This article provides a detailed explanation of fundamental performance limitations in tracking control for SISO systems with non-minimum phase zeros, where the reference signal is defined as a general class rather than a specific waveform. 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, Performance Limitation of Tracking Control Problem for a Class of References, IEEE Transactions on Automatic Control, Vol. 56, No. 11, pp. 2723–2727 (2011)

Free PDF (Kumamoto University Repository)

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This paper is co-work with Prof. Toru Asai (Osaka University).

Contents

Why Performance Limitations Matter

When designing a control system, an engineer naturally asks: how well can my system possibly track a given reference signal? Even with the best controller, there are fundamental limits to tracking accuracy that arise from the plant's own characteristics — specifically, its non-minimum phase zeros (unstable zeros in the right half plane).

The classical results on performance limitations, such as those by Chen, Qiu, Toker (2000) and Su, Qiu, Chen (2003), analyze these limits for specific reference signals such as step inputs or sinusoidal signals. However, the results are tied to the particular waveform chosen, making it difficult to separate the contribution of the plant from that of the reference signal.

This paper takes a different approach: it defines a general class of reference signals and derives the tracking performance limitation for this entire class. The key result is a closed-form expression that clearly separates the roles of the plant (through a matrix determined by unstable zeros) and the reference signal (through a vector of evaluated values).

This kind of analysis is useful for:

Understanding which plants are inherently harder to control, regardless of the reference signal
Comparing the difficulty of tracking different reference signals for a fixed plant
Designing reference signals (e.g., choosing the phase of a sinusoidal component) to improve achievable tracking performance

Problem Formulation

Plant and Reference Signal

Consider a SISO plant with transfer function $P(s)$ , which has $m _{p}$ unstable zeros $z _{1}, \ldots, z _{m _{p}}$ in the open right half plane (ORHP). All unstable zeros are assumed to be distinct.

The reference signal belongs to the class:

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

where $\mathcal{N}$ is the set of strictly proper real rational functions. In other words, $\mathcal{R}$ consists of all rational Laplace transforms whose inverse Laplace transforms are bounded signals. This includes step functions, sinusoidal functions, decaying exponentials, and their linear combinations.

The reference signal $\hat{r}(s)$ may also have $m _{r}$ unstable zeros $z _{m _{p}+1}, \ldots, z _{m _{a}}$ , where $m _{a} = m _{p} + m _{r}$ . The total relative degree is $h _{a} = h _{p} + h _{r}$ .

Importantly, no further details about the reference signal's poles on the imaginary axis are assumed — neither their positions nor their number.

2DOF Control System and Performance Index

The paper considers a two-degree-of-freedom (2DOF) control system. The tracking error is $e(t) = y(t) - r(t)$ , and the performance index is:

$\displaystyle J = \inf_{C(s)} \lVert e \rVert_2^2$

The goal is to find the minimum $L _{2}$ norm of the tracking error over all internally stabilizing controllers.

Set of Achievable Outputs

Rather than parameterizing all stabilizing controllers (the standard Youla parameterization approach), this paper takes a different route: it parameterizes the set of all achievable output signals that can be produced by an internally stable system while tracking the reference asymptotically.

An output $\hat{y}(s)$ is achievable if and only if:

$\hat{y} - \hat{r} \in \mathcal{S}$ (the error is stable, ensuring asymptotic tracking)
The relative degree of $\hat{y}(s)$ is at least $h _{a}$
$\hat{y}(z _{i}) = 0$ for all $i = 1, \ldots, m _{a}$ (interpolation conditions at the unstable zeros)

These conditions are derived from the requirement that both $G(s)$ and $P(s)^{-1}G(s)$ belong to $\mathcal{S}$ , where $G(s) = \hat{y}(s)/\hat{r}(s)$ is the closed-loop transfer function.

An advantage of 2DOF systems is that the achievable set does not depend on the unstable poles of the plant. This is in contrast to 1DOF systems.

Parameterization of the Error Set (Theorem 1)

Instead of working with the output set directly, the paper parameterizes the error set $\mathcal{E}$ via $\hat{e}(s) = \hat{y}(s) - \hat{r}(s)$ .

Theorem 1 gives:

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

where $U(s)$ and $V(s)$ are explicitly constructed rational functions. They are built through the following recursion starting from initial functions $K^{(0)}(s)$ and $L^{(0)}(s)$ :

$\displaystyle K^{(0)}(s) = -\sum_{i=1}^{m_p} \hat{r}(z_i) \left(\frac{z_i + a}{s+a}\right)^{m_a - 1} \prod_{\mu=1, \mu \neq i}^{m_a} \frac{s - z_\mu}{z_i - z_\mu}$

$\displaystyle L^{(0)}(s) = \frac{\prod_{\ell=1}^{m_{a}}(s - z_{\ell})}{(s+a)^{m_{a}}}$

with the recursion:

$\displaystyle K^{(k+1)}(s) = K^{(k)}(s) + \alpha_{k} L^{(k)}(s), \quad L^{(k+1)}(s) = \frac{1}{s+a} L^{(k)}(s)$

where $\alpha _{k} = -\lim _{s \to \infty} s^{k} (\hat{r}(s) + K^{(k)}(s))$ , and $a > 0$ is an arbitrary positive constant. Finally, $U(s) = K^{(h _{a})}(s)$ and $V(s) = L^{(h _{a})}(s)$ .

