Disturbance Observer

The beneficial property (2) is obtained under the assumption that the transfer functions in (1) are stable for all possible uncertain plants $P(s)$. Therefore, it is necessary to design $Q(s)$ (or, choose $\tau$ and $a_{i}$’s) such that the closed-loop system remains stable for all possible $P(s)$.

In order to deal with uncertain plants having parametric uncertainty, consider the set ${\mathcal{P}}$ of uncertain transfer functions

where the intervals $[\underline{\alpha}_{i},\overline{\alpha}_{i}]$, $[\underline{\beta}_{i},\overline{\beta}_{i}]$, and $[\underline{g},\overline{g}]$ are known, and $\underline{g}>0$.

The value of $\tau^{*}$ can be computed from the knowledge of the bounds of the intervals in ${\mathcal{P}}$ (Shim and Jo, 2009), but it may also be conservatively chosen based on repeated simulations in practice. Smaller $\tau^{*}$ implies larger bandwidth of Q-filter, which is desired in the sense that the property (2) holds for larger frequency range.

The proof of Theorem 1 proceeds by observing the closed-loop system in Fig. 2 has $2n+m$ poles where $m$ is the order of $C(s)$. Then one can inspect the behavior of those poles by changing the design parameter $\tau$. For this, let us denote the poles by $\lambda_{i}(\tau)$, $i=1,\dots,2n+m$. Then, it can be shown that

where $\lambda_{i}^{*}$, $i=1,\dots,\nu$, are the roots of $p_{f}(s)$, and

where $\lambda_{i}^{*}$, $i=\nu+1,\dots,n$, are the zeros of $P(s)$, and $\lambda_{i}^{*}$, $n+1,\dots,2n+m$, are the poles of the nominal closed-loop with $P_{n}(s)$ and $C(s)$.

This argument shows that, if there is no $\lambda_{i}^{*}$ on the imaginary axis, the conditions (a), (b), and (c) are also necessary for robust stability with sufficiently small $\tau$. It is also seen by (3) that, when $p_{f}(s)$ is Hurwitz, the poles $\lambda_{i}(\tau)$, $i=1,\dots,\nu$, escapes to the negative infinity as $\tau\to 0$. Therefore, with large bandwidth of $Q(s)$, the closed-loop system shows two-time scale behavior. In fact, it turns out that $p_{f}(s)$ is the characteristic polynomial of the fast dynamics called ‘boundary-layer system’ in the singular perturbation analysis with $\tau$ being the parameter of singular perturbation.

In order to satisfy the condition (c) of Theorem 1, one can choose $\{a_{i}:i=1,\cdots,\nu-1\}$ such that

be a Hurwitz polynomial, and then pick $a_{0}>0$ sufficiently small. Then, the polynomial $p_{f}(s)$ remains Hurwitz for all variation of $g\in[\underline{g},\overline{g}]$.

This can be justified by, for example, the circle criterion. With $G(s):=a_{0}/(s^{\nu}+a_{\nu-1}s^{\nu-1}+\cdots+a_{1}s)$ and a static gain $(g/g^{*})$ that belongs to the sector $[\underline{g}/g^{*},\overline{g}/g^{*}]$, the characteristic polynomial of the closed-loop system becomes $s^{\nu}+a_{\nu-1}s^{\nu-1}+\cdots+a_{1}s+(g/g^{*})a_{0}$. Therefore, if the Nyquist plot of $G(s)$ does not enter nor encircle the closed disk in the complex plane whose diameter is the line segment connecting $-g^{*}/\underline{g}$ and $-g^{*}/\overline{g}$, then $p_{f}(s)$ is Hurwitz for all variation of $g\in[\underline{g},\overline{g}]$. Since $G(s)$ has one pole at the origin and the rest poles have negative real parts, its Nyquist plot is bounded to the direction of real axis. Therefore, by choosing $a_{0}$ sufficiently small, the Nyquist plot is disjoint from and does not encircle the disk.

DOB for nonlinear systems inherits all the ingredients and properties of the DOB for linear systems. The DOB can be constructed for a class of systems that can be represented by the Byrnes-Isidori normal form in certain coordinates as

where $u\in{\mathbb{R}}$ is the input, $y\in{\mathbb{R}}$ is the measured output, $x=[x_{1},\cdots,x_{\nu}]^{T}\in{\mathbb{R}}^{\nu}$ and $z\in{\mathbb{R}}^{n-\nu}$ are the states of the $n$-th order system, and unknown external disturbances are denoted by $d$ and $d_{z}$. The disturbances and their first time-derivatives are assumed to be bounded. The functions $f$ and $g$, and the vector field $h$ contain uncertainty. The corresponding nominal model of the plant is considered as

where $\bar{u}$ represents the input to the nominal model, and the nominal $f_{n}$, $g_{n}$, and $h_{n}$ are known. Let us assume all functions and vector fields are smooth.

The underlying idea of the DOB is that, if one can apply the input $u_{desired}$ to the plant $P$, which is generated by

then the plant $P$ with (4) behaves identical to the nominal model $P_{n}$ plus $\dot{z}=h(x,z,d_{z})$. Then, the plant $P$ with (4) interacts with the baseline controller $C$, and so, it is stabilized, while the plant’s zero dynamics $\dot{z}=h(x,z,d_{z})$ becomes stand-alone. However, the zero dynamics is ISS, and so, the state $z$ is bounded under the bounded inputs $x$ and $d_{z}$.

This idea is, however, not implementable because $d$, $f$, and $g$, as well as the states $x$ and $z$ in (4), are unknown. It turns out that the role of the DOB is to estimate the state $x$ and the signal $u_{desired}$. In the linear case, the structure of the DOB in Fig. 2 actually does this job (Shim et al, 2016).

