# Robust Adaptive Model Predictive Control with Persistent Excitation
Conditions

Xiaonan Lu<EMAIL_ADDRESS>Mark Cannon<EMAIL_ADDRESS>Department of Engineering Science, University of Oxford, OX1 3PJ, UK

###### Abstract

For constrained linear systems with bounded disturbances and parametric
uncertainty, we propose a robust adaptive model predictive control strategy
with online parameter estimation. Constraints enforcing persistently exciting
closed loop control actions are introduced for a set-membership parameter
identification scheme. The algorithm requires the online solution of a convex
program, satisfies constraints robustly, and ensures recursive feasibility and
input-to-state stability. Almost sure convergence to the actual system
parameters is demonstrated under assumptions on stabilizability, reachability,
and tight disturbance bounds.

Keywords: model predictive control, robust adaptive control, constrained
systems, persistently exciting control

,

##  1 Introduction

The performance of Model Predictive Control (MPC) relies on an accurate model
of the controlled system. To reduce model uncertainty, adaptive MPC algorithms
have been proposed that allow model parameters to be estimated online. In
system identification and adaptive control, persistent excitation (PE)
conditions play a key role in establishing convergence of parameter estimates
(Green & Moore, 1986; Shimkin & Feuer, 1987). By incorporating constraints to
ensure appropriate PE conditions, a constrained MPC strategy can impose a
lower bound on the expected rate of parameter convergence. As a result,
adaptive MPC has the potential to estimate system parameters while controlling
the system subject to constraints. Various approaches have been proposed
(Mayne, 2014), but robust, computationally tractable adaptive MPC remains an
open topic under research.

Adaptive MPC strategies usually have the dual purpose of regulating the system
via feedback and providing sufficient excitation for identification of system
parameters. Different adaptive MPC approaches place varying emphasis on these
two competing objectives. Some focus on robust constraint satisfaction and
stability (e.g. through constraint tightening (Di Cairano, 2016), min-max cost
formulations (Adetola et al., 2009; Wang et al., 2017) or tube MPC (Lorenzen
et al., 2019; Lu & Cannon, 2019)) but omit persistent excitation conditions in
the problem formulation. On the other hand, some approaches consider a nominal
MPC problem and force the control law to be persistently exciting, but fail to
ensure constraint satisfaction and closed loop system stability (Goodwin &
Sin, 1984; Marafioti et al., 2014).

Other approaches aim to achieve the dual objectives of system regulation and
sufficient excitation simultaneously. For example, Weiss & Di Cairano (2014)
select the control input from the Robust Admissible Invariant (RAI) set and
balance the conflicting objectives by using an augmented cost function. The
resulting control law is more likely to be persistently exciting, but this is
not guaranteed. Tanaskovic et al. (2014) avoid imposing PE conditions by
considering the discrepancy between the nominal and the actual models.
However, the proposed algorithm involves solving a nonconvex, infinite-
dimensional optimization and can only be simplified for specific examples.
Gonzalez et al. (2014) separate the two objectives using a dual mode control
strategy that injects persistent excitation into the system whenever the state
enters a target region for parameter identification. The proposed algorithm is
only applicable to open-loop stable linear systems, and the existence of the
target region is example-dependent. Hernandez Vicente & Trodden (2019) propose
an algorithm that satisfies PE condition and state and input constraints
recursively, but the system model cannot be adapted online.

In addition, although the importance of persistent excitation conditions have
been widely acknowledged in adaptive control literature (Narendra & Annaswamy,
1987), few strategies incorporate these conditions in a convex optimization
formulation. For example, Marafioti et al. (2014) expresses the conditions for
persistent excitation as nonconvex quadratic inequalities in terms of the
control input. Similarly, Hernandez Vicente & Trodden (2019) demonstrate that
a PE condition can be satisfied using a periodic solution computed offline,
but this solution might not be optimal. Other approaches (Lu & Cannon, 2019;
Lu et al., 2021) use linearization of the PE condition around a reference
trajectory to determine sufficient conditions for persistency of excitation,
but are unable to ensure closed loop satisfaction of PE conditions through
recursively feasible constraints.

In this work we consider linearly constrained linear systems with parametric
uncertainty and bounded additive disturbances. Building on Lu et al. (2021),
we propose an adaptive MPC algorithm that combines set-based parameter
identification, robust regulation, and recursively feasible constraints. The
algorithm is input-to-state practically stable and provides convergence of
parameter estimates almost surely. This paper has three main contributions:

  * •

We show that a robust MPC law can be made persistently exciting by adding a
random signal to the terminal control law and incorporating additional
conditions in the optimization of predicted performance.

  * •

We provide a recursively feasible set of conditions for ensuring a
persistently exciting control law. These consist of convex constraints in the
MPC optimization and a posterior check performed on a nonconvex condition via
a set of sampled convex conditions.

  * •

The proposed algorithm is computationally tractable while achieving the dual
objectives of robustly stabilizing the system and providing persistent
excitation for online parameter estimation.

The remainder of the paper consists of four main sections. Section 2 begins by
introducing the class of constrained uncertain linear systems under
consideration. Then we briefly recap the set-based parameter identification
method and introduce the definition of persistent excitation that is employed
in this work. Following this we provide conditions such that the system under
a linear feedback law (with or without injected noise) satisfies the required
PE condition and hence that the resulting estimated parameter set converges
almost surely to the true parameter value. Section 3 explains the tube-based
robust MPC formulation, including the parameterized control law, the initial
and terminal conditions, and the cost function. We then define a sequence of
PE constraints and show how to convert the resulting nonconvex constraints
into convex constraints in the MPC optimization and how to perform a posterior
check using sampling. The robust adaptive algorithm is then summarized and the
section concludes with a set of results that demonstrate recursive
feasibility, input-to-state practical stability (ISpS) and satisfaction of PE
conditions in closed loop operation. Section 4 provides a numerical example.
We compare the proposed algorithm with robust MPC without PE constraints and
hence demonstrate that the proposed algorithm results in faster convergence of
the estimated parameter set, a higher PE coefficient value and improved
performance.

Notation: The set of real numbers is denoted $\mathbb{R}$. The non-negative
(or positive) integers are denoted $\mathbb{N}_{\geq 0}$ (or
$\mathbb{N}_{>0}$), and $\mathbb{N}_{[p,q]}$ denotes $\\{n\in\mathbb{N}:{p\leq
n\leq q}\\}$. The identity matrix is ${\mathcal{{I}}}$. The $i$th element of a
vector $a$ is $[a]_{i}$ and $\lVert{a}\rVert$ denotes the Euclidean norm. The
p-dimensional closed unit ball is
$\mathbb{B}^{p}=\\{x\in\mathbb{R}^{p}:\|x\|\leq 1\\}$. The $i$th row of a
matrix $A$ is $[A]_{i}$, $\operatorname{vec}(A)$ is the vector formed by
stacking the columns of $A$, the matrix inequality $A\geq 0$ applies
elementwise, and $A\succeq 0$ (or $A\succ 0$) indicates that $A$ is positive
semidefinite (positive definite). For
$\mathcal{X},\mathcal{Y}\subset\mathbb{R}^{n}$,
$A\mathcal{X}=\\{Ax:x\in\mathcal{X}\\}$,
$\mathcal{X}\oplus\mathcal{Y}=\\{x+y:x\in\mathcal{X},\,y\in\mathcal{Y}\\}$ and
$\operatorname{Co}\\{\mathcal{X},\mathcal{Y}\\}$ denotes the convex hull of
$\mathcal{X}$ and $\mathcal{Y}$. A polytope is a convex and compact polyhedral
subset of $\mathbb{R}^{n}$. The $k$ steps ahead predicted value of a variable
$y$ is denoted $y_{k}$, and the complete notation $y_{k|t}$ indicates the $k$
steps ahead prediction at time $t$. Probabilities and expectations conditioned
on the system state $x_{t}$ are denoted
$\mathbb{P}(\cdot|x_{t})=\mathbb{P}_{t}(\cdot)$ and
$\mathbb{E}(\cdot|x_{t})=\mathbb{E}_{t}(\cdot)$ respectively, and
$\mathbb{P}(\cdot)$, $\mathbb{E}(\cdot)$ are respectively equivalent to
$\mathbb{P}_{0}(\cdot)$, $\mathbb{E}_{0}(\cdot)$.

