{"id":176,"date":"2021-07-02T10:09:43","date_gmt":"2021-07-02T10:09:43","guid":{"rendered":"https:\/\/www.mathgallery.com\/?page_id=176"},"modified":"2021-08-29T15:38:24","modified_gmt":"2021-08-29T15:38:24","slug":"zero-dynamics","status":"publish","type":"page","link":"https:\/\/www.mathgallery.com\/index.php\/zero-dynamics\/","title":{"rendered":"Zero Dynamics"},"content":{"rendered":"\n

<\/p>\n\n\n\n

In this section we describe the basic features of zero dynamics. The zero dynamics can be seen as the remaining dynamics of a system when we steer the output to be identically zero. We will give a brief account for some definitions and results which can be found in e.g. Isidori [5]. First we interpret the zero dynamics of a single-input single-output (SISO) system, especially in the linear case, both in the frequency domain and in the time domain and then we extend the concept to multi-input multi-output (MIMO) systems.<\/p>\n\n\n\n

Definition 2.1:<\/strong> The single-input single-output nonlinear system<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[\dot x&=&f(x)+g(x)u\]\"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\"\[y&=&h(x)\]\"<\/p><\/p>\n\n\n\n

is said to have relative degree \"r\" at a point \"x^0\" if<\/p>\n\n\n\n

\\begin{itemize}<\/p>\n\n\n\n

\\item[i] \"L_gL^k_fh(x)=0\" for all \"x\" in a neighborhood of \"x^0\" and all \"0\le k<r-1\",<\/p>\n\n\n\n

\\item[ii] \"L_gL^{r-1}_fh(x^0)\not= 0\",<\/p>\n\n\n\n

where \"L_fh(x)\" is the so-called derivative of \"h\" along \"f\", defined by<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[L_fh(x)=\sum_{i=1}^n\frac {\partial h}{\partial x_i}f_i(x).\]\"<\/p><\/p>\n\n\n\n

\\end{itemize}<\/p>\n\n\n\n

The derivative of order \"k\", \"L_f^k h(x)\", satisfies the recursion<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[L_f^k h(x)=\frac {\partial (L_f^{k-1}h)}{\partial x}f(x),\]\"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\"\[L_f^0 h(x)=h(x)\]\"<\/p><\/p>\n\n\n\n

<\/p>\n\n\n\n

Similarly, we have<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[L_gL_f h(x)=\frac {\partial (L_f h)}{\partial x}g(x).\]\"<\/p><\/p>\n\n\n\n

For the linear system<\/p>\n\n\n\n

<\/a>

(1) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}\dot x&=&Ax+Bu\\y&=&Cx\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

we have

  <\/span>   <\/span>\"\[f(x)=Ax, g(x)=B, h(x)=Cx\]\"<\/p><\/p>\n\n\n\n

and therefore

  <\/span>   <\/span>\"\[L_f^kh(x)=CA^kx,~~ L_gL_f^kh(x)=CA^kB.\]\"<\/p><\/p>\n\n\n\n

Thus, the integer \"r\" is characterized by the conditions

(2) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}CA^kB&=&0,{\rm for~all~}0\le k<r-1,\CA^{r-1}B\not= 0.\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

It can be shown that the integer \"r\" satisfying these conditions is exactly equal to the difference between the degree of the denominator polynomial and the degree of the numerator polynomial,

  <\/span>   <\/span>\"\[r=\deg (q(s))-\deg (p(s)),\]\"<\/p>
<\/p>\n\n\n\n

that is, the number of poles and the number of zeros, respectively, of the transfer function

  <\/span>   <\/span>\"\[G(s)=C(sI-A)^{-1}B=\frac {p(s)}{q(s)}\]\"<\/p><\/p>\n\n\n\n

of the linear system (1<\/a>).<\/p>\n\n\n\n

In particular, any linear system with \"n\"-dimensional state-space and relative degree \"r\", which is strictly less than \"n\", as zeros in its transfer function. If, on the contrary, \"r=n\", the transfer function has no zeros.<\/p>\n\n\n\n

We can also interpret the relative degree in the time domain as follows. Consider the SISO system (??) and assume that it is in state \"x(t_0)=x^0\" at time \"t_0\". Calculating the value of the output \"y(t)\" and of its derivatives \"y^{(k)}(t)\", \"k=1,2,…\", at \"t=t_0\", using definition 2.1, we obtain

