Multi-Objective Complementary Control

There are usually multiple objectives or specifications required to be achieved by various control systems in practice [1, 2, 3, 4, 5, 6, 7, 8, 9], such as optimality of tracking performance, robustness to some unknown/uncertain disturbances or parameter variations, passivity, etc. Various formulated multi-objective (MO) control problems have received considerable attention in the control community, for example, the MO $\mathcal{H}_{2}/\mathcal{H}_{\infty}$ control [10, 11, 12, 13, 14, 15, 4], where optimization of performance in the $\mathcal{H}_{2}$ norm and robustness in the $\mathcal{H}_{\infty}$ norm are considered simultaneously and are heavily studied in 1990’s. Other kinds of MO optimal control problems have also been extensively explored, for example, in [1, 3, 15, 7, 6], where the concept of Pareto optimality is used with respect to multiple optimization criteria.

It can be observed that the common feature of these traditional MO control problems is the fact that a single-controller structure is applied which poses challenges for the design methodologies, especially, when the involved objectives are inherently conflicting. A typical example is the well-known conflicting pair of robustness and optimal performance in a traditional robust control design. When conflicting objectives could not be achieved at their best by the single controller, the trade-off is normally the only choice, which results in a compromised single controller, as can be seen in many mixed $\mathcal{H}_{2}/\mathcal{H}_{\infty}$ control results [10, 11, 12, 14, 15, 4]. There are some research reported for controller designs in the two-degree-of-freedom (2DoF) form [16, 17, 18]. The reference [16] is concerned with 2DoF optimal design for a quadratic cost. In particular, the class of all stabilizing 2DoF controllers which give finite cost is characterized. In [17], the class of all stabilizing 2DoF controllers which achieve a specified closed-loop transfer matrix is characterized in terms of a free stable parameter. Inspired by the generalized internal model control (GIMC) developed in [19], a bi-objective high-performance control design for antenna servo systems is studied in [18]. Performance limitation problems are also studied for 2DoF controllers in [20, 21], with tracking and/or regulation as a sole objective. However, there is, essentially, a lack of a fundamental and general framework that can assemble two controllers together in a complementary way for MO control design purposes. That said, it is desired to develop a new framework for MO control problems that can overcome the curse of trade-offs and achieve non-compromised MO performances, which is the main goal of this paper.

This work is closely related to the GIMC structure [19], a multi-block implementation of the famous Youla-Kučera parameterization of all stabilizing 1DoF and 2DoF controllers [22]. In this implementation, the Youla-type parameter $\bm{Q}(s)$ becomes an explicit design factor driven by the residual signal, instead of functioning as an optional parameter to deliver a specific stabilizing controller. Although this GIMC structure provides hope for the desired two-controller complementary structure to address MO control problems, no further details are given for the systematic design of $\bm{Q}(s)$ in [19]. Motivated by the GIMC, in this paper a new two-controller design framework called ‘Multi-Objective Complementary Control’ (MOCC) is proposed and explored in detail with rigorous performance analysis, aiming to achieve MO performances while alleviating trade-offs. In particular, state-space formulas are provided for the MOCC framework. A tracking control setting is utilized as a platform to demonstrate the said advantages of MOCC, that is, overcoming the trade-off in the conventional MO control to achieve nominal tracking performance and robustness in a complementary way. Furthermore, a data-driven optimization approach is also sketched for the design of Youla-type operator $\bm{Q}$ to turn a robust controller into an optimal one, when performing the tracking task repetitively over a finite time interval in the presence of an unknown but repetition-invariant disturbance.

This paper is organized as follows: in Section II, the motivation of this paper is explained in detail using a tracking control case as well as some preliminaries of GIMC and Youla-Kučera parameterization; Section III presents a new two-controller structure which enables the MOCC design in the later section; in Section IV, a tracking control problem is addressed in the new framework with a rigorous performance analysis, illustrating the design features and the advantages of MOCC; the data-driven performance optimization is sketched in Section V; simulation results can be found in Section VI and conclusions in Section VII.