##  2 Persistent excitation conditions

In this section we introduce our assumptions on the system model before
discussing set-based identification methods. The conditions for persistence of
excitation are considered and we show that these are satisfied under linear
feedback with mild assumptions.

###  2.1 Problem formulation

The system state $x_{t}\in\mathbb{R}^{n_{x}}$, control input
$u_{t}\in\mathbb{R}^{n_{u}}$ and unknown disturbance input
$w_{t}\in\mathbb{R}^{n_{w}}$, satisfy

$x_{t+1}=A(\theta^{\ast})x_{t}+B(\theta^{\ast})u_{t}+Fw_{t}.$ (1)

at all times $t\in\mathbb{N}_{\geq 0}$. Matrices $A(\theta^{\ast})$,
$B(\theta^{\ast})$ depend on an unknown constant parameter
$\theta^{\ast}\in\mathbb{R}^{p}$.

######  Assumption 1 (Additive disturbance).

The disturbance sequence $\\{w_{t}\in\mathcal{W},\,t\in\mathbb{N}_{\geq 0}\\}$
is independent and identically distributed (i.i.d.), $\mathbb{E}(w_{t})=0$,
$\mathbb{E}(w_{t}w_{t}^{\top})\succeq\epsilon_{w}{\mathcal{{I}}}$,
$\epsilon_{w}>0$, and $\mathcal{W}$ is a known convex polyhedral set.

######  Assumption 2 (Model parameters).

(a). $A(\theta),B(\theta)$ are defined in terms of known matrices
$A_{i},B_{i}$, ${i\in\mathbb{N}_{[0,p]}}$:

$(A(\theta),B(\theta))=(A_{0},B_{0})+\smash{\sum_{i=1}^{p}(A_{i},B_{i})}[\theta]_{i},\
\ \forall\theta\in\Theta_{0}.$

(b). $\Theta_{0}$ is a known polytopic set containing $\theta^{\ast}$:

$\theta^{\ast}\in\Theta_{0}=\\{\theta:M_{\Theta}\theta\leq\mu_{0}\\}=\operatorname{Co}\\{\theta_{0}^{(1)},\dots,\theta_{0}^{(m)}\\}.$

(c). The pair $\bigl{(}A(\theta^{\ast}),B(\theta^{\ast})\bigr{)}$ is
reachable.
(d). $(A(\theta),B(\theta))=(A(\theta^{\ast}),B(\theta^{\ast}))$ if and only
if $\theta=\theta^{\ast}$.

###  2.2 Set-based parameter identification

Parameter estimation methods include recursive least squares (Heirung et al.,
2017), comparison sets (Aswani et al., 2013), set-membership identification
(Tanaskovic et al., 2014; Lorenzen et al., 2019) and neural network training
(Akpan & Hassapis, 2011; Reese & Collins, 2016). Here we use a set-membership
approach to enable robust satisfaction of constraints. Set-based parameter
identification was proposed in (Chisci et al., 1998; Veres et al., 1999) and
it was shown in Lu et al. (2021) that the estimated parameter set converges to
the true parameter value with probability 1 if the associated regressor is
persistently exciting (PE).

At times $t\in\mathbb{N}_{>0}$ we use observations of the state $x_{t}$ to
determine a set $\Delta_{t}$ of consistent model parameters, known as the
unfalsified parameter set. This is combined with $\Theta_{t-1}$ to construct a
new parameter set estimate $\Theta_{t}$.

The model (1) can be rewritten as

$x_{t+1}=\Phi(x_{t},u_{t})\theta^{\ast}+\phi(x_{t},u_{t})+Fw_{t}$

where $\Phi_{t}$ and $\phi_{t}$ are known at time $t$ and are defined by

$\displaystyle\Phi_{t}$
$\displaystyle=\Phi(x_{t},u_{t})=\begin{bmatrix}A_{1}x_{t}+B_{1}u_{t}&\cdots&A_{p}x_{t}+B_{p}u_{t}\end{bmatrix}$
(2) $\displaystyle\phi_{t}$
$\displaystyle=\phi(x_{t},u_{t})=A_{0}x_{t}+B_{0}u_{t}.$ (3)

Given $x_{t}$, $x_{t-1}$, $u_{t-1}$ and the disturbance set $\mathcal{W}$, the
unfalsified parameter set at time $t$ is given by

$\Delta_{t}=\\{\theta:x_{t}-A(\theta)x_{t-1}-B(\theta)u_{t-1}\in
F\mathcal{W}\\}$ (4)

The parameter set $\Theta_{t}$ may be updated using $\Delta_{t}$ by various
methods, including minimal (Chisci et al., 1998), fixed-complexity (Lorenzen
et al., 2019), and limited-complexity (Tanaskovic et al., 2014) update laws.
In each case, $\Theta_{t}$ is non-increasing and
$\Theta_{t}\subseteq\Theta_{t-1}$ for all $t\in\mathbb{N}_{>0}$.

For a fixed-complexity parameter set update law, the parameter set estimate
$\Theta_{t}$ is defined as
$\Theta_{t}=\Theta(\mu_{t})=\\{\theta:M_{\Theta}\theta\leq\mu_{t}\\}$ where
$M_{\Theta}\in\mathbb{R}^{r\times p}$ is an a priori chosen matrix and
$\mu_{t}\in\mathbb{R}^{r}$ is determined so that $\Theta_{t}$ is the smallest
set containing the intersection of $\Theta_{t-1}$ and the unfalsified sets
$\Delta_{j}$ for $j=t-N_{\mu}+1,\ldots,t$,

$\mu_{t}:=\min_{\mu\in\mathbb{R}^{r}}{\operatorname{vol}}\bigl{(}\Theta(\mu)\bigr{)}\
\
\text{s.t.}\\!\smash{\bigcap_{j=t-N_{\mu}+1}^{t}}\\!\\!\Delta_{j}\cap\Theta_{t-1}\\!\subseteq\Theta(\mu)$
(5)

where $\Delta_{j}:=\mathbb{R}^{p}$ for $j\leq 0$ and $N_{\mu}$ is the
parameter update window length. Note that $\mu_{t}$ can be computed by solving
a set of linear programs.

We briefly recap the definition of persistent excitation (PE). The regressor
$\Phi_{t}$ in (2) is persistently exciting if there exist a horizon $N_{u}$
and a scalar $\epsilon_{\Phi}>0$ such that

$\sum_{k=t}^{t+N_{u}-1}\Phi_{k}^{\top}\Phi_{k}\succeq\epsilon_{\Phi}{\mathcal{{I}}}$
(6)

for all $t\in\mathbb{N}_{\geq 0}$. In the current work however, we define
persistent excitation using the expectation:

$\sum_{k=t}^{t+N_{u}-1}\mathbb{E}\bigl{\\{}\Phi_{k}^{\top}\Phi_{k}\bigr{\\}}\succeq\epsilon_{\Phi}{\mathcal{{I}}},$
(7)

which is required to hold for some $\epsilon_{\Phi}>0$, at an infinite number
of discrete time instants $t\in\mathbb{N}_{\geq 0}$. We refer to the interval
$\mathbb{N}_{[t,t+N_{u}-1]}$ as a PE window.