(3) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}y(t_0)&=&h(x(t_0))=h(x^0),\\y'(t)&=&\frac {\partial h}{\partial x}\frac {dx}{dt}=\frac {\partial h}{\partial x}(f(x(t))+g(x(t))u(t))=L_fh(x(t))+L_gh(x(t))u(t).\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

If the relative degree \"r\" is larger than 1 for all \"t\" in a neighborhood of \"t_0\", we have \"L_gh(x(t))=0\" and therefore

  <\/span>   <\/span>\"\[y'(t)=L_fh(x(t)).\]\"<\/p><\/p>\n\n\n\n

The derivative of second order is

  <\/span>   <\/span>\"\[y''(t)=\frac {\partial (L_fh)}{\partial x}\frac {dx}{dt}=L_f^2h(x(t))+L_gL_fh(x(t))u(t),\]\"<\/p><\/p>\n\n\n\n

and if the relative degree is larger than 2 for all \"t\" in a neighborhood of \"t_0\", we have \"L_gL_fh(x(t))=0\", so that

  <\/span>   <\/span>\"\[y''(t)=L_f^2h(x(t)).\]\"<\/p><\/p>\n\n\n\n

In the \"k\"th step we have

(4) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}y^{(k)}(t)&=&L_f^kh(x(t)) {\rm ~~for~all~}k<r{\rm~and~all~}t{\rm~near}~t_0,\\y^{(r)}(t_0)&=&L_f^rh(x^0)+L_gL_f^{r-1}h(x^0)u(t_0).\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

Thus, the relative degree \"r\" is exactly equal to the number of times the output \"y(t)\" has to be differentiated at \"t=t_0\" in order to have the value \"u(t_0)\" of the input explicitly appearing in the expression for the differentiated output. If

  <\/span>   <\/span>\"\[L_gL_f^kh(x)=0\]\"<\/p><\/p>\n\n\n\n

for all \"k\ge 0\" and all \"x\" in a neighborhood of \"x^0\" (in which case no relative degree can be defined there), then the output is unaffected by the input for \"t\" close to \"t_0\" and<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[y(t)=\sum_{k=0}^{\infty} L_f^kh(x^0)\frac {(t-t_0)^k}{k!}\]\"<\/p><\/p>\n\n\n\n

is a function depending only on the initial state and not on the input.<\/p>\n\n\n\n

Now we consider the so-called normal form of a system, which we obtain by making a special coordinate trans-formation. In this form, it is easy to see some of the properties of solutions of certain control problems. From Isidori [5, pp.149-150], we state the following<\/p>\n\n\n\n

Proposition 2.1:<\/strong> Suppose that a SISO system has relative degree \"r\" at a point \"x^0\", where \"r\le n\". Set

(5) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}\phi_1(x)&=&h(x)\\\phi_2(x)&=&L_fh(x)\\&\vdots&\\\phi_r(x)&=&L_f^{r-1}h(x).\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

If \"r\" is strictly less than \"n\", it is always possible to find \"n-r\" more functions

  <\/span>   <\/span>\"\[\phi_{r+1}(x),…,\phi_n(x)\]\"<\/p><\/p>\n\n\n\n

such that the mapping

(6) <\/span>   <\/span>\"\begin{equation*}\Phi(x)=\left(\begin{array}{c}\phi_1(x)\\ \vdots\\ \phi_n(x)\end{array}\right)\end{equation*}\"<\/p><\/p>\n\n\n\n

has a Jacobian matrix which is nonsingular at \"x^0\" and therefore can be used as a coordinate transformation in a neighborhood of \"x^0\". The value at \"x^0\" of these additional functions can be fixed arbitrarily. Moreover, it is always possible to choose \"\phi_{r+1}(x),…,\phi_n(x)\" in such a way that<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[L_g\phi_i(x)=0\]\"<\/p><\/p>\n\n\n\n

for all \"r+1\le i \le n\" and all \"x\" in a neighborhood of \"x^0\".<\/p>\n\n\n\n

