Integrated Adaptive Control and Reference Governors for Constrained Systems with State-Dependent Uncertainties

The contributions of this paper are as follows. Firstly, for constrained control under uncertainties, we develop an ${\mathcal{L}_{1}}$-RG framework for linear systems with matched nonlinear uncertainties that could depend on both time and states, and with both input and state constraints. Our adaptive robust RG framework leverages an ${\mathcal{L}_{1}}$ adaptive controller (${\mathcal{L}_{1}}$AC) to estimate and compensate for the uncertainties, and to guarantee uniform bounds on the error between actual states and inputs and those of a nominal (i.e., uncertainty-free) closed-loop system. These uniform bounds characterize tubes in which actual states and control inputs are guaranteed to stay despite the uncertainties. A reference governor designed for the nominal system with constraints tightened using these uniform bounds guarantees robust constraint satisfaction in the presence of uncertainties. Additionally, we show that these uniform bounds on state and input errors, and thus the conservatism induced by constraint tightening can be arbitrarily reduced in theory by tuning the filter bandwidth and estimation sample time parameters of the ${\mathcal{L}_{1}}$AC. Secondly, as a separate contribution to ${\mathcal{L}_{1}}$ adaptive control, we propose a novel scaling technique that allows deriving separate tight uniform bounds on each state and adaptive control input, as opposed to a single bound for all states, or adaptive control inputs in existing ${\mathcal{L}_{1}}$AC solutions [18]. The ability to provide such separate tight bounds makes an ${\mathcal{L}_{1}}$AC particularly attractive to be integrated with an RG for simultaneous constraint enforcement and improved trajectory tracking. Thirdly, we validate the efficacy of the proposed ${\mathcal{L}_{1}}$-RG framework on a flight control example and we compare it with both baseline and robust RG solutions in simulations.

Compared to existing literature, in particular, robust/adaptive MPC, ${\mathcal{L}_{1}}$-RG has the following novel aspects:

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  Thanks to the uncertainty compensation and transient performance guarantees available for the ${\mathcal{L}_{1}}$AC, ${\mathcal{L}_{1}}$-RG, (under suitable assumptions,) does not require uncertainty propagation along the prediction horizon. This uncertainty propagation is generally required in all existing robust and adaptive MPC approaches, and incurs conservatism, which is avoided by ${\mathcal{L}_{1}}$-RG.

• •

  ${\mathcal{L}_{1}}$-RG simultaneously improves tracking performance and enforces the constraints, while existing robust/disturbance-observer-based RG or robust/adaptive MPC solutions except a few such as [15, 19], focus on constraint satisfaction only.

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  Within ${\mathcal{L}_{1}}$-RG, the uniform bounds on the state and input errors (used for constraint tightening) and thus the conservatism induced by constraint tightening can be made arbitrarily small, which cannot be achieved by existing methods.

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  ${\mathcal{L}_{1}}$-RG is able to handle uncertainties that can nonlinearly depend on both time and states. Such a case has not been considered by previous adaptive MPC solutions that are based on uncertainty compensation. For instance, the solution in [15], which also leverages an ${\mathcal{L}_{1}}$AC, only treats parametric uncertainties and state constraints.

The paper is structured as follows. Section II formally states the problem. Section III provides an overview of the proposed solution and discusses preliminaries related to RG and ${\mathcal{L}_{1}}$AC design. Section IV introduces a scaling technique to derive separate and tight performance bounds for an ${\mathcal{L}_{1}}$AC, while Section V presents synthesis and performance analysis of the proposed ${\mathcal{L}_{1}}$-RG framework. Section VI includes validation of the proposed ${\mathcal{L}_{1}}$-RG framework on a flight control problem in simulations.