######  Assumption 3 (Tight disturbance bound).

For any point $w^{0}$ on the boundary of $\mathcal{W}$ and any $\epsilon>0$,
the disturbance sequence $\\{w_{0},w_{1},\ldots\\}$ satisfies
$\mathbb{P}\bigl{\\{}\lVert{w_{t}-w^{0}}\rVert<\epsilon\bigr{\\}}\geq
p_{w}(\epsilon)>0$, for all $t\in\mathbb{N}_{\geq 0}$.

If the regressor in (2) satisfies the PE condition (6), then under Assumptions
1 and 3 the parameter set estimate $\Theta_{t}$ under a fixed-complexity
update law with $N_{\mu}\geq N_{u}$ converges to $\\{\theta^{\ast}\\}$ with
probability 1. This is shown in Lu et al. (2021, Theorem 3 and Corollary 3).
Here we use a version of this result that applies to (7).

######  Lemma 1.

Under Assumptions 1 and 3, if $\Phi_{t}$ satisfies the PE condition (7), then
the minimal complexity parameter set estimate $\Theta_{t}$ with $N_{\mu}\geq
N_{u}$ converges to $\\{\theta^{\ast}\\}$ as $t\to\infty$ with probability 1.

Proof. If (7) holds, then there exists (with probability 1) an infinite
sequence $\\{\kappa_{i},\,i\in\mathbb{N}_{\geq 0}\\}$ satisfying, for some
$\epsilon_{\Phi}>0$ and any given $\theta\in\mathbb{R}^{p}$,

$\sum_{k=\kappa_{i}}^{\kappa_{i}+N_{u}-1}\|\Phi_{k}\theta\|^{2}\geq\epsilon_{\Phi}\lVert\theta\rVert^{2}.$

Choose $\kappa_{i}$ so that $\kappa_{i+1}\geq\kappa_{i}+N_{u}-1$ and therefore
$\Theta_{\kappa_{i}+N_{u}-1}\supseteq\Theta_{\kappa_{i+1}}$. Then we can prove
the Lemma using Theorem 3 and Corollary 3 of Lu et al. (2021) with $t$
replaced by $\kappa_{t}+N_{u}-1$, since these results imply that any
$\theta\in\Theta_{0}$ such that $\theta\neq\theta^{\ast}$ is necessarily
excluded from $\Theta_{\kappa_{i}}$ with a probability that converges to $1$
as $i\to\infty$. It follows that $\Theta_{t}\to\\{\theta^{\ast}\\}$ as
$t\to\infty$ with probability 1. ∎

Assumption 1 on the disturbance sequence $\\{w_{t},t\in\mathbb{N}_{\geq 0}\\}$
is common in robust MPC and set-membership identification formulations.
Clearly, if the disturbance sequence contains temporal correlation that is
representable as an i.i.d. disturbance sequence driving a known linear filter,
then this can be incorporated in the dynamics (1) without violating Assumption
1. In addition, the assumption of zero-mean disturbances can be relaxed
without affecting the convergence result (Lemma 1). For example, by redefining
the model parameters $A(\theta),B(\theta),F$ and $\theta$, the mean value of
the disturbance can be estimated with the other unknown model parameters via
the parameter set update law. Assumption 3 may be difficult to verify, but we
note that this assumption can be relaxed at the expense of some residual
uncertainty in the parameter set estimate (Lu et al., 2021, Sec. 5.3). We also
note that, although $x_{t}$ is assumed to be exactly known in the definition
(4) of the unfalsified set $\Delta_{t}$, this assumption can be relaxed to
allow the use of noisy measurements or state estimates (see e.g. Lu et al.,
2021, Sec. 5.4).

In the next section we show that the PE condition (7) can be satisfied under a
given linear feedback law, which provides useful insight into the properties
of predicted control laws.

###  2.3 Regressor under linear feedback law

Consider the system (1) under the action of a linear feedback law
$u_{t}=Kx_{t}$. To simplify notation we define
$A_{K}(\theta)=A(\theta)+B(\theta)K$ and $A_{K,i}=A_{i}+B_{i}K$,
$i\in\mathbb{N}_{[0,p]}$.

######  Assumption 4 (Stability).

For $z_{t}\in\mathbb{R}^{n_{x}}$ and $t\in\mathbb{N}_{\geq 0}$,
$z_{t+1}\in\operatorname{Co}\\{A_{K}(\theta)z_{t},$ $\theta\in\Theta_{0}\\}$
is quadratically stable.

######  Assumption 5.

The gain $K$ is chosen so that:
(a). $A_{K}(\theta)=A_{K}(\theta^{\ast})$ if and only if
$\theta=\theta^{\ast}$.
(b). The pair $\bigl{(}A_{K}(\theta^{\ast}),F\bigr{)}$ is reachable.

Assumption 4 is equivalent to feasibility of the linear matrix inequalities
(LMIs): $P-\smash{A_{K}(\theta_{0}^{(j)})^{\top}PA_{K}(\theta_{0}^{(j)})}\succ
0$, $j\in\mathbb{N}_{[1,m]}$, $P=P^{\top}\succ 0$ (e.g. Blanchini & Miani,
2008, Theorem 6.21).

Assumption 5(b) is more difficult to verify because $\theta^{\ast}$ is
unknown. However, a sufficient condition is that $(A_{K}(\theta_{t}),F)$ is
reachable for all sequences $\\{\theta_{0},\theta_{1},\ldots\\}$ such that
$\theta_{t}\in\Theta_{0}$ for all $t$, which can be verified by checking the
feasibility of a set of LMIs defined by the vertices of $\Theta_{0}$.

######  Theorem 2.

If $N_{u}>n_{x}$, then under Assumptions 1, 2 and 5, the regressor $\Phi_{t}$
in (2) with $u_{t}=Kx_{t}$ satisfies (7) for some $\epsilon_{\Phi}>0$ and all
$t\in\mathbb{N}_{\geq 0}$,

Proof. Let $N_{u}>n_{x}$. Under the feedback law $u_{t}=Kx_{t}$, the system
(1) becomes $x_{t+1}=A_{K}(\theta^{\ast})x_{t}+Fw_{t}$, and the definition of
the regressor $\Phi_{t}$ yields, for all $\theta\in\mathbb{R}^{p}$,

$\displaystyle\sum_{k=t}^{t+N_{u}-1}\theta^{\top}\mathbb{E}_{t}\bigl{[}\Phi(x_{k},Kx_{k})^{\top}\Phi(x_{k},Kx_{k})\bigr{]}\theta$
(8)
$\displaystyle\quad=\operatorname{tr}\Bigl{\\{}\sum_{i=1}^{p}A_{K,i}^{\top}\,[\theta]_{i}\sum_{j=1}^{p}A_{K,j}\,[\theta]_{j}\sum_{k=t}^{t+N_{u}-1}\mathbb{E}_{t}(x_{k}x_{k}^{\top})\Bigr{\\}}.$

But $w_{k}$ is i.i.d. and
$\mathbb{E}(w_{k}w_{k}^{\top})\succeq\epsilon_{w}{\mathcal{{I}}}$ (Assumption
1), so
$\mathbb{E}_{t}(x_{k}x_{k}^{\top})\succeq\epsilon_{w}\sum_{j=0}^{k-t-1}\bigl{[}A_{K}(\theta^{\ast})\bigr{]}^{j}FF^{\top}\bigl{[}A_{K}(\theta^{\ast})^{\top}\bigr{]}^{j}$
for all $k>t$. In addition, the pair $\bigl{(}A_{K}(\theta^{\ast}),F\bigr{)}$
is reachable (Assumption 5), so a positive scalar $\epsilon_{\mathcal{R}}$
exists such that
$\sum_{j=0}^{n_{x}-1}\bigl{[}A_{K}(\theta^{\ast})\bigr{]}^{j}FF^{\top}\bigl{[}A_{K}(\theta^{\ast})^{\top}\bigr{]}^{j}\succeq\epsilon_{\mathcal{R}}{\mathcal{{I}}}$.
It follows that
$\mathbb{E}_{t}(x_{k}x_{k}^{\top})\succeq\epsilon_{w}\epsilon_{\mathcal{R}}{\mathcal{{I}}}$
for all $k\geq t+n_{x}$ and hence