Now, letting
<\/p>\n\n\n

  <\/span>   <\/span>\"\[z_i=\phi_i(x), 1\leq i\leq n,\]\"<\/p>\n\n\n\n

we obtain for \"z_1,…,z_{r-1}\" <\/p>\n\n\n\n

(7) <\/span>   <\/span>\"\begin{equation*} \begin{array}{ccl} \frac {dz_1}{dt}&=\frac {\partial \phi_1}{\partial x}\frac {dx}{dt}= \frac {\partial h}{\partial x}\frac {dx}{dt}=L_fh(x(t))=\phi_2(x(t))=z_2(t)\\ &\vdots\\ \frac {dz_{r-1}}{dt}&=\frac {\partial \phi_{r-1}}{\partial x}\frac {dx}{dt}= \frac {\partial L_f^{r-2}h}{\partial x}\frac {dx}{dt}=L_f^{r-1}h(x(t))= \phi_r(x(t))=z_r(t). \end{array} \end{equation*}\"<\/p> <\/p>\n\n\n\n

For \"z_r\" we obtain <\/p>\n\n\n\n

<\/a>

(8) <\/span>   <\/span>\"\begin{equation*} \frac {dz_r}{dt}=L_f^rh(x(t))+L_gL_f^{r-1}h(x(t))u(t). \end{equation*}\"<\/p> <\/p>\n\n\n\n

We substitute<\/p>\n\n\n

  <\/span>   <\/span>\"\[x(t)=\Phi^{-1}(z(t))\]\"<\/p>\n\n\n\n

into the right-hand side of (8<\/a>) and let <\/p>\n\n\n\n

(9) <\/span>   <\/span>\"\begin{equation*} \begin{array}{ccl} a(z)&:=L_gL_f^{r-1}h(\Phi^{-1}(z))\\ b(z)&:=L_f^rh(\Phi^{-1}(z)) \end{array} \end{equation*}\"<\/p> <\/p>\n\n\n\n

so that we have <\/p>\n\n\n\n

(10) <\/span>   <\/span>\"\begin{equation*} \frac {dz_r}{dt}=b(z(t))+a(z(t))u(t), \end{equation*}\"<\/p> <\/p>\n\n\n\n

where, at the point \"z^0=\Phi(x^0)\", \"a(z^0)\not= 0\" by definition. Thus, the coefficient \"a(z)\" is nonzero for all \"z\" in a neighborhood of \"z^0\". Now we choose \"\phi_{r+1}(x),…,\phi_n(x)\" in such a way that <\/p>\n\n\n\n

  <\/span>   <\/span>\"\[L_g\phi_i(x)=0,r+1\leq i\leq n\]\"<\/p><\/p>\n\n\n\n

so that

(11) <\/span>   <\/span>\"\begin{equation*}\begin{array}{cl}\frac {dz_i}{dt}&=\frac {\partial \phi_i}{\partial x}(f(x(t))+g(x(t))u(t))=\\&=L_f\phi_i(x(t))+L_g\phi_i(x(t))u(t)=L_f\phi_i(x(t)).\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

Setting

  <\/span>   <\/span>\"\[q_i(z):=L_f\phi_i(\Phi^{-1}(z))\]\"<\/p><\/p>\n\n\n\n

for all \"r+1\le i\le n\", we can write

(12) <\/span>   <\/span>\"\begin{equation*}\frac {dz_i}{dt}=q_i(z(t)).\end{equation*}\"<\/p><\/p>\n\n\n\n

Since \"y=h(x)\", we immediately get the output \"y=z_1\" in the new coordinates. Now we define the vectors
<\/a>

(13) <\/span>   <\/span>\"\begin{equation*}\xi :=\left(\begin{array}{c} z_1\\ \vdots\\ z_r\\\end{array}\right),\eta :=\left(\begin{array}{c} z_{r+1}\\ \vdots\\ z_n\end{array}\right)\end{equation*}\"<\/p><\/p>\n\n\n\n

so that the normal form can be written
<\/a>

(14) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}\dot z_1&=z_2\\\dot z_2&=z_3\\&\vdots\\\dot z_{r-1}&=z_r\\\dot z_r&=b(\xi ,\eta )+a(\xi ,\eta )u\\\dot \eta &=q(\xi ,\eta )\\y&=z_1.\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