Notations: Let $\mathbb{R}$, $\mathbb{R}_{+}$ and $\mathbb{Z}_{+}$ denote the set of real, non-negative real, and non-negative integer numbers, respectively. $\mathbb{R}^{n}$ and $\mathbb{R}^{m\times n}$ denote the $n$-dimensional real vector space and the set of real $m$ by $n$ matrices, respectively. $\mathbb{Z}_{i}$ and $\mathbb{Z}_{1}^{n}$ denote the integer sets $\{i,i+1,\cdots\}$ and $\{1,2,\cdots,n\}$, respectively. $I_{n}$ denotes an identity matrix of size $n$, and $0$ is a zero matrix of a compatible dimension. $\left\lVert\cdot\right\rVert$ and $\left\lVert\cdot\right\rVert_{\infty}$ denote the $2$-norm and $\infty$-norm of a vector or a matrix, respectively. The $\mathcal{L}_{\infty}$- and truncated $\mathcal{L}_{\infty}$-norm of a function $x:\mathbb{R}_{+}\rightarrow\mathbb{R}^{n}$ are defined as $\left\lVert x\right\rVert_{\mathcal{L}_{\infty}}\triangleq\sup_{t\geq 0}\left\lVert x(t)\right\rVert_{\infty}$ and $\left\lVert x\right\rVert_{\mathcal{L}_{\infty}^{[0,T]}}\triangleq\sup_{0\leq t\leq T}\left\lVert x(t)\right\rVert_{\infty}$, respectively. The Laplace transform of a function $x(t)$ is denoted by $x(s)\triangleq\mathfrak{L}[x(t)]$. For a vector $x$, $x_{i}$ denotes the $i$th element of $x$. Given a positive scalar $\rho$, $\Omega(\rho)\triangleq\{z\in\mathbb{R}^{n}:\left\lVert z\right\rVert_{\infty}\leq\rho\}$ denotes a high dimensional ball set of radius $\rho$ and centered at the origin, while its dimension $n$ can be deduced from the context. For a high-dimensional set $\mathcal{X}$, $\textup{int}(\mathcal{X})$ denotes the interior of $\mathcal{X}$ and $\mathcal{X}_{i}$ denotes the projection of $\mathcal{X}$ onto the $i$th coordinate. For given sets $\mathcal{X},{\mathcal{Y}}\subset\mathbb{R}^{n}$, $\mathcal{X}\oplus{\mathcal{Y}}\triangleq\{x+y:x\in\mathcal{X},y\in{\mathcal{Y}}\}$ is the Minkowski set sum and $\mathcal{X}\ominus{\mathcal{Y}}\triangleq\{z:z+y\in\mathcal{X},\forall y\in{\mathcal{Y}}\}$ is the Pontryagin set difference.

Figure 1 depicts the proposed ${\mathcal{L}_{1}}$-RG framework. As shown in Fig. 1, ${\mathcal{L}_{1}}$-RG is comprised of two integrated components. The first one is an ${\mathcal{L}_{1}}$AC designed to compensate for the uncertainty $f(t,x)$ and to guarantee uniform bounds on the errors between actual states and inputs, and those of the nominal closed-loop system:

The second component is a RG designed for the nominal system 9a with tightened constraints computed using the uniform bounds guaranteed by the ${\mathcal{L}_{1}}$AC.

More formally, we will design the ${\mathcal{L}_{1}}$AC to ensure

where $x(t)$ and $u(t)$ are the vectors of states and of the total control inputs of the closed-loop system 5:

where $u(t)$ is given by 4 and $u_{\textup{n}}(t)$ by 9b, and $\tilde{\mathcal{X}}$ and $\tilde{\mathcal{U}}$ are some pre-computed hyperrectangular sets dependent on the properties of $f(t,x)$ and of the ${\mathcal{L}_{1}}$AC. The details will be given in Theorem 3 in Section III-C. Define

Then for robust constraint enforcement, one just needs to design a RG for the nominal system 9 with tightened constraints given by

