This article provides a detailed explanation of tracking performance limitations for one-degree-of-freedom (1DOF) control systems, extending the achievable-output-set approach to systems where unstable poles also affect the fundamental limit. Related articles, related papers, and links are placed at the bottom.

Author: Hiroshi Okajima, Associate Professor, Kumamoto University, Japan — 20 years of control engineering research

This article is based on the following paper.

Okajima and T. Asai, Tracking Performance Limitation for 1-DOF Control Systems Using a Set of Attainable Outputs, SICE Journal of Control, Measurement, and System Integration, Vol. 8, No. 5, pp. 348–353 (2015) (Open Access)

This paper is co-work with Prof. Toru Asai (Professor, Chubu University).

Contents

Why 1DOF Performance Limitations Matter

In the companion paper (IEEE TAC 2011, also explained in this blog article), the tracking performance limitation was analyzed for 2DOF control systems with a general class of reference signals. The key result was that the optimal performance depends only on the plant's unstable zeros and is independent of the plant's unstable poles.

However, in many practical situations, we use 1DOF (single-degree-of-freedom) feedback controllers, where the same controller must handle both tracking and stability. In this case, the unstable poles of the plant also degrade the achievable tracking performance. The classical result by Chen, Qiu, and Toker (2000) analyzed the 1DOF limitation only for step references. For other reference signals such as sinusoidal inputs, the 1DOF limitation was not known.

This paper extends the 2DOF results to 1DOF systems for the same general class of reference signals. The main contributions are:

An explicit parameterization of all admissible error signals for 1DOF systems
A closed-form expression for the optimal tracking performance that clearly separates the contributions of the plant (unstable zeros and poles) and the reference signal
A quantitative characterization of how unstable poles worsen the performance compared to the 2DOF case

Problem Formulation

Plant and Reference Signal

Consider an SISO plant with transfer function $P(s)$ , which has $m _{p}$ unstable zeros $z _{1}, \ldots, z _{m _{p}}$ and $p$ unstable poles $p _{1}, \ldots, p _{p}$ in the open right half plane. All unstable poles and zeros are assumed to be distinct, and there is no unstable pole/zero cancellation. The relative degree of $P(s)$ is $h _{p}$ .

The reference signal belongs to the general class:

$\displaystyle \mathcal{R} = \{ \hat{r}(s) \in \mathcal{N} : \mathcal{L}^{-1}[\hat{r}(s)$ \in L_\infty \} ]

where $\mathcal{N}$ is the set of strictly proper real rational functions. This class includes step functions, sinusoidal functions, decaying exponentials, and their linear combinations — without specifying the waveform explicitly.

The reference signal $\hat{r}(s)$ may have $m _{r}$ unstable zeros, giving a total of $m _{a} = m _{p} + m _{r}$ unstable zeros and total relative degree $h _{a} = h _{p} + h _{r}$ .

1DOF Control System

The paper considers the 1DOF feedback system where the closed-loop transfer function from the reference to the output is:

$\displaystyle G(s) = \frac{P(s) C(s)}{1 + P(s) C(s)}$

The tracking error is $e(t) = y(t) - r(t)$ , and the performance index is:

$\displaystyle J = \inf \lVert e \rVert_2^2$

where the infimum is taken over all internally stabilizing controllers $C(s)$ .

Parameterization of the Admissible Error Set for 2DOF Systems (Review)

The starting point is the parameterization of the error set $\mathcal{E} _{s}$ for 2DOF systems, established in the prior work (Theorem 1 in the 2DOF paper). This set contains all error signals $\hat{e}(s)$ that can be produced by internally stable systems satisfying the tracking condition, without the constraint that the sensitivity function $P(s)(1 - G(s))$ must be stable.

The set $\mathcal{E} _{s}$ is parameterized as:

$\displaystyle \mathcal{E}_s = U_s(s) + V_s(s) \mathcal{S}$

where $U _{s}(s)$ and $V _{s}(s)$ are explicitly constructed stable proper rational functions. They encode the interpolation constraints at the unstable zeros and the relative degree constraints. For the explicit recursion formulas, see Theorem 1 in the paper or the 2DOF blog article.