Note that it is always possible to choose arbitrarily the value at \"x^0\" of the last \"n-r\" coordinates. Therefore, without loss of generality, we can assume that \"\xi =0\" and \"\eta =0\" at \"x^0\". In order to have \"y(t)=0\" for all \"t\" we must have

  <\/span>   <\/span>\"\[\dot z_1(t)=\dot z_2(t)=…=\dot z_r(t)=0,\]\"<\/p><\/p>\n\n\n\n

that is, \"\xi (t)=0\" for all \"t\". Furthermore, the input \"u(t)\" must satisfy

  <\/span>   <\/span>\"\[0=b(0,\eta (t))+a(0,\eta (t))u(t),\]\"<\/p><\/p>\n\n\n\n

where \"a(0,\eta (t))\not= 0\" if \"\eta (t)\" is close to \"0\". The remaining dynamics<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[\dot \eta (t)=q(0,\eta (t)),\eta (0)=\eta^0\]\"<\/p><\/p>\n\n\n\n

are called the zero dynamics of the system.<\/p>\n\n\n\n

In order to interpret the concept of zero dynamics in the frequency domain and in the time domain, we consider a SISO linear system with relative degree \"r\" and transfer function
<\/a>

(15) <\/span>   <\/span>\"\begin{equation*}G(s)=K\frac {s^{n-r}+b_{n-r-1}s^{n-r-1}+…+b_1s+b_0}{s^n+a_{n-1}s^{n-1}+…+a_1+a_0}=K\frac {p(s)}{q(s)}\end{equation*}\"<\/p><\/p>\n\n\n\n

Suppose that the polynomials \"p(s)\" and \"q(s)\" are coprime, i.e. the polynomials do not have any common zeros, and consider the minimal realization
<\/a>

(16) <\/span>   <\/span>\"\begin{equation*}\begin{array}{cl}\dot x&=Ax+Bu\\y&=Cx\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

where
<\/a>

(17) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}A=&\left(\begin{array}{ccccc}0& 1& 0& \ldots & 0\\0& 0& 1& \ldots & 0\\\vdots & \vdots & \vdots &\ddots & \vdots \\-a_0& -a_1& -a_2& \ldots & -a_{n-1}\end{array}\right),\\B=&\left(\begin{array}{c}0\\ 0\\ \vdots \\0\\ K\end{array}\right),\\C=&\left(\begin{array}{cccccccc} b_0& b_1& \ldots & b_{n-r-1}& 1& 0& \ldots & 0\end{array}\right).\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

In the frequency domain we consider the equation

  <\/span>   <\/span>\"\[Y(s)=G(s)U(s),\]\"<\/p><\/p>\n\n\n\n

where \"Y(s)\" and \"U(s)\" are the Laplace transforms of the output \"y(t)\" and the input \"u(t)\", respectively. Thus, we have

  <\/span>   <\/span>\"\[G(s)=K\frac {p(s)}{q(s)}=\frac {Y(s)}{U(s)}\]\"<\/p><\/p>\n\n\n\n

Now we multiply the numerator and denumerator of the transfer function by an arbitrary nonzero variable \"x\", so that

  <\/span>   <\/span>\"\[K\frac {p(s)x}{q(s)x}=\frac {Y(s)}{U(s)},\]\"<\/p><\/p>\n\n\n\n

and we observe that \"Y(s)=0\" exactly when \"p(s)x=0\". Taking the inverse Laplace transform of

  <\/span>   <\/span>\"\[p(s)x=s^{n-r}x+b_{n-r-1}s^{n-r-1}x+…+b_1sx+b_0x=0\]\"<\/p><\/p>\n\n\n\n

and defining the new coordinates according to

(18) <\/span>   <\/span>\"\begin{equation*}\begin{array}{rccl}z_0&=&x\\z_1&=&\frac {dx}{dt}\\z_2&=&\frac {d^2x}{dt^2}\\&\vdots&\\z_{n-r-1}&=&\frac {d^{n-r-1}x}{dt^{n-r-1}}\\z_{n-r}&=&\frac {d^{n-r}x}{dt^{n-r}}=-b_0z_0-b_1z_1-…-b_{n-r-1}z_{n-r-1}\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