$\sum_{k=t}^{t+N_{u}-1}\mathbb{E}_{t}(x_{k}x_{k}^{\top})\succeq\epsilon_{w}\epsilon_{\mathcal{F}}{\mathcal{{I}}}$

where $\epsilon_{\mathcal{F}}=(N_{u}-n_{x})\epsilon_{\mathcal{R}}$. Therefore
(8) becomes

$\displaystyle\sum_{k=t}^{t+N_{u}-1}\theta^{\top}\mathbb{E}_{t}\bigl{[}\Phi(x_{k},Kx_{k})^{\top}\Phi(x_{k},Kx_{k})\bigr{]}\theta$
$\displaystyle\quad\geq\epsilon_{w}\epsilon_{\mathcal{F}}\bigl{\|}\operatorname{vec}\Bigl{(}\smash{\sum_{i=1}^{p}A_{K,i}}\,[\theta]_{i}\Bigr{)}\bigr{\|}^{2}\geq\sigma^{2}\epsilon_{w}\epsilon_{\mathcal{F}}\lVert{\theta}\rVert^{2}$

where we have used the identities
$\operatorname{tr}(A^{\top}A)=\lVert{\operatorname{vec}(A)}\rVert^{2}$ and
$\operatorname{vec}\bigl{(}\sum_{i=1}^{p}A_{K,i}\,[\theta]_{i}\bigr{)}=\bigl{[}\operatorname{vec}(A_{K,1})\
\ \cdots\ \ \operatorname{vec}(A_{K,p})\bigr{]}\theta$. Here $\sigma$ is the
minimum singular value of the matrix $[\operatorname{vec}(A_{K,1})\ \cdots\
\operatorname{vec}(A_{K,p})]$, which is non-zero due to Assumption 5. It
follows that (7) holds for all $t\geq 0$ with
$\epsilon_{\Phi}=\sigma^{2}\epsilon_{w}\epsilon_{\mathcal{F}}>0$ for any
$N_{u}>n_{x}$. ∎

Lemma 1 and Theorem 2 imply that the minimal and fixed-complexity parameter
set estimates under linear feedback converge to $\\{\theta^{\ast}\\}$ with
probability 1 if Assumptions 1-5 hold.

###  2.4 Regressor under linear feedback with injected noise

Theorem 2 links the convergence of parameter set estimates to properties of
the disturbance sequence (Assumptions 1 and 3) and $(A_{i},B_{i})$ and $K$
(Assumptions 2 and 5). To relax these conditions, specifically Assumptions
5(a) and (b), consider a feedback law with injected noise:

$u_{t}=Kx_{t}+s_{t}.$ (9)

Here $s_{t}\in\mathbb{R}^{n_{u}}$ is a random variable with a known
probability distribution that provides additional excitation to encourage
convergence of parameter estimates.

######  Assumption 6.

The sequence $\\{s_{t}\in\mathcal{S},\,t\in\mathbb{N}_{\geq 0}\\}$ is i.i.d.
with $\mathbb{E}(s_{t})=0$,
$\mathbb{E}(s_{t}s_{t}^{\top})\succeq\epsilon_{s}{\mathcal{{I}}}$,
$\epsilon_{s}>0$, $s_{t}$ is independent of $x_{t}$ and $w_{t}$, and
$\mathcal{S}$ is a known polytopic set.

Similarly to Theorem 2, we next give a bound on the expectation of the PE
condition (6) under feedback law (9).

######  Theorem 3.

If $N_{u}>n_{x}$, then under Assumptions 1, 2 and 6, the regressor $\Phi_{t}$
in (2) with $u_{t}=Kx_{t}+s_{t}$ satisfies (7) for some $\epsilon_{\Phi}>0$
and all $t\in\mathbb{N}_{\geq 0}$.

Proof. Let $N_{u}>n_{x}$ be given. The control law (9) implies
$x_{t+1}=A_{K}(\theta^{\ast})x_{t}+B(\theta^{\ast})s_{t}+Fw_{t}$, and since
$\bigl{(}A_{K}(\theta^{\ast}),B(\theta^{\ast})\bigr{)}$ is reachable
(Assumption 2(c)) we have

$\mathbb{E}_{t}(x_{k}x_{k}^{\top})\succeq\epsilon_{s}\\!\sum_{j=0}^{k-t-1}A_{K}(\theta^{\ast})^{j\\!}B(\theta^{\ast})B(\theta^{\ast})^{\\!\top\\!\\!}{A_{k}(\theta^{\ast})^{j\\!}}^{\top}\\!\\!\succeq\epsilon_{s}\epsilon_{\mathcal{R}}\mathcal{I}$

for all $k\geq t+n_{x}$, where $\epsilon_{\mathcal{R}}>0$.

Hence Assumption 6 implies

$\sum_{k=t}^{t+N_{u}-1}\\!\\!\\!\mathbb{E}_{t}\biggl{\\{}\begin{bmatrix}x_{k}\\\
s_{k}\end{bmatrix}\\!\begin{bmatrix}x_{k}^{\top\\!}&s_{k}^{\top\\!}\end{bmatrix}\biggr{\\}}\succeq\\!\sum_{k=0}^{N_{u}-1}\\!\\!\begin{bmatrix}\mathbb{E}_{t}[x_{k}x_{k}^{\top}]\\!\\!&0\\\
0&\\!\\!\epsilon_{s}{\mathcal{{I}}}\end{bmatrix}\succeq\epsilon_{s}\epsilon_{\mathcal{F}}{\mathcal{{I}}}$

where
$\epsilon_{\mathcal{F}}=\min\\{(N_{u}-n_{x})\epsilon_{\mathcal{R}},1\\}$.
Furthermore, since $A_{i}x_{k}+B_{i}u_{k}=A_{K,i}x_{k}+B_{i}s_{k}=[A_{K,i}\
B_{i}][x_{k}^{\top}\ s_{k}^{\top}]^{\top}$, an argument similar to the proof
of Theorem 2 yields

$\displaystyle\sum_{k=t}^{t+N_{u}-1}\theta^{\top}\mathbb{E}_{t}\bigl{[}\Phi(x_{k},Kx_{k}+s_{k})^{\top}\Phi(x_{k},Kx_{k}+s_{k})\bigr{]}\theta$
$\displaystyle\quad\geq\epsilon_{s}\epsilon_{\mathcal{F}}\operatorname{tr}\Bigl{\\{}\sum_{i=1}^{p}\begin{bmatrix}A_{K,i}^{\top}\\\
B_{i}^{\top}\end{bmatrix}[\theta]_{i}\sum_{j=1}^{p}\begin{bmatrix}A_{K,j}&B_{j}\end{bmatrix}[\theta]_{j}\Bigr{\\}}$
$\displaystyle\quad\geq\sigma^{2}\epsilon_{s}\epsilon_{\mathcal{F}}\lVert{\theta}\rVert^{2}$

for all $\theta\in\mathbb{R}^{p}$, where $\sigma$ is the minimum singular
value of the matrix whose $i$th column is
$[\operatorname{vec}(A_{K,i})^{\top}\
\operatorname{vec}(B_{i})^{\top}]^{\top}$ for $i\in\mathbb{N}_{[1,p]}$ and
$\sigma>0$ by Assumption 2(d). Hence (7) holds with
$\epsilon_{\Phi}=\sigma^{2}\epsilon_{s}\epsilon_{\mathcal{F}}>0$. ∎

######  Remark 4.

In comparison to Theorem 2, Theorem 3 is less restrictive because it does not
rely on Assumption 5 and Assumption 6 can be easily satisfied because $s_{t}$
is user-defined.