We now introduce the RG for the nominal system 9 to enforce the constraints 13. We use the discrete-time RG approach of [3] that uses a discrete-time model:

where $\mathrm{x_{\textup{n}}}(k)$, $\mathrm{v}(k)$ and $\mathrm{u_{\textup{n}}}(k)$ denotes the vectors of states, of reference command inputs, and of nominal control inputs, respectively, and $\hat{A}_{m}$ and $\hat{B}_{v}$ are computed from $A_{m}$ and $B_{v}$ in 9 assuming a sampling time, $T_{d}$. When doing the discretization, we ensure that the discrete-time system 14 has the same states as the continuous-time system at all sampling instants. This can be achieved by using the zero-order hold discretization, since

which indicates that $v(t)$ is piecewise constant. The constraints 13 are imposed in discrete-time as

where $\hat{\mathcal{X}}_{\textup{n}}$ and $\hat{\mathcal{U}}_{\textup{n}}$ are tightened versions of $\mathcal{X}_{\textup{n}}$ and ${\mathcal{U}}_{\textup{n}}$, respectively, introduced to avoid inter-sample constraint violations, and are defined by

while

with $\mathcal{V}$ denoting the set of all possible reference commands output by the RG.

The following lemma formally guarantees that no inter-sample constraint violations will happen for the continuous-time system 9 when the constraints for the discrete-time system 21 are satisfied at all sampling instants.

Define

Then, the constraints 16 can be rewritten as

where $\times$ denotes the cross product.

Similar to most RG schemes, the RG scheme we adopt here computes at each time instant a command $\mathrm{v}(k)$ such that, if it is constantly applied from the time instant $k$ onward, the ensuing output will always satisfy the constraints. More formally, we define the maximal output admissible set $O_{\infty}$ [21] as the set of all states $\mathrm{x_{\textup{n}}}$ and inputs $\mathrm{v}$, such that the predicted response from the initial state $\mathrm{x_{\textup{n}}}$ and with a constant input $\mathrm{v}$ satisfies the constraints 21, i.e.,

where for system 14 the output prediction $\hat{\mathrm{y}}_{\textup{n}}^{c}(k|\mathrm{x_{\textup{n}}},\mathrm{v})$ is given by

Define $\tilde{O}_{\infty}$ as a slightly tightened version of $O_{\infty}$ obtained by constraining the command $\mathrm{v}$ so that the associated steady-state output $\bar{\mathrm{y}}_{\textup{n}}^{c}=(D_{c}+C_{c}(I_{n}\!-\!\hat{A}_{m})^{-1}\hat{B}_{v})\mathrm{v}$ satisfies constraints with a nonzero (typically small) margin $\epsilon>0$, i.e.,

where $O^{\epsilon}\triangleq\{(\mathrm{v},\mathrm{x_{\textup{n}}}):\bar{\mathrm{y}}_{\textup{n}}^{c}\in(1-\epsilon){\mathcal{Y}}_{\textup{n}}\}.$ Clearly, ${\tilde{O}}_{\infty}$ can be made arbitrarily close to $O_{\infty}$ by decreasing $\epsilon$. Based on the currently available state $\mathrm{x_{\textup{n}}}(k)$ at an instant $k$, the RG computes $\mathrm{v}(k)$ so that

It is proven in [21] that if $\hat{A}_{m}$ is Schur, $(\hat{A}_{m},C_{c})$ is observable, ${\mathcal{Y}}_{\textup{n}}$ is compact with $0$ in the interior, and $\epsilon>0$ is sufficiently small, then the set ${\tilde{O}}_{\infty}$ is finitely determined, i.e., there exists a finite index $k^{\star}$ such that

Moreover, ${\tilde{O}}_{\infty}$ is positively invariant, which means that if $(\mathrm{v}(k),\mathrm{x_{\textup{n}}}(k))\in{\tilde{O}}_{\infty}$ and $\mathrm{v}(k)$ is applied to the system at time $k$, then $(\mathrm{v}(k),\mathrm{x_{\textup{n}}}(k+1))\in{\tilde{O}}_{\infty}$. Furthermore, if ${\mathcal{Y}}_{\textup{n}}$ is convex, then ${\tilde{O}}_{\infty}$ is also convex.