we have

(19) <\/span>   <\/span>\"\begin{equation*}\begin{array}{rccl}\dot z_0&=&z_1\\\dot z_1&=&z_2\\&\vdots&\\\dot z_{n-r-1}&=&z_{n-r}=\frac {d^{n-r}x}{dt^{n-r}}=-b_0z_0-b_1z_1-…-b_{n-r-1}z_{n-r-1}\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

and we can write the system as

  <\/span>   <\/span>\"\[\dot z=\bar{A}z,\]\"<\/p><\/p>\n\n\n\n

where
<\/a>

(20) <\/span>   <\/span>\"\begin{equation*}\bar{A}=\left(\begin{array}{rcccl}0& 1& 0& \ldots & 0\\0& 0& 1& \ldots & 0\\\vdots & \vdots & \ddots & \vdots \\-b_0& -b_1& -b_2& \ldots & -b_{n-r-1}\end{array}\right).\end{equation*}\"<\/p><\/p>\n\n\n\n

The eigenvalues of (20<\/a>) are exactly the zeros of the transfer function (15<\/a>). To see this, we transform \"(sI-\bar A)\" to triangular form, i.e. the matrix

  <\/span>   <\/span>\"\[\left(\begin{array}{ccccc}s& -1& 0& \ldots & 0\\0& s& -1& \ldots & 0\\\vdots & \vdots & \vdots &\ddots & \vdots \\b_0& b_1& b_2& \ldots & s+b_{n-r-1}\end{array}\right)\]\"<\/p><\/p>\n\n\n\n

is transformed to

  <\/span>   <\/span>\"\[\left(\begin{array}{ccccc}s& -1& 0& \ldots & 0\\0& s& -1& \ldots & 0\\\vdots & \vdots & \vdots &\ddots & \vdots \\0& 0& 0& \ldots & \alpha \end{array}\right),\]\"<\/p><\/p>\n\n\n\n

where

  <\/span>   <\/span>\"\[\alpha=s+b_{n-r-1}+\frac {b_{n-r-2}}{s}+\ldots+\frac {b_1}{s^{n-r-2}}+\frac {b_0}{s^{n-r-1}},\]\"<\/p><\/p>\n\n\n\n

from which we see that the determinant

  <\/span>   <\/span>\"\[\det (sI-\bar A)=s^{n-r-1}(s+b_{n-r-1}+\frac {b_{n-r-2}}{s}+\ldots+\frac {b_1}{s^{n-r-2}}+\frac {b_0}{s^{n-r-1}})=0\]\"<\/p><\/p>\n\n\n\n

is exactly the numerator polynomial \"p(s)\" of the transfer function (15<\/a>). This is the reason for the name zero dynamics; in fact, zero dynamics can be seen as a generalization of the transmission zeros of linear systems, as we show in the end of this section.<\/p>\n\n\n\n

Now we consider the time domain representation (16<\/a>) of a SISO linear system with
matrices given by (17<\/a>) and calculate its normal form. For the first \"r\" coordinates we have to take

(21) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}z_1&=&Cx=b_0x_1+b_1x_2+…+b_{n-r-1}x_{n-r}+x_{x-r+1}\\z_2&=&CAx=b_0x_2+b_1x_3+…+b_{n-r-1}x_{n-r+1}+x_{n-r+2}\\&\vdots&\\z_r&=&CA^{r-1}x=b_0x_r+b_1x_{r+1}+…+b_{n-r-1}x_{n-1}+x_n.\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

The remaining \"n-r\" coordinates are chosen so that the conditions of proposition 2.1 are satisfied. We can, for example, choose

(22) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}z_{r+1}&=&x_1\\z_{r+2}&=&x_2\\&\vdots&\\z_n&=&x_{n-r}.\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

This is an admissible choice since the corresponding transformation matrix \"z=\Phi (x)\" has a nonsingular Jacobian \"\partial \Phi /\partial x\". Thus, we have

(23) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}\dot z_1&=&z_2\\\dot z_2&=&z_3\\&\vdots&\\\dot z_r&=&R\xi +S\eta +Ku\\\dot \eta &=&P\xi +Q\eta\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

where \"R\" and \"S\" are row vectors, \"P\" and \"Q\" matrices of suitable dimensions and \"\xi\" and \"\eta\" are defined as in (13<\/a>). The zero dynamics of this system are