Restriction to 1DOF Systems (Lemma 1 and Theorem 2)

For 1DOF systems, the error set $\mathcal{E}$ is a proper subset of $\mathcal{E} _{s}$ , because the additional stability condition $P(s)(1 - G(s)) \in \mathcal{S}$ imposes extra constraints. Specifically, the sensitivity function must have zeros at the unstable poles of $P(s)$ .

Lemma 1: An error signal $\hat{e}(s)$ belongs to $\mathcal{E}$ if and only if $\hat{e}(s) \in \mathcal{E} _{s}$ and

$\displaystyle \hat{e}(p_i) = 0, \quad i = 1, \ldots, p$

This condition means that the error signal must vanish at each unstable pole of the plant. These are additional interpolation constraints beyond those already imposed by the unstable zeros.

Using Lemma 1, the paper derives the explicit parameterization of $\mathcal{E}$ :

Theorem 2: The admissible error set for 1DOF systems is:

$\displaystyle \mathcal{E} = U(s) + V(s) \mathcal{S}$

where $U(s)$ and $V(s)$ are constructed from $U _{s}(s)$ and $V _{s}(s)$ by a second recursion that enforces the pole interpolation conditions. Specifically, starting from $\tilde{K}^{(0)}(s) = U _{s}(s)$ and $\tilde{L}^{(0)}(s) = V _{s}(s)$ , the recursion is:

$\displaystyle \tilde{K}^{(k+1)}(s) = \tilde{K}^{(k)}(s) + \beta_k \tilde{L}^{(k)}(s)$

$\displaystyle \tilde{L}^{(k+1)}(s) = \frac{s - p_{k+1}}{s + b} \tilde{L}^{(k)}(s)$

where $\beta _{k} = -\tilde{K}^{(k)}(p _{k+1}) / \tilde{L}^{(k)}(p _{k+1})$ and $b > 0$ is an arbitrary positive constant. After $p$ steps, $U(s) = \tilde{K}^{(p)}(s)$ and $V(s) = \tilde{L}^{(p)}(s)$ .

The parameterization depends only on the unstable poles and zeros (including infinite zeros) of $P(s)$ and $\hat{r}(s)$ , the values $\hat{r}(z _{i})$ , and high-frequency behavior of $\hat{r}(s)$ . It is independent of the neutrally stable poles of the reference signal and the choices of $a > 0$ and $b > 0$ .

Performance Limitation (Theorem 3)

With the parameterization of $\mathcal{E}$ established, the minimization of $\lVert e \rVert _{2}$ reduces to a standard H2 model-matching problem. The solution is given explicitly.

Theorem 3: The optimal tracking performance for 1DOF systems is:

$\displaystyle J = \rho^* M \rho$

where $M$ is the same matrix as in the 2DOF case, defined by the plant's unstable zeros:

$\displaystyle (M)_{ij} = \frac{\bar{w}_i \bar{q}_i q_j w_j}{\bar{z}_i + z_j}$

and the vector $\rho$ is:

$\displaystyle \rho = \begin{pmatrix} \sigma_{1} \hat{r}(z_{1}) \cr \vdots \cr \sigma_{m_{p}} \hat{r}(z_{m_{p}}) \end{pmatrix}$

The key quantities are:

$\displaystyle q_{i} = (z_{i} + \bar{z}_{i}) \prod_{n=1, n \neq i}^{m_{p}} \frac{z_{i} + \bar{z}_{n}}{z_{i} - z_{n}}$

$\displaystyle w_{i} = \prod_{n=1}^{p} \frac{z_{i} + \bar{p}_{n}}{z_{i} - p_{n}}$

$\displaystyle \sigma_{i} = \prod_{n=m_{p}+1}^{m_{a}} \frac{z_{i} + \bar{z}_{n}}{z_{i} - z_{n}}$

The crucial difference from the 2DOF result is the factor $w _{i}$ , which captures the interaction between the unstable zeros and unstable poles of the plant. Since $\lvert (z _{i} + \bar{p} _{n}) / (z _{i} - p _{n}) \rvert \geq 1$ always holds, the 1DOF performance is always worse than or equal to the 2DOF performance: $J \geq J _{s}$ .