Notations: Throughout this paper, the symbol of transfer matrices or systems will be in bold to distinguish them from constant matrices. A system and its transfer function are denoted by the same symbol and whenever convenient, the dependence on the frequency variable $s$ or $j\omega$ for a transfer function may be omitted. The set $\mathbb{R}^{n}$ consists of all $n$-dimensional real vectors. The unsubscripted norm $\|\cdot\|$ denotes the standard Euclidean norm on vectors. For a matrix or vector $X$, $X^{\prime}$ denotes its transpose. For a rational transfer matrix $\bm{T}(s)$, $\bm{T}^{*}$ denotes $\bm{T}^{\prime}(-s)$ or complex conjugation of $\bm{T}(j\omega)$. $\mathcal{RL}_{\infty}$ ($\mathcal{RH}_{\infty}$) denotes the space of all rational (and stable) functions with the norm $\displaystyle{\|\bm{T}(s)\|_{\infty}=\sup_{\omega}\bar{\sigma}\{\bm{T}(j\omega)\}}$ for any $\bm{T}(s)\in\mathcal{RL}_{\infty}(\mathcal{RH}_{\infty})$, where $\bar{\sigma}\{\cdot\}$ represents the largest singular value. A rational proper transfer matrix in terms of state-space data is simply denoted by $\left[\begin{array}[]{c|c}A&B\\ \hline\cr C&D\end{array}\right]:=C(sI-A)^{-1}B+D$.

Transfer functions shall be utilized to motivate our idea, though this paper mainly works on state space. Consider a tracking control system shown in Fig. 1. Let $\bm{G}(s)=\bm{\tilde{M}}^{-1}(s)\bm{\tilde{N}}(s)$ be a linear time-invariant (LTI) plant with a left coprime factorization $\bm{\tilde{M}}(s)\in{\cal RH}_{\infty}$ and $\bm{\tilde{N}}(s)\in{\cal RH}_{\infty}$, $\bm{C}(s)=\bm{\tilde{V}}^{-1}(s)\bm{\tilde{U}}(s)$ be a stabilizing controller with a left coprime factorization $\bm{\tilde{V}}(s)\in{\cal RH}_{\infty}$ and $\bm{\tilde{U}}(s)\in{\cal RH}_{\infty}$, $\bm{r}(s)$ be a reference signal to be tracked, and $\bm{w}(s)$ be an uncertain or unknown disturbance signal. It is well-known that all stabilizing controllers can be characterized in the format of Youla-Kučera parameterization

for the same plant $\bm{G}(s)$, with $\bm{Q}(s)\in{\cal RH}_{\infty}$ and ${\rm det}(\bm{\tilde{V}}(\infty)-\bm{Q}(\infty)\bm{\tilde{N}}(\infty))\neq 0$ [19], [22, Chapter 5], [23]. The tracking error $\bm{e}(s)=\bm{r}(s)-\bm{y}(s)$ with either $\bm{C}(s)$ or $\bm{K}(s)$ is given by

It can be seen that the single controller $\bm{C}(s)$ (or $\bm{K}(s)$) has to be designed to deal with both the tracking performance for the reference signal $\bm{r}(s)$ and the robustness against the disturbance $\bm{w}(s)$ simultaneously. In practice, trade-off usually should be carried out to compromise the design difficulty of the single $\bm{C}(s)$ (or $\bm{K}(s)$) between the tracking performance and the disturbance attenuation. On the other hand, considering the challenge posed for the selection of $\bm{Q}(s)$ under a given performance expectation of $\bm{K}(s)$, one can conclude that finding a $\bm{K}(s)$ is not necessarily easier than designing $\bm{C}(s)$ when both performance and robustness need to be addressed.