  <\/span>   <\/span>\"\[\dot \eta =Q\eta .\]\"<\/p><\/p>\n\n\n\n

According to our choice of the last \"n-r\" coordinates (i.e. the elements of \"\eta\"), we have

(24) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}\frac {dz_{r+1}}{dt}&=&\frac {dx_1}{dt}=x_2(t)=z_{r+2}(t)\\&\vdots&\\\frac {dz_{n-1}}{dt}&=&\frac {dx_{n-r-1}}{dt}=x_{n-r}(t)=z_n(t)\\\frac {dz_n}{dt}&=&\frac {dx_{n-r}}{dt}=x_{n-r+1}(t)=-b_0x_1(t)-…-b_{n-r-1}x_{n-r}(t)+z_1(t)=\\&=&-b_0z_{r+1}(t)-…-b_{n-r-1}z_n(t)+z_1(t),\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

where \"z_1=0\" since the output is zero. We see that \"Q=\bar A\", where \"\bar A\" is given by (20<\/a>).<\/p>\n\n\n\n

We can quite easily extend the concepts of relative degree and zero dynamics to the multi-input multi-output (MIMO) case. Consider the multivariable nonlinear system with the same number \"m\" of inputs and outputs
<\/a>

(25) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}\dot x&=&f(x)+\sum_{j=1}^m g_j(x)u_j\\y_1&=&h_1(x)\\&\vdots&\\y_m&=&h_m(x),\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

in which \"f(x), g_1(x),\ldots, g_m(x)\" are smooth vector fields, and \"h_1(x),\ldots, h_m(x)\" are smooth functions, defined on an open set of \"R^n\".<\/p>\n\n\n\n

Definition 2.2:<\/strong> A multivariable nonlinear system of the form (25<\/a>) is said to have a (vector) relative degree \"(r_1,…,r_m)\" at a point \"x^0\" if<\/p>\n\n\n\n

\\begin{itemize}
\\item[(i)] \"L_{g_j}L_f^k h_i (x)=0\" for all \"1\le j\le m\", \"1\le i\le m\", \"0\le k<r_i-1\", and for all \"x\"
in a neighborhood of \"x^0\".
\\item[(ii)] the \"m \times m\" matrix

(26) <\/span>   <\/span>\"\begin{equation*}P(x)=\left(\begin{array}{ccc}L_{g_1}L_f^{r_1-1}h_1(x)& \ldots & L_{g_m}L_f^{r_1-1}h_1(x)\\L_{g_1}L_f^{r_2-1}h_2(x)& \ldots & L_{g_m}L_f^{r_2-1}h_2(x)\\\vdots & \ddots & \vdots \\L_{g_1}L_f^{r_m-1}h_m(x)& \ldots & L_{g_m}L_f^{r_m-1}h_m(x)\end{array}\right)\end{equation*}\"<\/p><\/p>\n\n\n\n

is nonsingular at \"x=x^0\".<\/p>\n\n\n\n

Note that each integer \"r_i\" is associated with the \"i\"th output of the system. Furthermore, in a neighborhood of \"x^0\", the row vector

  <\/span>   <\/span>\"\[\left(\begin{array}{cccc}L_{g_1}L_f^kh_i(x)& L_{g_2}L_f^kh_i(x)& \ldots & L_{g_m}L_f^kh_i(x)\end{array}\right)\]\"<\/p><\/p>\n\n\n\n

is zero for all \"k<r_i-1\" and, for \"k=r_i-1\", this vector is nonzero (i.e. has at least one nonzero element). Thus, for each \"i\" there is at least one \"j\" such that the SISO system having output \"y_i\" and input \"u_j\" has exactly relative degree \"r_i\" at \"x^0\" and, for any other possible choice of \"j\", the corresponding relative degree at \"x^0\" – if any – is necessarily higher than or equal to \"r_i\". Similarly to the SISO case, \"r_i\" is the number of times \"y_i(t)\", the \"i\"th output vector, has to be differentiated at \"t=t_0\" in order to have at least one component of the input vector \"u(t_0)\" explicitly appearing.<\/p>\n\n\n\n