The separation structure is preserved: $M$ depends only on the plant parameters (unstable zeros and poles), while $\rho$ depends on the reference signal evaluated at the plant's unstable zeros. The factor $w _{i}$ quantifies how much the unstable poles worsen the performance, and this degradation factor is independent of the choice of reference signal.

Numerical Example

The paper illustrates the result with a plant having one unstable zero $z > 0$ and one unstable pole $p > 0$ . In this case, Theorem 3 yields:

$\displaystyle J = 2z \left| \frac{z + p}{z - p} \right|^2 |\hat{r}(z)|^2$

For comparison, the 2DOF performance limitation is:

$\displaystyle J_s = 2z |\hat{r}(z)|^2$

The ratio $J / J _{s} = \lvert (z+p)/(z-p) \rvert^{2} > 1$ shows the exact penalty due to the unstable pole. This penalty factor depends only on the plant parameters and is independent of the reference signal.

For the sinusoidal reference $r(t) = \sin(t)$ with $z = 1$ :

Plant	Performance
Stable plant (z=1)	$J _{s} = 2 \lvert \hat{r}(1) \rvert^{2}$
Unstable (z=1, p=2)	$J = 18 \lvert \hat{r}(1) \rvert^{2}$
Unstable (z=1, p=10)	$J = \frac{242}{81} \lvert \hat{r}(1) \rvert^{2}$

The case $p = 10$ gives a smaller penalty than $p = 2$ because the unstable pole is farther from the unstable zero. When $p$ is far from $z$ , the factor $\lvert (z+p)/(z-p) \rvert^{2}$ approaches 1, and the 1DOF performance approaches the 2DOF performance.

This work is part of a broader research program on analytical performance limitations and the achievable-output-set approach:

2DOF Performance Limitation (Predecessor) — H. Okajima and T. Asai, Performance Limitation of Tracking Control Problem for a Class of References, IEEE Transactions on Automatic Control, Vol. 56, No. 11, pp. 2723–2727 (2011). Establishes the achievable-output-set approach and derives the performance limitation for 2DOF systems, where the result is independent of the plant's unstable poles. See also the blog article.

Disturbance Rejection Performance Limitation — H. Okajima, T. Asai and N. Matsunaga, Disturbance rejection performance limit for a class of disturbance signals, Proceedings of the IEEE Conference on Decision and Control 2015. Extends the achievable-output-set approach to disturbance rejection problems.

Model Error Compensator (MEC) — When the performance limitation analysis reveals that a plant is inherently difficult to control, the Model Error Compensator can be used to add robustness to existing control systems without redesigning the controller from scratch.

Blog Articles (blog.control-theory.com)

Research Web Pages (www.control-theory.com)

Publications / Model Error Compensator

Video

YouTube: Control Engineering Channel / Portal

Paper Information

Okajima and T. Asai, "Tracking Performance Limitation for 1-DOF Control Systems Using a Set of Attainable Outputs", SICE Journal of Control, Measurement, and System Integration, Vol. 8, No. 5, pp. 348–353, 2015. DOI: 10.9746/jcmsi.8.348 (Open Access)

Co-author: Toru Asai (Professor, Chubu University)

Predecessor paper (2DOF): H. Okajima and T. Asai, "Performance Limitation of Tracking Control Problem for a Class of References", IEEE Transactions on Automatic Control, Vol. 56, No. 11, pp. 2723–2727 (2011). Support Page

Self-Introduction

Hiroshi Okajima — Associate Professor, Graduate School of Science and Technology, Kumamoto University. Member of SICE, ISCIE, and IEEE.

www.control-theory.com / Blog / YouTube / GitHub

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