An interesting controller architecture called GIMC was proposed in [19] which provides a promising approach to overcome the trade-off in the traditional design. The GIMC is shown in Fig. 2, featuring the parameter $\bm{Q}(s)\in{\cal RH}_{\infty}$ as a separate design factor in the structure, instead of being buried in $\bm{K}(s)$. By direct algebra, the tracking error $\bm{e}(s)$ in Fig. 2 can be derived as

It is clear from the above expression that different from the control structure in Fig. 1, there are potentially two separate design degrees of freedom for tracking control in GIMC: $\bm{C}(s)$ for tracking performance and $\bm{Q}(s)$ for disturbance attenuation. In addition, if there is no uncertainty, i.e., $\bm{w}=0$, then $\bm{e}=(I+\bm{G}\bm{C})^{-1}\bm{r}$ and thus the nominal tracking error signal is recovered. These attractive features motivate us to revisit MO control problems through the design of $\bm{C}(s)$ and $\bm{Q}(s)$, instead of the traditional design of a single controller $\bm{C}(s)$ or $\bm{K}(s)$. Noting that, for a given $\bm{C}(s)$, if $\bm{Q}(s)$ can be determined then $\bm{K}(s)$ is derived and vice verse. Hence, design of $\bm{C}(s)$ and $\bm{Q}(s)$ can be reduced to the design of $\bm{C}(s)$ and $\bm{K}(s)$ even if $\bm{K}(s)$ is not explicitly seen in Fig 2. It is noted that how to coordinate $\bm{C}(s)$ and $\bm{K}(s)$ using $\bm{Q}(s)$ such that specified performances are achieved and rigorous performance analysis for GIMC have not systematically and profoundly been investigated so far. In [24], a solution to $\bm{Q}(s)$ is derived in state space for a robust LQG control problem through the $\mathcal{H}_{\infty}$ design with a modified control structure from GIMC, although it is still not clear how to systematically use $\bm{Q}(s)$ to coordinate $\bm{C}(s)$ and $\bm{K}(s)$ in the Youla-Kučera parameterization.

In the following sections of this paper, a two-controller structure $\bm{K_{CQ}}$ is proposed in state space, motivated from the analysis above. In particular, how $\bm{Q}$ can be constructed from two independently designed controllers $\bm{C}$ and $\bm{K}$ to coordinate $\bm{C}$ and $\bm{K}$ is explored systematically. To showcase the superior performance of this new control design framework, a tracking control problem is presented to demonstrate how $\bm{C}$ and $\bm{K}$ can be coordinated through $\bm{Q}$ to address MO tracking performance and robustness in a complementary way, together with a rigorous performance analysis.

This section presents a new stabilizing control structure motivated by GIMC. Consider the following finite-dimensional linear time-invariant (FDLTI) system

where $x\in\mathbb{R}^{n}$, $u\in\mathbb{R}^{m_{2}}$ and $y\in\mathbb{R}^{p_{2}}$ are the state, control input, and sensor output, respectively. All matrices are real constant and have compatible dimensions. The following assumption is required throughout the remainder of this paper.

A full-order Luenberger observer for system (3) can be given by

where $f=C_{2}\hat{x}-y$ is the residual signal and $L$ is any matrix such that $A+LC_{2}$ is stable. Note that the Luenberger observer (4) is very related to the GIMC structure in Fig. 2. Specifically, by letting $\bm{G}(s):=\left[\begin{array}[]{c|c}A&B_{2}\\ \hline\cr C_{2}&0\end{array}\right]$ in the GIMC structure, a state-space realization of $\bm{\tilde{M}}(s)$ and $\bm{\tilde{N}}(s)$ can be obtained as

and it is not difficult to observe that $\bm{f}(s)=\bm{\tilde{N}}(s)\bm{u}(s)-\bm{\tilde{M}}(s)\bm{y}(s)$ is the residual signal generated by the Luenberger observer (4).

For the system $\bm{G}$ in (3), consider a dynamic output-feedback stabilizing controller $\bm{C}$:

Clearly, $(A_{c},B_{c},C_{c})$ is stabilizable and detectable, since $\bm{C}$ stabilizes $\bm{G}$ if and only if $\bm{G}$ stabilizes $\bm{C}$ [22, Section 5.1]. Then there exists a matrix $L_{c}$, such that $A_{c}+L_{c}C_{c}$ is stable. Motivated from the GIMC structure in Fig. 2, we present the following composite controller $\bm{K_{CQ}}$:

based on the observer (4) and the controller ${\bm{C}}$ in (7), where ${\bm{Q}}(\cdot)$ is a stable dynamic operator driven by the residual signal $f$ to be designed in the following form:

with $A_{q}$ stable.