The proposed ${\mathcal{L}_{1}}$-RG framework can leverage most of existing RG schemes developed for uncertainty-free systems. As an illustration and demonstration in Section VI, we choose the scalar RG introduced in [22, 23]. The scalar RG computes at each time instant $k$ a command $\mathrm{v}(k)$ which is the best approximation of the desired set-point $\mathrm{r}(k)$ along the line segment connecting $\mathrm{v}(k-1)$ and $\mathrm{r}(k)$ that ensures $(\mathrm{v}(k),\mathrm{x_{\textup{n}}}(k))\in{\tilde{O}}_{\infty}$. More specifically, the scalar RG solves at each discrete time $k$, the following optimization problem:

where $\kappa(k)$ is a scalar adjustable bandwidth parameter and $\mathrm{v}(k)=\mathrm{v}(k-1)+\kappa(k)(\mathrm{r}(k)-\mathrm{v}(k-1))$ is the modified reference command to be applied to the system. If there is no danger of constraint violation, $\kappa(k)=1$ and $\mathrm{v}(k)=\mathrm{r}(k)$ so that the RG does not interfere with the desired operation of the system. If $\mathrm{v}(k)=\mathrm{r}(k)$ would cause a constraint violation, the value of $\kappa(k)$ is decreased by the RG. In the extreme case, $\kappa(k)=0$, $\mathrm{v}(k)=\mathrm{v}(k-1)$, which means that the RG momentarily isolates the system from further variations of the reference command for constraint enforcement. Due to the positive invariance of ${\tilde{O}}_{\infty}$, $\mathrm{v}(k)=\mathrm{v}(k-1)$ always satisfies the constraints, which ensures recursive feasibility under the condition that at $t=0$ a command $\mathrm{v}(0)$ is known such that $\left(\mathrm{v}(0),\mathrm{x_{\textup{n}}}(0)\right)\in{\tilde{O}}_{\infty}$. Response properties of the scalar RG, including conditions for the finite-time convergence of $\mathrm{v}(k)$ to $\mathrm{r}(k)$ are detailed in [23].

We now present an ${\mathcal{L}_{1}}$AC that guarantees the bounds in (10), without considering the state and control constraints in 2. We first recall some basic definitions and facts from control theory, and introduce some definitions and lemmas.

The following lemma follows directly from Definition 1.

A unique feature of an $\mathcal{L}_{1}$AC is a low-pass filter ${\mathcal{C}}(s)$ (with DC gain ${\mathcal{C}}(0)=I_{m}$) that decouples the estimation loop from the control loop, thereby allowing for arbitrarily fast adaptation without sacrificing the robustness [18]. For simplicity, we can select ${\mathcal{C}}(s)$ to be a first-order transfer function matrix

where $k^{j}_{f}$ ($j\in\mathbb{Z}_{1}^{m}$) is the bandwidth of the filter for the $j$th input channel. We now introduce a few notations that will be used later:

where $A_{m},B_{v}$ correspond to system 9 and $B$ to 1. Also, letting $x_{\textup{in}}(t)$ be the state of the system $\dot{x}_{\textup{in}}(t)=A_{m}x_{\textup{in}}(t),\ x_{\textup{in}}(0)=x_{0},$ we have $x_{\textup{in}}(s)\triangleq(sI_{n}-A_{m})^{-1}x_{0}$. Defining $\rho_{\textup{in}}\triangleq\left\lVert s(sI_{n}-A_{m})^{-1}\right\rVert_{\mathcal{L}_{1}}\max_{x_{0}\in\mathcal{X}_{0}}\left\lVert x_{0}\right\rVert_{\infty}$, and further considering that $A_{m}$ is Hurwitz and $\mathcal{X}_{0}$ is compact, we have $\left\lVert x_{\textup{in}}\right\rVert_{\mathcal{L}_{\infty}}\leq\rho_{\textup{in}}$ according to Lemma 2.