It follows from Lemma 1 and Theorem 3 that Assumptions 1-3 and 6 ensure that
the minimal and fixed-complexity parameter set estimates under
$u_{t}=Kx_{t}+s_{t}$ converge to $\\{\theta^{\ast}\\}$ with probability 1.

##  3 Adaptive robust model predictive control

The noise input $s_{t}$ added to the feedback law in (9) can cause poor
tracking performance and may violate constraints on the states and control
inputs of the system (1). However, a receding horizon control law
incorporating injected noise can avoid these undesirable effects while
exploiting the PE properties it provides. Consider a predicted control law
parameterized at time $t$ in terms of decision variables
$\mathbf{v}_{t}=\\{v_{0|t},\ldots,v_{N-1|t}\\}$:

$\displaystyle
u_{k|t}=\begin{cases}Kx_{k|t}+v_{k|t},&k\in\mathbb{N}_{[0,N-1]}\\\
Kx_{k|t}+s_{k|t},&k\in\mathbb{N}_{[N,N+N_{u}-1]}\end{cases}$ (10)

where $N$ is the conventional MPC prediction horizon (see e.g. Kouvaritakis &
Cannon, 2016). The sequence
$\mathbf{s}_{t}=\\{\smash{s_{k|t}},\,k\in\mathbb{N}_{[0,N+N_{u}-1]}\\}$ is a
random sequence such that $\mathbf{s}_{t}$ is independent of $\mathbf{s}_{q}$
for all $t\neq q$, $t,q\in\mathbb{N}_{\geq 0}$ and $\mathbf{s}_{t}$ satisfies
Assumption 6 (i.i.d. with $\mathbb{E}(s_{k|t})=0$,
$\mathbb{E}(s_{k|t}\smash{s_{k|t}^{\top}})\succeq\epsilon_{s}{\mathcal{{I}}}\succ
0$, $s_{k|t}\in\mathcal{S}$, $\mathbf{s}_{t}$ independent of $x_{t}$ and
$w_{t}$). The feedback gain $K$ is assumed to be stabilizing (Assumption 4).

We assume linear state and control input constraints

$x_{k|t}\in\mathcal{X},\quad u_{k|t}\in\mathcal{U},\quad\forall
k\in\mathbb{N}_{[0,N+N_{u}-1]},$ (11)

where $\mathcal{X}$, $\mathcal{U}$ are given polytopes. To enforce these
constraints we define a terminal set $\mathcal{X}_{T}$ satisfying

$\displaystyle\mathcal{X}_{T}\subseteq\mathcal{X},\quad
K\mathcal{X}_{T}\oplus\mathcal{S}\subseteq\mathcal{U},$ (12) $\displaystyle
A_{K}(\theta)\mathcal{X}\oplus B(\theta)\mathcal{S}\oplus
F\mathcal{W}\subseteq\mathcal{X}_{T}\quad\forall\theta\in\Theta_{t}.$ (13)

###  3.1 Tube MPC formulation

To ensure satisfaction of constraints (11), we construct a sequence of sets
denoted
$\mathbf{X}_{t}=\\{\mathcal{X}_{k|t},\,k\in\mathbb{N}_{[0,N+N_{u}-1]}\\}$,
satisfying, for all $\theta\in\Theta_{t}$ and $k\in\mathbb{N}_{[0,N-1]}$,

$\displaystyle\mathcal{X}_{k|t}\subseteq\mathcal{X},\quad
K\mathcal{X}_{k|t}\oplus\\{v_{k|t}\\}\subseteq\mathcal{U},$ (14)
$\displaystyle
A_{K}(\theta)\mathcal{X}_{k|t}\oplus\\{B(\theta)v_{k|t}\\}\oplus
F\mathcal{W}\subseteq\mathcal{X}_{k+1|t},$ (15)

with initial and terminal conditions,

$\displaystyle\mathcal{X}_{0|t}$ $\displaystyle=\\{x_{t}\\},$ (16)
$\displaystyle\mathcal{X}_{k|t}$ $\displaystyle=\mathcal{X}_{T},\
k\in\mathbb{N}_{[N,N+N_{u}-1]},$ (17)

where $\mathcal{X}_{T}$ satisfies (12)-(13).

The tube $\mathbf{X}_{t}$ is an online optimization variable, and, depending
on the computational resources available, different tube formulations may be
employed. Here we assume homothetic (Lorenzen et al., 2019) or fixed-
complexity polytopic tubes (Lu & Cannon, 2019) that have convenient vertex
representations:
$\mathcal{X}_{k|t}=\smash{\operatorname{Co}\\{x_{k|t}^{(1)},\ldots,x_{k|t}^{(\nu)}\\}}$,
where $\nu$ is a fixed number of vertices.

Various objective functions have been proposed for adaptive MPC. Examples
include min-max piecewise linear Lu & Cannon (2019), min-max quadratic Adetola
et al. (2009), and nominal quadratic Lorenzen et al. (2019) cost functions.
Here we consider a nominal cost defined for a given nominal parameter vector
$\bar{\theta}_{t}\in\Theta_{t}$ by

$J(x_{t},{\bf
v}_{t},\bar{\theta}_{t})=\sum_{k=0}^{N-1}l(\bar{x}_{k|t},Kx_{k|t}+v_{k|t})+V_{N|t}(\bar{x}_{N|t})$

with $\bar{x}_{0|t}=x_{t}$,
$\bar{x}_{k+1|t}=A_{K}(\bar{\theta}_{t})\bar{x}_{k|t}+B(\bar{\theta}_{t})v_{k|t}$,
$k\in\mathbb{N}_{[0,N-1]}$. The stage cost $l(\cdot,\cdot)$ and terminal cost
$V_{N|t}(\cdot)$ are assumed to be positive definite quadratic functions
satisfying, for given $\bar{\theta}_{t}\in\Theta_{0}$ and all
$x\in\mathbb{R}^{n_{x}}$,

$V_{N|t}(x)=V_{N|t}\bigl{(}A_{K}(\bar{\theta}_{t})x\bigr{)}+l(x,Kx).$ (18)

In the remainder of the paper we use $(\mathbf{v}_{t}^{o},\mathbf{X}_{t}^{o})$
to denote the optimal decision variables $(\mathbf{v}_{t},\mathbf{X}_{t})$ for
the problem of minimizing $J(x_{t},\mathbf{v}_{t},\bar{\theta}_{t})$ subject
to constraints (14)-(17) and additional constraints to ensure the PE condition
(7), as we discuss next in Section 3.2.

###  3.2 PE condition

To define a recursively feasible set of constraints, we construct a series of
$N_{pe}$ overlapping PE windows extending from the past and across the
prediction horizon, where

$N_{pe}=\begin{cases}N+t,&t<N_{u}-1\\\ N+N_{u}-1,&t\geq N_{u}-1.\end{cases}$

Consider the PE conditions defined at time $t\in\mathbb{N}_{>0}$ for given
$\mathbf{X}_{t}$, $\mathbf{v}_{t}$, $\mathbf{s}_{t}$, and all
$\kappa\in\mathbb{N}_{[N-N_{pe},N-1]}$ by

$\sum_{k=\kappa}^{\kappa+N_{u}-1}\\!\\!\\!\Phi(x_{k|t},Kx_{k|t}+q_{k|t})^{\\!\top}\Phi(x_{k|t},Kx_{k|t}+q_{k|t})\succeq\beta_{\kappa|t}{\mathcal{{I}}}$
(19)

for all $x_{k|t}\in\mathcal{X}_{k|t}$,
$k\in\mathbb{N}_{[\kappa,\kappa+N_{u}-1]}$, where $q_{k|t}:=v_{k|t}$ if $k<N$
and $q_{k|t}:=s_{k|t}$ if $k\geq N$. The maximum value of $\beta_{\kappa|t}$
satisfying (19) defines a PE coefficient corresponding to a PE window starting
at time $t+\kappa$. With a slight abuse of notation, we allow $\kappa<0$ and
thus include past states and control inputs in this definition by defining
$\mathcal{X}_{k|t}:=\\{x_{k+t}\\}$, $Kx_{k|t}+v_{k|t}:=u_{k+t}$ whenever
$k<0$.

