Zero Dynamics

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.

Definition 2.1: The single-input single-output nonlinear system

    \[\dot x&=&f(x)+g(x)u\]

    \[y&=&h(x)\]

is said to have relative degree r at a point x^0 if

\begin{itemize}

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

\item[ii] L_gL^{r-1}_fh(x^0)\not= 0,

where L_fh(x) is the so-called derivative of h along f, defined by

    \[L_fh(x)=\sum_{i=1}^n\frac {\partial h}{\partial x_i}f_i(x).\]

\end{itemize}

The derivative of order k, L_f^k h(x), satisfies the recursion

    \[L_f^k h(x)=\frac {\partial (L_f^{k-1}h)}{\partial x}f(x),\]

    \[L_f^0 h(x)=h(x)\]

Similarly, we have

    \[L_gL_f h(x)=\frac {\partial (L_f h)}{\partial x}g(x).\]

For the linear system

(1)   \begin{equation*}\begin{array}{ccl}\dot x&=&Ax+Bu\\y&=&Cx\end{array}\end{equation*}

we have

    \[f(x)=Ax, g(x)=B, h(x)=Cx\]

and therefore

    \[L_f^kh(x)=CA^kx,~~ L_gL_f^kh(x)=CA^kB.\]

Thus, the integer r is characterized by the conditions

(2)   \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*}

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,

    \[r=\deg (q(s))-\deg (p(s)),\]


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

    \[G(s)=C(sI-A)^{-1}B=\frac {p(s)}{q(s)}\]

of the linear system (1).

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.

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)   \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*}

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

    \[y'(t)=L_fh(x(t)).\]

The derivative of second order is

    \[y''(t)=\frac {\partial (L_fh)}{\partial x}\frac {dx}{dt}=L_f^2h(x(t))+L_gL_fh(x(t))u(t),\]

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

    \[y''(t)=L_f^2h(x(t)).\]

In the kth step we have

(4)   \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*}

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

    \[L_gL_f^kh(x)=0\]

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

    \[y(t)=\sum_{k=0}^{\infty} L_f^kh(x^0)\frac {(t-t_0)^k}{k!}\]

is a function depending only on the initial state and not on the input.

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

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

(5)   \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*}

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

    \[\phi_{r+1}(x),…,\phi_n(x)\]

such that the mapping

(6)   \begin{equation*}\Phi(x)=\left(\begin{array}{c}\phi_1(x)\\ \vdots\\ \phi_n(x)\end{array}\right)\end{equation*}

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

    \[L_g\phi_i(x)=0\]

for all r+1\le i \le n and all x in a neighborhood of x^0.

Now, letting

    \[z_i=\phi_i(x), 1\leq i\leq n,\]

we obtain for z_1,…,z_{r-1}

(7)   \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*}

For z_r we obtain

(8)   \begin{equation*} \frac {dz_r}{dt}=L_f^rh(x(t))+L_gL_f^{r-1}h(x(t))u(t). \end{equation*}

We substitute

    \[x(t)=\Phi^{-1}(z(t))\]

into the right-hand side of (8) and let

(9)   \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*}

so that we have

(10)   \begin{equation*} \frac {dz_r}{dt}=b(z(t))+a(z(t))u(t), \end{equation*}

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

    \[L_g\phi_i(x)=0,r+1\leq i\leq n\]

so that

(11)   \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*}

Setting

    \[q_i(z):=L_f\phi_i(\Phi^{-1}(z))\]

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

(12)   \begin{equation*}\frac {dz_i}{dt}=q_i(z(t)).\end{equation*}

Since y=h(x), we immediately get the output y=z_1 in the new coordinates. Now we define the vectors

(13)   \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*}

so that the normal form can be written

(14)   \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*}

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

    \[\dot z_1(t)=\dot z_2(t)=…=\dot z_r(t)=0,\]

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

    \[0=b(0,\eta (t))+a(0,\eta (t))u(t),\]

where a(0,\eta (t))\not= 0 if \eta (t) is close to 0. The remaining dynamics

    \[\dot \eta (t)=q(0,\eta (t)),\eta (0)=\eta^0\]

are called the zero dynamics of the system.