The idea of estimating $x$ and $u_{desired}$ is realized in the semi-global sense. Suppose that $S_{0}\subset{\mathbb{R}}^{n+m}$ be a compact set for possible initial conditions $(x(0),z(0),\eta(0))$, and $U$ be a compact subset of region of attraction for the nominal closed-loop system $P_{n}$ and $C$, which is assumed to contain a compact set $S$ that is strictly larger than $S_{0}$. Let $S_{z}$ and $U_{x}$ be the projections of $S$ and $U$ to the $z$-axis and the $x$-axis, respectively, and let $M_{d}$ and $M_{d_{z}}$ be an upper bounds for the norms of $d$ and $d_{z}$, respectively. Uncertain $f$, $g$, and $h$ are supposed to belong to the sets ${\mathcal{F}}$, ${\mathcal{G}}$, and ${\mathcal{H}}$, respectively, which are defined as follows. Let ${\mathcal{H}}$ be a collection of uncertain $h$. Then there is a compact set $Z_{h}\subset{\mathbb{R}}^{n-\nu}$ to which the state $z(t)$ belongs, where $z(t)$ is the solution to each ISS system $\dot{z}=h(u_{1},z,u_{2})$, $h\in{\mathcal{H}}$, with any $z(0)\in S_{z}$ and any bounded inputs $u_{1}(t)\in U_{x}$ and $\|u_{2}(t)\|\leq M_{d_{z}}$. Assume that ${\mathcal{H}}$ is such that there is a compact set $Z\supset\cup_{h\in{\mathcal{H}}}Z_{h}$. The set ${\mathcal{F}}$ is a collection of uncertain $f$ such that, for every $f\in{\mathcal{F}}$, there are uniform bounds $M_{f}$ and $M_{df}$ such that $|f(x,z)|\leq M_{f}$ and $\|\partial f(x,z)/\partial(x,z)\|\leq M_{df}$ on $U_{x}\times Z$. The set ${\mathcal{G}}$ is a collection of uncertain $g$ such that, for every $g\in{\mathcal{G}}$, there are uniform bounds $\underline{g}$, $\overline{g}$, and $M_{dg}$ such that $\|\partial g(x,z)/\partial(x,z)\|\leq M_{dg}$ and

The DOB is illustrated in Fig. 3 and is given by

where

and $g^{*}$ is any constant between $\underline{g}$ and $\overline{g}$, $\bar{u}$ is the output of the baseline controller $C$, and two saturations are globally bounded, continuously differentiable functions such that ${\rm sat}_{x}(x)=x$ for all $x\in U_{x}$ and ${\rm sat}_{\phi}(\phi)=\phi$ for all $\phi\in S_{\phi}$ where the interval $S_{\phi}$ is given by

Suppose that $(\bar{z}(0),q(0),p(0))\in S_{0z}\times S_{qp}$ where $S_{0z}$ is the projection of $S_{0}$ to the $z$-axis, and $S_{qp}$ is any compact set in ${\mathbb{R}}^{2\nu}$.

In practice, choosing the sets like $U$, $Z$, and $S_{\phi}$ may not be easy. If so, conservative choice of them works. Based on repeated simulations or numerical computations, one can take sufficiently large compact sets for them.

If everything is linear, then the above controller $D$ becomes effectively the same as the state-space realization of the linear DOB in Fig. 2.

Under this assumption, the polynomial

is Hurwitz for all $(x,z)\in U_{x}\times Z$ and for all uncertain functions $g\in{\mathcal{G}}$, as discussed in the section ‘Design of $Q(s)$ for robust stability.’

To prove the theorem, one can convert the closed-loop system into the standard singular perturbation form with $\tau$ being the singular perturbation parameter. Then, it can be seen that the quasi-steady-state system on the slow manifold is simply the nominal closed-loop system $P_{n}$ and $C$ without external disturbance $d$ plus the actual zero dynamics $\dot{z}=h(x,z,d_{z})$. Since the quasi-steady-state system is assumed to be stable by Assumption 1, the overall system is stable with sufficiently small $\tau$ if the boundary-layer system is also stable. Then, it turns out that the boundary-layer system is linear since all the slow variables such as $x(t)$ and $z(t)$ are treated as frozen parameters, and the characteristic polynomial of the linear boundary-layer system is $p_{f}(s)$. For details, see (Back and Shim, 2008).

It can be also seen that, on the slow manifold, all the saturation functions in the DOB become inactive. The role of the saturations is to prevent the peaking phenomenon (Sussmann and Kokotovic, 1991) from transferring into the plant. Without saturations, the region of attraction may shrink in general as $\tau$ gets smaller, as in (Kokotovic and Marino, 1986), and only local stability is achieved. On the other hand, even if the plant is protected from the peaking components by the saturation functions, some internal components must peak for fast convergence of the DOB states. In this regard, the role of the dead-zone nonlinearity in Fig. 3 is to allow peaking inside the DOB structure.

Additional benefit of the DOB with saturation functions is robustness of transient response. If the baseline controller $C$ is designed for the nominal model $P_{n}$ to achieve desired transients such as small overshoot or fast settling time for example, then, similar transients can be obtained for the actual plant $P$ under external disturbances by adding the nonlinear DOB to the baseline controller. This holds true also for linear plants.

Since $y=x_{1}$, this theorem ensures robust transient response that $y(t)$ remains close to its nominal counterpart $y_{N}(t)$ from the initial time $t=0$. However, nothing can be said regarding the state $z(t)$ except that it is bounded by the ISS property of the zero dynamics.

Theorem 3 is basically an application of Tikhonov’s theorem (Hoppensteadt, 1966).