The PE conditions in (19) are nonconvex in $(\mathbf{v}_{t},\mathbf{X}_{t})$,
and hence are unsuitable as constraints in an online MPC optimization.
However, following Lu et al. (2021), we can linearize these conditions around
a reference trajectory
$(\hat{\mathbf{x}}_{t},\hat{\mathbf{u}}_{t})=\\{(\hat{x}_{k|t},\hat{u}_{k|t}),\,k\in\mathbb{N}_{[0,N+N_{u}-1]}\\}$.
For a given nominal parameter vector $\bar{\theta}_{t}$ and sequences
$\hat{\mathbf{v}}_{t}$, $\mathbf{s}_{t}$, the reference trajectory is defined
by

$\displaystyle\hat{x}_{0|t}=x_{t}$ (20a)
$\displaystyle\hat{x}_{k+1|t}=A(\bar{\theta}_{t})\hat{x}_{k|t}+B(\bar{\theta}_{t})\hat{u}_{k|t},\,k\in\mathbb{N}_{[0,N+N_{u}-2]}\\!\\!$
(20b)
$\displaystyle\hat{u}_{k|t}=\begin{cases}K\hat{x}_{k|t}+\hat{v}_{k|t},&k\in\mathbb{N}_{[0,N-1]}\\\
K\hat{x}_{k|t}+{s}_{k|t},&k\in\mathbb{N}_{[N,N+N_{u}-1]}.\end{cases}$ (20c)

We define the sequence $\hat{\mathbf{v}}_{t}$ for $t\in\mathbb{N}_{>0}$ using
the solution of the MPC optimization at time $t-1$, denoted
$\mathbf{v}^{o}_{t-1}=\\{v_{0|t-1}^{o},\ldots,v_{N-1|t-1}^{o}\\}$, and
$s_{N-1|t}$:

$\hat{v}_{k|t}=\begin{cases}v^{o}_{k+1|t-1},&k\in\mathbb{N}_{[0,N-2]}\\\
s_{N-1|t},&k=N-1.\end{cases}$ (21)

A nominal parameter vector $\bar{\theta}_{t}\in\Theta_{t}$ can be obtained for
example by projecting an RLS parameter estimate onto $\Theta_{t}$ (Lorenzen et
al., 2019) or by projecting $\bar{\theta}_{t-1}$ onto $\Theta_{t}$,

$\bar{\theta}_{t}=\operatorname*{argmin}_{\theta\in\Theta_{t}}\|\bar{\theta}_{t-1}-\theta\|.$
(22)

Linearizing (19) by neglecting quadratic terms in the decision variables
$(\mathbf{v}_{t},\mathbf{X}_{t})$ yields a set of LMIs in $\mathbf{v}_{t}$,
the vertices, ${x_{k|t}^{(j)}}$, of $\mathbf{X}_{t}$, and additional
optimization variables
$\boldsymbol{\beta}_{t}^{\prime}=\\{\beta_{\kappa|t}^{\prime},\,\kappa\in\mathbb{N}_{[N-N_{pe},N-1]}\\}$,
given for $\kappa\in\mathbb{N}_{[N-N_{pe},N-1]}$ by

$\sum_{k=\kappa}^{-1}\Phi_{k+t}^{\top}\Phi_{k+t}+\sum_{k=\max\\{0,\kappa\\}}^{\kappa+N_{u}-1}M_{k|t}\succeq{\beta}_{\kappa|t}^{\prime}{\mathcal{{I}}}$
(23)

where $M_{k|t}=M_{k|t}^{\top}\in\mathbb{R}^{p\times p}$,
$k\in\mathbb{N}_{[0,N+N_{u}-2]}$ satisfies

$\displaystyle M_{k|t}$
$\displaystyle\preceq\hat{\Phi}_{k|t}^{\top}\hat{\Phi}_{k|t}+\hat{\Phi}_{k|t}^{\top}\Phi(x^{(j)}_{k|t}-\hat{x}_{k|t},Kx^{(j)}_{k|t}+q_{k|t}-\hat{u}_{k|t})$
$\displaystyle\quad+\Phi(x^{(j)}_{k|t}-\hat{x}_{k|t},Kx^{(j)}_{k|t}+q_{k|t}-\hat{u}_{k|t})^{\top}\hat{\Phi}_{k|t},$

for all $j\in\mathbb{N}_{[1,\nu]}$, with
$\hat{\Phi}_{k|t}=\Phi(\hat{x}_{k|t},\hat{u}_{k|t})$.

To increase the probability that the MPC law satisfies the PE condition (7),
we include (23) in the MPC optimization at times $t>0$ with the following
constraints on $\boldsymbol{\beta}_{t}^{\prime}$

$\smash{\beta_{\kappa|t}^{\prime}\geq\hat{\beta}^{\prime}_{\kappa|t},\ \
\forall\kappa\in\mathbb{N}_{[N-N_{pe},N-1]},}$ (24)

where $\smash{\hat{\beta}_{\kappa|t}^{\prime}}$ is a lower bound on the
maximum value of $\smash{\beta_{\kappa|t}^{\prime}}$ satisfying (23) for
$(\mathbf{v}_{t},\mathbf{X}_{t})$ satisfying (14)-(17). Thus we determine
$\smash{\hat{\boldsymbol{\beta}}\mbox{}^{\prime}_{t}}=\\{\hat{\beta}_{\kappa|t}^{\prime},\,\kappa\in\mathbb{N}_{[N-N_{pe},N-1]}\\}$
by finding $\smash{\hat{\beta}_{\kappa|t}^{\prime}}$ in (24) as the solution
of

$\hat{\beta}_{\kappa|t}^{\prime}:=\max_{\beta_{\kappa|t}^{\prime}\in\mathbb{R}}\beta_{\kappa|t}^{\prime}\
\ \text{s.t.}\ \ (\ref{eq:PE_windows})$ (25)

with $\smash{q_{k|t}:=\hat{v}_{k|t}}$ if $k<N$ and $\smash{q_{k|t}:=s_{k|t}}$
if $k\geq N$, and with $x_{k|t}^{(j)}:=x_{k+1|t-1}^{(j)\,o}$, $\forall
j\in\mathbb{N}_{[1,\nu]}$, $k>0$, where $\smash{x_{k|t-1}^{(j)\,o}}$ denotes a
vertex of $\mathbf{X}_{t-1}^{o}$.

To ensure that (7) holds with non-zero probability for all $t$, we propose a
check on the solution of the online MPC optimization based on the nonconvex
condition

$\beta_{\kappa|t}\geq\hat{\beta}_{\kappa|t},\qquad\forall\kappa\in\mathbb{N}_{[N-N_{pe},N-1]},$
(26)

where $\beta_{\kappa|t}$ is defined in (19) and $\hat{\beta}_{\kappa|t}$ has
the same definition except that $q_{k|t}:=\hat{v}_{k|t}$ if $k<N$.