The normal form of a multi-input multi-output system can be written
<\/a>

(27) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}z&=&\left(\begin{array}{cccc}z^1& z^2& \ldots& z^m\end{array}\right)\\\dot z_1^i&=&z_2^i\\\dot z_2^i&=&z_3^i\\&\vdots&\\\dot z^i_{r_i-1}&=&z^i_{r_i}\\\dot z^i_{r_i}&=&b_i(z,\eta )+\sum_{j=1}^{m}a_{ij}(z,\eta )u_j\\\dot \eta &=&q(z,\eta )\\y_i&=&z_1^i,i=1,…,m,\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

where

(28) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}a_{ij}(z,\eta)&=&L_{g_j}L_f^{r_i-1}h_i(\Phi^{-1}(z,\eta)),{\rm for~all~}1\le i,j\le m,\\b_i(z,\eta)&=&L_f^{r_i}h_i(\Phi^{-1}(z,\eta)), {\rm for~all~}1\leq i\leq m.\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

The the zero dynamics of (27<\/a>) can be written
<\/a>

(29) <\/span>   <\/span>\"\begin{equation*}\dot \eta =q(0,\eta ).\end{equation*}\"<\/p><\/p>\n\n\n\n

However, instead of considering the conditions under which a transformation exists which transforms a system to its normal form, we will state a more general definition which is independent of the relative degree.<\/p>\n\n\n\n

Definition 2.3:<\/strong> Consider the system (25<\/a>) with equal number \"m\" of inputs and outputs. Let the state \"x\" be defined on an open set \"N\" of \"R^n\". Suppose that \"x^0\in N\", \"f(x^0)=0\", \"h(x^0)=0\" and that there exists a smooth connected submanifold <\/p>\n\n\n\n

\"Z^{\ast}\" of \"N\" which contains \"x^0\" and satisfies\\begin{itemize}
\\item[(i)] \"h(x)=0\quad \forall x\in Z^{\ast}\" ,
\\item[(ii)] at each point \"x\in Z^{\ast}\" there exists a unique \"u^{\ast}\in R^m\" such that \"f(x)+g(x)u^{\ast}\" is tangent to \"Z^{\ast}\".\\item[(iii)] \"Z^{\ast}\" is maximal with respect to (i) and (ii).\\end{itemize}
Then \"Z^{\ast}\" is called the zero dynamics manifold and

  <\/span>   <\/span>\"\[\dot x=f(x)+g(x)u^{\ast},\quad x\in Z^{\ast}\]\"<\/p>
is the zero dynamics of (25<\/a>).<\/p>\n\n\n\n

Even for multivariable systems we can interpret the zero dynamics as a nonlinear analogue of the “zeros” of a linear system. Consider the linear system
<\/a>

(30) <\/span>   <\/span>\"\begin{equation*}\begin{array}{ccl}\dot x&=&Ax+Bu\y&=&Cx.\end{array}\end{equation*}\"<\/p><\/p>\n\n\n\n

In the linear case, the conditions of definition 2.3 give the problem of finding the maximal controlled invariant subspace \"V^{\ast}\" such that \"y=Cx=0\". Clearly, we must have

  <\/span>   <\/span>\"\[V^{\ast}\subseteq \ker C.\]\"<\/p><\/p>\n\n\n\n

We are looking for a state feedback \"u^{\ast}=Fx\" such that \"Ax+Bu^{\ast}\" is tangent to \"V^{\ast}\", i.e. \"(A+BF)x\" is in \"V^{\ast}\" for all \"x\in V^{\ast}\". This is equivalent to the condition

  <\/span>   <\/span>\"\[AV^{\ast}\subseteq V^{\ast}+{\rm Im~}B.\]\"<\/p><\/p>\n\n\n\n

Thus, \"V^{\ast}\" is just the largest controlled invariant subspace of the system (30<\/a>). Now, making the additional assumptions

  <\/span>   <\/span>\"\[\dim ({\rm Im~}B)=m,\quad V^{\ast}\cap{\rm Im~}B=0,\]\"<\/p><\/p>\n\n\n\n

a unique control law \"u^{\ast}\" exists and we see that the largest controllability subspace \"R^{\ast}=0\" from the definition