The significant feature of the composite controller $\bm{K_{CQ}}$ is that if the output signal $u_{q}$ of ${\bm{Q}}$ is zero, then $u=u_{c}$ and $\bm{K_{CQ}}$ is the same as the controller $\bm{C}$ in (7). On the other hand, if $u_{q}\neq 0$, then the composite controller $\bm{K_{CQ}}$ behaves like a different one. Therefore, $\bm{K_{CQ}}$ can be viewed as a two-controller structure to stabilize the system $\bm{G}$ in (3), as shown in Fig. 4.

Now consider another dynamic output-feedback stabilizing controller ${\bm{K}}$ designed in the following form:

Then we give the solution of $\bm{Q}$ in state space such that $\bm{K_{CQ}}=\bm{K}$ when $u_{q}\neq 0$, which is presented in the following proposition.

In the sequel, two interesting cases: shared observer and static stabilizing controllers, are considered.

In general, a typical MO control problem specifies different objectives on different channels of the closed-loop system [10, 13, 14, 2, 15, 4, 7]. The objectives under consideration can be roughly divided into two classes: performance involving commands and robustness against unknown disturbances. Performance is meant for transient process, tracking accuracy, passivity requirement [26], etc, while robustness is meant to keep healthy performance in an uncertain or partially unknown environment. It has been pointed out in the Introduction that the traditional design technique using a single controller to address multiple objectives of the closed-loop system usually renders trade-off solutions with compromised performance and robustness in some sense, and for 2DoF controllers there does not exist a general and systematic design method so far to address both performance and the robustness. In contrast, we shall show that MO control problems can be handled effectively by applying the two-controller structures presented in the previous section with two independently designed controllers operating in a naturally complementary way, leading to a MO complementary control (MOCC) framework.

In the sequel, a specific robust tracking control problem is presented to demonstrate the advantages of the proposed MOCC framework. The MO tracking problem under consideration has two objectives: tracking performance and robustness. In the new control structure, $\bm{C}$ is designed to address the tracking performance without disturbances, and $\bm{K}$ is designed to address the robustness with respect to unknown/uncertain disturbances.

Consider a perturbed FDLTI system (3), described by

where $x\in\mathbb{R}^{n}$ is the system state, $u\in\mathbb{R}^{m_{2}}$ is the control input, $y\in\mathbb{R}^{p_{2}}$ is the measured output, $w\in\mathbb{R}^{m_{1}}$ is the unknown disturbance input, $r\in\mathbb{R}^{p_{2}}$ known or measurable reference signal, and $z\in\mathbb{R}^{p_{1}}$ consisting of tracking error and control input is an output variable evaluating the tracking performance.

Assume that an admissible tracking controller ${\bm{C}}$ is designed as the following form

for the system in (72) with $w=0$, i.e.,

as shown in Fig. 7.

On the other hand, assume that a robust controller ${\bm{K}}$ having the form of

is designed for the system in (72) with $r=0$, i.e,

as shown in Fig. 8.

Then by applying the two-controller structure in Fig. 4, a MO tracking controller $\bm{K_{CQ}^{T}}$ (the superscript “$T$” stands for tracking) can be obtained shown in Fig. 9, where the state-space model of $\bm{Q}$ is constructed as that in Proposition 1:

with $(A_{q},B_{q},C_{q},D_{q})$ given by (23). Note that the difference between the controller $\bm{K_{CQ}}$ in Fig. 4 and the tracking controller $\bm{K_{CQ}^{T}}$ in Fig. 9 is that for $\bm{K_{CQ}^{T}}$, the reference signal $r$ is also as an input to generate the control signal $u_{c}$. Then it follows from Proposition 1 that for the tracking control system with $\bm{K_{CQ}^{T}}$, the transfer matrix from $y$ to $u$ is $\bm{K}(s)$.

Next, we shall conduct a performance analysis for the resulting tracking controller $\bm{K_{CQ}^{T}}$. It will be shown that the tracking controller $\bm{K_{CQ}^{T}}$ consisting of ${\bm{C}}$ for the nominal tracking performance and ${\bm{K}}$ for the robustness leads to a decoupled design and the two controllers ${\bm{C}}$ and ${\bm{K}}$ operate in a complementary way.