A command governor approach to set-theoretic model reference adaptive control for enforcing partially adjustable performance guarantees

Abstract

In feedback control, the presence of system uncertainties cause system state trajectories to deviate from their ideal responses. In practice, a subset of these trajectories can be more critical than the rest due to physical and/or performance characteristics associated with a problem of interest. Hence, it is desired not only to have performance guarantees on the entire system state trajectories but also to be able to adjust the resulting worst-case performance bound for that critical subset. Yet, in model reference adaptive control of uncertain dynamical systems, assigning performance bounds on a subset of system trajectories is not trivial. This paper addresses this gap by proposing a new control architecture that has the capability to enforce a user-defined performance bound on the selected subset of dynamical system trajectories, entitled as partially adjustable performance guarantees. The proposed architecture is predicated on a set-theoretic treatment and utilizes a two-level constructive design framework. In particular, we first form an auxiliary state dynamics in order to construct the auxiliary system error vector between uncertain dynamical system states and this auxiliary dynamics states. This construction aids a control designer to weigh each element of the auxiliary system error vector independently, while enforcing performance bounds on the norm of this error vector. Then, a command governor mechanism is designed for driving a feasible user-selected subset of system states to a close (and user-controllable) neighborhood of the corresponding reference model states. This results in adjustable performance guarantees on a subset of system error trajectories.

Appendix: Necessary definitions

For completeness, this appendix presents two key definitions used in our main results.

Definition 1

(Projection operator) Let a convex hypercube in ${\mathbb {R}}^n$ be defined as $\Omega = \big \{\theta \in {\mathbb {R}}^n : (\theta ^\mathrm {min}_i \le \theta _i \le \theta ^\mathrm {max}_i )_{i=1,2,\ldots ,n}\big \}$, where $(\theta ^\mathrm {min}_i,$$\theta ^\mathrm {max}_i)$ represent the minimum and maximum bounds for the ith component of the n-dimensional parameter vector $\theta$. Furthermore, let $\Omega _\nu = \big \{\theta \in {\mathbb {R}}^n : (\theta ^\mathrm {min}_i + \nu \le \theta _i \le$$\theta ^\mathrm {max}_i - \nu )_{i=1,2,\ldots ,n}\big \}$ be the second hypercube for a sufficiently small positive constant $\nu$, where $\Omega _\nu \subset \Omega$. The definition of the projection operator ${\mathrm {Proj}}:{\mathbb {R}}^n \times {\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$ is then component-wise given by

$$\begin{aligned} \mathrm {Proj}(\theta ,y) \triangleq \left\{ \begin{array}{l@{\quad }l} \left( \frac{\theta ^\mathrm {max}_i - \theta _i}{\nu }\right) y_i, &{} \text {if } \theta _i> \theta ^\mathrm {max}_i - \nu \text { and } y_i > 0,\\ \left( \frac{\theta _i - \theta ^\mathrm {min}_i}{\nu }\right) y_i, &{} \text {if } \theta _i< \theta ^\mathrm {min}_i + \nu \text { and } y_i < 0, \\ y_i, &{} \text {otherwise }, \end{array}\right. \end{aligned}$$

(A.1)

where $y \in {\mathbb {R}}^n$ [16].

Based on Definition 1, it is well-known that

$$\begin{aligned} \left( \theta -\theta ^{*}\right) ^\mathrm {T}\left( \mathrm {Proj} \left( \theta ,y\right) -y\right) \le 0, \end{aligned}$$

(A.2)

holds (see [16, 38] for details). One can also generalize (A.2) to matrices using $\mathrm {Proj}_\mathrm {m}(\Theta , Y) = \bigl (\mathrm {Proj}(\mathrm {col}_{1}(\Theta ), \mathrm {col}_{1}(Y)),\ldots ,$$\mathrm {Proj}(\mathrm {col}_{m}(\Theta ),$$\mathrm {col}_{m}(Y))\bigl )$, where $\Theta \in {\mathbb {R}}^{n \times m}$, $Y\in {\mathbb {R}}^{n \times m}$, and $\mathrm {col}_{i}(\cdot )$ denotes ith column operator. In this case $\mathrm {tr} \ \bigl [(\Theta - \Theta ^{*})^\mathrm {T}(\mathrm {Proj}_\mathrm {m}(\Theta , Y) - Y)\bigl ]= \sum ^{m}_{i=1}\bigl [\mathrm {col}_{i}(\Theta - \Theta ^{*})^\mathrm {T}$$(\mathrm {Proj}(\mathrm {col}_{i}(\Theta ), \mathrm {col}_{i}(Y)) - \mathrm {col}_{i}(Y))\bigl ] \le 0$ follows from (A.2).

Definition 2

(Generalized restricted potential function) For $z \in {\mathbb {R}}^p$ and $H\in {\mathbb {R}}^{p \times p}_{+}$, $\phi (\Vert z\Vert _\mathrm {H})$, $\phi : {\mathbb {R}}\rightarrow {\mathbb {R}}$, is called a generalized restricted potential function (generalized barrier Lyapunov function) on the set

$$\begin{aligned} \mathcal {D}_\epsilon \triangleq \{z : \Vert z\Vert _\mathrm {H} \in [0,\epsilon ) \}, \end{aligned}$$

(A.3)

where $\epsilon \in {\mathbb {R}}_+$ is a-priori user-defined constant, when the next statements hold [5]:

• (i) If $\Vert z\Vert _\mathrm {H}=0$, then $\phi (\Vert z\Vert _\mathrm {H})=0$.

• (ii) If $z \in \mathcal {D}_\epsilon \text { and } \Vert z\Vert _\mathrm {H}\ne 0$, then $\phi (\Vert z\Vert _\mathrm {H})> 0$.

• (iii) If $\Vert z\Vert _\mathrm {H}\rightarrow \epsilon$, then $\phi (\Vert z\Vert _\mathrm {H})\rightarrow \infty$.

• (iv) $\phi (\Vert z\Vert _\mathrm {H})$ is continuously differentiable on $\mathcal {D}_\epsilon$.

• (v) If $z \in \mathcal {D}_\epsilon$, then $\phi _d (\Vert z\Vert _\mathrm {H}) >0$, where

  $$\begin{aligned} \phi _d (\Vert z\Vert _\mathrm {H}) \triangleq \frac{\mathrm {d}\phi (\Vert z\Vert _\mathrm {H})}{\mathrm {d}\Vert z\Vert ^2_\mathrm {H}}. \end{aligned}$$

  (A.4)

• (vi) If $z \in \mathcal {D}_\epsilon$, then

  $$\begin{aligned} 2 \phi _d (\Vert z\Vert _\mathrm {H}) \Vert z\Vert _\mathrm {H}^2 -\phi (\Vert z\Vert _\mathrm {H})> 0. \end{aligned}$$

  (A.5)