Condition (26) is difficult to check directly because (19) is nonconvex in
$x_{k|t}$ and $q_{k|t}$. Therefore we check instead an approximate condition
using sampling. For each $k\in\mathbb{N}_{[0,N+N_{u}-1]}$, we draw $N_{s}$
samples $\smash[b]{\\{{x}_{k|t}^{(1)},\ldots,{x}_{k|t}^{(N_{s})}\\}}$
uniformly distributed on $\mathcal{X}_{k|t}$ and compute the corresponding
regressor products defined by

$\Psi^{(i)}_{k|t}(\mathbf{v}_{t})=\Phi(x^{(i)}_{k|t},Kx^{(i)}_{k|t}+v_{k|t})^{\top}\Phi(x^{(i)}_{k|t},Kx^{(i)}_{k|t}+v_{k|t})$

for $i\in\mathbb{N}_{[1,N_{s}]}$. We then impose the condition
$\smash{\beta^{s}_{\kappa|t}\geq\hat{\beta}^{s}_{\kappa|t}}$ with sampled
states, where $\beta^{s}_{\kappa|t}$ and $\hat{\beta}^{s}_{\kappa|t}$ are
defined as

$\displaystyle\beta^{s}_{\kappa|t}$ $\displaystyle:=\max_{\beta}\beta\
\text{s.t.}\min_{\mathbf{i}\in\mathbb{N}_{[1,N_{s}]}\times\cdots\times\mathbb{N}_{[1,N_{s}]}}\\!\\!\\!\\!\sum_{k=\kappa}^{\kappa+N_{u}-1}\\!\\!\\!\Psi^{(i_{k})}_{k|t}(\mathbf{v}^{o}_{t})\succeq\beta{\mathcal{{I}}}$
$\displaystyle\hat{\beta}^{s}_{\kappa|t}$ $\displaystyle:=\max_{\beta}\beta\
\text{s.t.}\min_{\mathbf{i}\in\mathbb{N}_{[1,N_{s}]}\times\cdots\times\mathbb{N}_{[1,N_{s}]}}\\!\\!\\!\\!\sum_{k=\kappa}^{\kappa+N_{u}-1}\\!\\!\\!\Psi^{(i_{k})}_{k|t}(\hat{\mathbf{v}}_{t})\succeq\beta{\mathcal{{I}}}$

where $\mathbf{i}=(i_{\kappa},\ldots,i_{\kappa+N_{u}-1})$. Since the samples
$x^{(i)}_{k|t}$ are uniformly distributed over $\mathcal{X}_{k|t}$, and the
product $\smash{\Psi^{(i)}_{k|t}(\mathbf{v}_{t})}$ is continuous in
$\smash{x^{(i)}_{k|t}}$, we can infer that (26) is satisfied with a
probability of at least $p_{s}$, where $p_{s}>0$, whenever
$\beta^{s}_{\kappa|t}\geq\hat{\beta}^{s}_{\kappa|t}$. The adaptive MPC
algorithm is summarized in Algorithm 1.

Algorithm 1 Adaptive MPC with PE constraints

At $t=0$: Choose $\Theta_{0}$ and $\mathcal{S}$. Determine $K$ satisfying
Assumption 4 and $\mathcal{X}_{T}$, $V_{N|0}$ satisfying (12)-(13) and (18).
Define $N$, $N_{u}$, $N_{w}$, $N_{\theta}$. Obtain $\bar{\theta}_{0}$ and
$x_{0}$ and compute the solution, $(\mathbf{v}^{o}_{0},\mathbf{X}^{o}_{0})$,
of the quadratic program (QP):

$\displaystyle\mathcal{P}_{0}:\
\operatorname*{minimize}_{\mathbf{v}_{0},\mathbf{X}_{0}}\
J(x_{0},\mathbf{v}_{0},\bar{\theta}_{0})\ \ \text{s.t.\
(\ref{eq:tube_xu_constraint})-(\ref{eq:terminal_constraint})}.$ (27)

Apply the control input $u_{0}=Kx_{0}+v_{0|0}^{o}$.

At times $t=1,2,\ldots$:

  1. (a).

Obtain the current state $x_{t}$.

  2. (b).

Update $\bar{\theta}_{t}$, $\Theta_{t}$ and $V_{N|t}$ using(5), (22) and (18).

  3. (c).

Generate the noise sequence $\mathbf{s}_{t}$ and compute
$\hat{\mathbf{x}}_{t}$, $\hat{\mathbf{u}}_{t}$, $\hat{\mathbf{v}}_{t}$,
$\smash{\hat{\boldsymbol{\beta}}\mbox{}^{\prime}_{t}}$ using (20), (21) and
(25).

  4. (d).

Compute the solution $(\mathbf{v}^{o}_{t},\mathbf{X}_{t}^{o})$ of the
semidefinite program (SDP):

$\displaystyle\mathcal{P}_{>0}:$
$\displaystyle\operatorname*{minimize}_{\mathbf{v_{t}},\mathbf{X}_{t},\boldsymbol{\beta}_{t}}\
J(x_{t},\mathbf{v}_{t},\bar{\theta}_{t})$ (28) $\displaystyle\ \text{s.t.\
(\ref{eq:terminala})-(\ref{eq:terminal_constraint}), (\ref{eq:PE_windows}) and
(\ref{eq:beta_bound_moving})}.$

  5. (e).

Generate $N_{s}$ samples of $\mathcal{X}_{k|t}$ and $N_{s}$ samples of
$\mathcal{X}^{o}_{k+1|t-1}$ for each
$k\in\smash{\mathbb{N}_{[\kappa,\kappa+N_{u}-1]}}$ and compute
$\beta^{s}_{\kappa|t}$ and $\hat{\beta}^{s}_{\kappa|t}$ for
$\kappa\in\mathbb{N}_{[N-N_{pe},N-1]}$. If
$\beta^{s}_{\kappa|t}<\hat{\beta}^{s}_{\kappa|t}$ for any $\kappa$, set
$\mathbf{v}_{t}^{o}:=\hat{\mathbf{v}}_{t}$ and
$\mathbf{X}_{t}^{o}:=\bigl{\\{}\\{x_{t}\\},\mathcal{X}^{o}_{2|t-1},\ldots,$
$\smash[t]{\mathcal{X}^{o}_{N-1|t-1},\mathcal{X}_{T},\ldots,\mathcal{X}_{T}\bigr{\\}}}$.

  6. (f).

Apply the control input $u_{t}=Kx_{t}+v_{0|t}^{o}$

######  Remark 5.

PE constraints are applied to $N_{pe}$ windows at each time step, and each
constraint (23) is a LMI. Therefore the computational load of online
optimization initially increases over time and levels off at time
$t=N+N_{u}-1$.

######  Remark 6.

Alternative parameter set and nominal parameter update laws can be used in
Algorithm 1 instead of (5) and (22), provided that
$\Theta_{t+1}\subseteq\Theta_{t}$ and $\bar{\theta}_{t}\in\Theta_{t}$.

###  3.3 Recursive Feasibility and Input-to-State Stability

######  Proposition 7 (Recursive feasibility).

The online MPC optimization in Algorithm 1 is feasible at all times $t\geq 1$
if (28) is feasible at $t=1$ and $\Theta_{t}\subseteq\Theta_{t-1}$ for all
$t$.

######  Proposition 8 (Stability).

Let Assumptions 1, 2 and 4 hold. Then the system (1) under Algorithm 1 is
input-to-state practically stable (ISpS) (Limon et al., 2009, Def. 6) in the
set of initial conditions $x_{0}$ for which $\mathcal{P}_{0}$ is feasible.

Proof. For $\mathbf{s}_{t}$, $\mathbf{v}^{o}_{t}$, $\hat{\mathbf{v}}_{t}$ in
Algorithm 1 steps (c) and (d) let $\zeta^{o}_{k|t}=v_{k|t}^{o}-s_{k|t}$ and
$\smash{\hat{\zeta}}_{k|t}=\smash{\hat{v}}_{k|t}-s_{k|t}$ for all
$k\in\mathbb{N}_{[0,N-1]}$.