The initial functions $K^{(0)}$ and $L^{(0)}$ handle the interpolation constraints at the unstable zeros, while the recursion handles the relative degree constraints. The parameterization structure is analogous to the KYJB parameterization of stabilizing controllers, but here it parameterizes signals rather than controllers.

Optimal Performance (Theorem 2)

Using the parameterization from Theorem 1, the minimization of the L2 norm of the error reduces to a standard model-matching problem. The solution yields a closed-form expression for the optimal performance.

Theorem 2: The optimal tracking performance is:

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

where $M$ is a matrix defined by:

$\displaystyle (M)_{k\ell} = \frac{\bar{q}_k q_\ell}{z_k + \bar{z}_\ell}, \quad q_i = \frac{\prod_{\mu=1}^{m_p}(z_i + \bar{z}_\mu)}{\prod_{\mu=1,\mu \neq i}^{m_p}(z_i - z_\mu)}$

and the vector $\rho$ is defined by:

$\displaystyle \rho = \begin{pmatrix} \hat{r}_c(z_1) \cr \vdots \cr \hat{r}_c(z_{m_p}) \end{pmatrix}, \quad \hat{r}_c(s) = \hat{r}(s) \prod_{\mu=m_p+1}^{m_a} \frac{s + \bar{z}_\mu}{s - z_\mu}$

The key insight of this result is:

The matrix $M$ depends only on the non-minimum phase zeros of the plant $P(s)$ . It characterizes the inherent difficulty of the plant.
The vector $\rho$ depends only on the reference signal (evaluated at the plant's unstable zeros, after conjugating the reference's own unstable zeros). It captures the interaction between the reference and the plant zeros.
The optimal performance is independent of the unstable poles of the plant (in the 2DOF case).

This separation is the main contribution of the paper: one can assess the plant difficulty (via $M$ ) and the reference difficulty (via $\rho$ ) independently.

If $P(s)$ has an unstable zero $z$ close to an imaginary-axis pole of the reference signal, $\hat{r} _{c}(z)$ becomes large, leading to poor tracking performance. This provides an intuitive understanding of why certain plant-reference combinations are fundamentally difficult.

Optimal Tracking Error (Theorem 3)

The paper also gives the explicit form of the optimal error signal.

Theorem 3: The optimal tracking error is:

$\displaystyle \hat{e}_{\mathrm{opt}}(s) = -\sum_{i=1}^{m_p} \hat{r}(z_i) \, q_i \frac{\prod_{\ell=1, \ell \neq i}^{m_a}(s - z_\ell)}{\prod_{\ell=1}^{m_a}(s + \bar{z}_\ell)}$

The optimal error depends only on the values $\hat{r}(z _{i})$ and the unstable zeros. No other details of $P(s)$ or $\hat{r}(s)$ are needed. This is a stronger result than what was available in earlier literature, where typically only the optimal norm — not the optimal signal itself — was derived.

Example: Choosing the Phase of a Reference Signal

To illustrate the usefulness of Theorem 2, the paper considers a plant with one unstable zero $z _{\alpha}$ and the reference signal:

$\displaystyle r(t) = \sin(t + \theta) + 5(1 - \exp(-100t)), \quad 0 \leq \theta \leq \pi/2$

The parameter $\theta$ can be freely chosen. The reference signal $\hat{r}(s)$ has no non-minimum phase zeros, so $\rho = \hat{r}(z _{\alpha})$ .

Using Theorem 2, one can directly compute how $J$ depends on $\theta$ :

When $z _{\alpha} \leq 1$ , the optimal choice is $\theta = 0$
When $z _{\alpha} > 1$ , the optimal choice is $\theta = \pi/2$

This demonstrates a practical application: by choosing the phase of a reference signal appropriately, one can improve the achievable tracking performance without changing the plant or the controller structure.

The paper also verifies consistency with the existing results of Su, Qiu, and Chen (2003) for the special case of trigonometric reference signals.

This work is part of a broader research program on analytical performance limitations and plant/reference interactions:

Extension to 1DOF Systems — H. 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). Extends the 2DOF results to one-degree-of-freedom systems, where the unstable poles of the plant also affect the performance limitation.

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, "Performance Limitation of Tracking Control Problem for a Class of References", IEEE Transactions on Automatic Control, Vol. 56, No. 11, pp. 2723–2727, 2011. DOI: 10.1109/TAC.2011.2159418. Free PDF. Support Page

Co-author: Toru Asai (Osaka University)

Conference version: H. Okajima and T. Asai, "Performance limitation of tracking control problem for a class of references", Proceedings of the 47th IEEE Conference on Decision and Control, pp. 3694–3699 (2008)

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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