  <\/span>   <\/span>\"\[R^{\ast}:=<A+BF|_{V^{\ast}}\cap \rm{Im~}B>.\]\"<\/p><\/p>\n\n\n\n

The restriction
<\/a>

(31) <\/span>   <\/span>\"\begin{equation*}(A+BF)\vert_{V^{\ast}}\end{equation*}\"<\/p><\/p>\n\n\n\n

identifies a linear dynamical system, defined on \"V^{\ast}\", whose dynamics are by definition the zero dynamics of the original system. Furthermore, the eigenvalues of (31<\/a>) coincide with the so-called transmission zeros of the system.<\/p>\n\n\n\n

To see this, consider the following proposition, see Wonham [2, pp. 111-112], from which the definition of transmission zeros readily follows.<\/p>\n\n\n\n

Proposition 2.2:<\/strong> Consider the system (30<\/a>). Let \"F\" be a map \"F:X\rightarrow U\" such that \"(A+BF)V^{\ast}\subseteq V^{\ast}\", where \"V^{\ast}\" is the largest controlled invariant subspace, and let \"R^{\ast}\" be the largest controllability subspace. Then

  <\/span>   <\/span>\"\[\sigma [(A+BF)\vert_{V^{\ast}}]=\sigma_F\cup \sigma^{\ast},\]\"<\/p><\/p>\n\n\n\n

where

  <\/span>   <\/span>\"\[\sigma_F:=\sigma [(A+BF)\vert_{R^{\ast}}]\]\"<\/p><\/p>\n\n\n\n

is freely assignable by suitable choice of \"F\", and

  <\/span>   <\/span>\"\[\sigma^{\ast}:=\sigma [(A+BF)\vert_{V^{\ast}/R^{\ast}}]\]\"<\/p><\/p>\n\n\n\n

is fixed for all \"F\". The fixed spectrum \"\sigma^{\ast}\" is called the set of transmission zeros of the linear system.<\/p>\n\n\n\n

If the numbers of inputs and outputs are not equal, we can drop the assumption that \"R^{\ast}\not= 0\" and simply
redefine the zero dynamics to be the dynamics of the system (30<\/a>) restricted to the subspace
<\/a>

(32) <\/span>   <\/span>\"\begin{equation*}(A+BF)\vert_{V^{\ast}/R^{\ast}},\end{equation*}\"<\/p><\/p>\n\n\n\n

which is the definition we will use in the sequel.<\/p>\n\n\n\n

Similarly to the SISO case, it can be shown that a MIMO system which has a relative degree \"(r_1,…,r_m)\" at the point \"x^0\" where \"r_1+…+r_m=n\" can be transformed into a fully linear and controllable system by means of feedback and transformation of coordinates. Such a system has a 0-dimensional zero dynamics manifold, i.e. has no zero dynamics, see Isidori [5, p. 382].<\/p>\n\n\n\n

We conclude this section with one more definition.<\/p>\n\n\n\n

Definition 2.4:<\/strong> If the eigenvalues of the zero dynamics (32<\/a>) lie off the imaginary axis, i.e. if

  <\/span>   <\/span>\"\[{\rm Re~}\sigma [(A+BF)\vert_{V^{\ast}/R^{\ast}}]\not= 0,\]\"<\/p><\/p>\n\n\n\n

then the zero dynamics are called hyperbolic.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this section we describe the basic features of zero dynamics. The zero dynamics can be seen as the remaining dynamics of a system when we steer the output to be identically zero. We will give a brief account for some definitions and results which can be found in e.g. Isidori [5]. First we interpret […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-176","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.mathgallery.com\/index.php\/wp-json\/wp\/v2\/pages\/176","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathgallery.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mathgallery.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathgallery.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathgallery.com\/index.php\/wp-json\/wp\/v2\/comments?post=176"}],"version-history":[{"count":36,"href":"https:\/\/www.mathgallery.com\/index.php\/wp-json\/wp\/v2\/pages\/176\/revisions"}],"predecessor-version":[{"id":432,"href":"https:\/\/www.mathgallery.com\/index.php\/wp-json\/wp\/v2\/pages\/176\/revisions\/432"}],"wp:attachment":[{"href":"https:\/\/www.mathgallery.com\/index.php\/wp-json\/wp\/v2\/media?parent=176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}