input that keeps the output at zero is given by: G(s) Un-r]. = 0= a(0, 2) + b(0, )uo(t) uo(t) where (t) evolves according to the zero dynamics. Problem: Consider the linear system with the following transfer function a(0,4) b(0,4) sn-r+Bn-r-18n-r-1 + +B18+ Bo sn + an-18-1+.. + α18 + a0 where r <n. Consider the minimal state space realization of G(s): * = Ax + Bu y = Cx where x = [1, 2,..., n] 0 1 0 0 0 1 : ⠀ DH B 0 0 1 -XX0 -α1 -02 -an-1 Show that the normal form of this linear system is given by A || Q 0 0 ⠀ || μ₁ = μ2 μ^₂ = μ3 ⠀ jr_1 = flr jr = Rp + S + u & = = Pµ+Q where R and S are row vectors of dimension r and n-r, respectively, P is an (nr) x r matrix and Q is a rx (n-r) matrix given by 1 0 0 1 ⠀ ⠀ 0 0 0 -Bo -B1 B2 C = [Bo Bi Note that the zero dynamics is given by 0 0 Bn-r-1 1 0.. : 1 -Bn-r-1. 0]. $ = Q4, whose eigenvalues are the zeros of the transfer function G(s). Hint: Differentiate the output y to obtain the normal form states ₁. Then, consider the other nr states ; to be i, i = 1,...,nr. Note that with this choice of states, z = =(x) is a diffeomorphism where z = [].