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

(15)   \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*}

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

(16)   \begin{equation*}\begin{array}{cl}\dot x&=Ax+Bu\\y&=Cx\end{array}\end{equation*}

where

(17)   \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*}

In the frequency domain we consider the equation

    \[Y(s)=G(s)U(s),\]

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

    \[G(s)=K\frac {p(s)}{q(s)}=\frac {Y(s)}{U(s)}\]

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

    \[K\frac {p(s)x}{q(s)x}=\frac {Y(s)}{U(s)},\]

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

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

and defining the new coordinates according to

(18)   \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*}

we have

(19)   \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*}

and we can write the system as

    \[\dot z=\bar{A}z,\]

where

(20)   \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*}

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

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

is transformed to

    \[\left(\begin{array}{ccccc}s& -1& 0& \ldots & 0\\0& s& -1& \ldots & 0\\\vdots & \vdots & \vdots &\ddots & \vdots \\0& 0& 0& \ldots & \alpha \end{array}\right),\]

where

    \[\alpha=s+b_{n-r-1}+\frac {b_{n-r-2}}{s}+\ldots+\frac {b_1}{s^{n-r-2}}+\frac {b_0}{s^{n-r-1}},\]

from which we see that the determinant

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

is exactly the numerator polynomial p(s) of the transfer function (15). 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.

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

(21)   \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*}

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

(22)   \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*}

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)   \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*}

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

    \[\dot \eta =Q\eta .\]

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

(24)   \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*}

where z_1=0 since the output is zero. We see that Q=\bar A, where \bar A is given by (20).

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

(25)   \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*}

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.

Definition 2.2: A multivariable nonlinear system of the form (25) is said to have a (vector) relative degree (r_1,…,r_m) at a point x^0 if

\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)   \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*}

is nonsingular at x=x^0.

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

    \[\left(\begin{array}{cccc}L_{g_1}L_f^kh_i(x)& L_{g_2}L_f^kh_i(x)& \ldots & L_{g_m}L_f^kh_i(x)\end{array}\right)\]

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 ith 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.

The normal form of a multi-input multi-output system can be written

(27)   \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*}

where

(28)   \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*}

The the zero dynamics of (27) can be written

(29)   \begin{equation*}\dot \eta =q(0,\eta ).\end{equation*}

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.

Definition 2.3: Consider the system (25) 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

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

    \[\dot x=f(x)+g(x)u^{\ast},\quad x\in Z^{\ast}\]


is the zero dynamics of (25).

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

(30)   \begin{equation*}\begin{array}{ccl}\dot x&=&Ax+Bu\y&=&Cx.\end{array}\end{equation*}

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

    \[V^{\ast}\subseteq \ker C.\]

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

    \[AV^{\ast}\subseteq V^{\ast}+{\rm Im~}B.\]

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

    \[\dim ({\rm Im~}B)=m,\quad V^{\ast}\cap{\rm Im~}B=0,\]

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

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

The restriction

(31)   \begin{equation*}(A+BF)\vert_{V^{\ast}}\end{equation*}

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) coincide with the so-called transmission zeros of the system.

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

Proposition 2.2: Consider the system (30). 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

    \[\sigma [(A+BF)\vert_{V^{\ast}}]=\sigma_F\cup \sigma^{\ast},\]

where

    \[\sigma_F:=\sigma [(A+BF)\vert_{R^{\ast}}]\]

is freely assignable by suitable choice of F, and

    \[\sigma^{\ast}:=\sigma [(A+BF)\vert_{V^{\ast}/R^{\ast}}]\]

is fixed for all F. The fixed spectrum \sigma^{\ast} is called the set of transmission zeros of the linear system.

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) restricted to the subspace

(32)   \begin{equation*}(A+BF)\vert_{V^{\ast}/R^{\ast}},\end{equation*}

which is the definition we will use in the sequel.

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].

We conclude this section with one more definition.

Definition 2.4: If the eigenvalues of the zero dynamics (32) lie off the imaginary axis, i.e. if

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

then the zero dynamics are called hyperbolic.