Since $\mathcal{P}_{>0}$ is necessarily feasible if $\mathcal{P}_{0}$ is
feasible, input-to-state practical stability follows from Theorem 1 and
Corollary 1 of Lu et al. (2021). This can be shown using the proofs of (Lu et
al., 2021, Theorem 1 and Corollary 1) with $v_{k|t}$ and $\hat{v}_{k|t}$
replaced respectively by $\zeta_{k|t}$ and $\smash{\hat{\zeta}_{k|t}}$, and
with disturbance input $w_{t}$ replaced by $Fw_{t}+B(\theta^{\ast})s_{0|t}$. ∎

Proposition 8 implies (Limon et al., 2009; Lu et al., 2021) the existence of a
$\mathcal{K}\mathcal{L}$-function $\eta$ and $\mathcal{K}$-functions $\psi$,
$\xi$ satisfying, for all $x_{0}$ such that $\mathcal{P}_{0}$ is feasible, the
bound

$\displaystyle\|x_{t}\|$
$\displaystyle\leq\eta(\|x_{0}\|,t)+\psi\bigl{(}\max_{\tau\in\mathbb{N}_{[0,t-1]}}\|Fw_{\tau}+B(\theta^{\ast})s_{0|\tau}\|\bigr{)}$
$\displaystyle\quad+\xi\bigl{(}\max_{\tau\in\mathbb{N}_{[0,t-1]}}\|\bar{\theta}_{\tau}-\theta^{\ast}\|\bigr{)}.$

###  3.4 Closed loop satisfaction of the PE condition

We define the closed loop PE coefficient $\epsilon_{t}$ at any time $t\geq
N_{u}-1$ as the maximum value of $\epsilon_{t}$ satisfying

$\sum_{k=t-N_{u}+1}^{t}\Phi(x_{k},u_{k})^{\top}\Phi(x_{k},u_{k})\succeq\epsilon_{t}{\mathcal{{I}}}.$
(29)

To ensure closed loop satisfaction of the PE condition, a more restrictive
version of Assumption 2(c) is needed.

######  Assumption 7.

The pair $\bigl{(}A_{K}(\theta_{t}),B(\theta_{t})\bigr{)}$ is reachable for
all sequences $\\{\theta_{0},\theta_{1},\ldots\\}$ such that
$\theta_{t}\in\Theta_{0}$ for all $t$.

This reachability assumption can be verified by checking whether a set of LMIs
defined by the vertices of $\Theta_{0}$ is feasible.

######  Theorem 9.

Under Assumptions 1, 2, 6 and 7, the closed loop PE coefficient $\epsilon_{t}$
in (29) under Algorithm 1 satisfies

$\mathbb{E}(\epsilon_{t})\succeq\epsilon_{\Phi}$ (30)

for some $\epsilon_{\Phi}>0$ with non-zero probability, for all $t\geq
N+N_{u}-1$.

Proof. Given the optimal control $\mathbf{v}_{t}^{o}$ at time $t$, the
predicted PE coefficient $\beta_{\kappa|t}$ in (19) can be rewritten as

$\displaystyle\beta_{\kappa|t}=\min_{x_{k|t}\in\mathcal{X}_{k|t}}\max_{\beta}\beta$
(31) $\displaystyle\text{s.t.
}\sum_{k=\kappa}^{\kappa+N_{u}-1}\Phi(x_{k},Kx_{k}+q_{k|t})^{\top}\Phi(x_{k},Kx_{k}+q_{k|t})\succeq\beta{\mathcal{{I}}}$

where $q_{k|t}:=v^{o}_{k|t}$ if $k<N$ and $q_{k|t}:=s_{k|t}$ if $k\geq N$.

Using the reference input $\hat{\bf v}_{t}$ defined in (21), the reference PE
coefficient $\hat{\beta}_{\kappa-1|t+1}$ in (25) of the same PE window is

$\displaystyle\hat{\beta}_{\kappa-1|t+1}=\min_{z_{k}\in\mathcal{Z}_{k}}\max_{\beta}\beta$
(32)
$\displaystyle\text{s.t.}\sum_{k=\kappa}^{\kappa+N_{u}-1}\Phi(z_{k},Kz_{k}+{q}_{k|t})^{\top}\Phi(z_{k},Kz_{k}+{q}_{k|t})\succeq\beta{\mathcal{{I}}}$

where $\mathcal{Z}_{k}=\mathcal{X}_{k-1|t+1}$ and $q_{k|t}:=\hat{v}^{o}_{k|t}$
if $k<N-1$ and $q_{k|t}:=s_{k-1|t+1}$ if $k\geq N-1$.

But $\Theta_{t+1}\subseteq\Theta_{t}$ and the constraints (16) imply
$\mathcal{Z}_{k}\subseteq\mathcal{X}_{k|t}$, for all
$k\in\mathbb{N}_{[1,N+N_{u}-1]}$. Given that the injected noise sequences
$\mathbf{s}_{t+1}$ and $\mathbf{s}_{t}$ are i.i.d with zero mean and the
regressor $\Phi(x,u)$ is affine in $(x,u)$, we can deduce that
$\mathbb{E}_{t}\bigl{[}\hat{\beta}_{\kappa-1|t+1}\bigr{]}\geq\mathbb{E}_{t}\bigl{[}\beta_{\kappa|t}\bigr{]}$.
Combining this with the bounds enforced by step (e) in Algorithm 1, we have

$\mathbb{E}_{t}\bigl{[}\beta_{\kappa-1|t+1}\bigr{]}\geq\mathbb{E}_{t}\bigl{[}\hat{\beta}_{\kappa-1|t+1}\bigr{]}\geq\mathbb{E}_{t}\bigl{[}\beta_{\kappa|t}\bigr{]}$
(33)

with probability $p_{s}>0$.

Next we derive a lower bound on $\mathbb{E}_{t}[\beta_{N|t}]$. Given
Assumption 7, there exists a scalar $\epsilon_{\mathcal{B}}>0$ satisfying
$\sum_{j=0}^{n_{x}-1}[A_{K}(\theta)]^{j}B(\theta)B(\theta)^{\top}[{A_{K}(\theta)^{\top}}]^{j}\succeq\epsilon_{\mathcal{B}}{\mathcal{{I}}}$,
and hence
$\sum_{k=N}^{N+N_{u}-1}\mathbb{E}_{T}[x_{k|t}x_{k|t}^{\top}]\succeq(N_{u}-n_{x})\epsilon_{\mathcal{B}}\mathbb{E}(s_{k}s_{k}^{\top})$.
The predicted PE coefficient $\beta_{N|t}$ therefore satisfies

$\displaystyle\mathbb{E}_{t}$
$\displaystyle[\beta_{N|t}]\overset{(\text{E1})}{=}\min_{{\theta}\in\mathbb{B}^{p}}\min_{x_{k|t}\in\mathcal{X}_{k|t}}\Bigg{\\{}\sum_{k=N}^{N+N_{u}-1}\mathbb{E}_{t}\Bigg{[}s_{k}^{\top}B({\theta})^{\top}B({\theta})s_{k}$
$\displaystyle\hskip
85.35826pt+\Big{(}A_{K}({\theta})x_{k|t}\Big{)}^{\top}\Big{(}A_{K}({\theta})x_{k|t}\Big{)}\Bigg{]}\Bigg{\\}}$
$\displaystyle\overset{(\text{I2})}{\geq}$
$\displaystyle\min_{{\theta}\in\mathbb{B}^{p}}\text{tr}\Bigg{\\{}\Big{(}A_{K}({\theta})\Big{)}^{\top}\Big{(}A_{K}({\theta})\Big{)}(N_{u}-n_{x})\epsilon_{\mathcal{B}}\mathbb{E}\Big{[}s_{k}s_{k}^{\top}\Big{]}$
$\displaystyle\hskip
85.35826pt+B({\theta})^{\top}B({\theta})\sum_{k=N}^{N+N_{u}-1}\mathbb{E}\Big{[}s_{k}s_{k}^{\top}\Big{]}\Bigg{\\}}$
$\displaystyle\overset{(\text{I3})}{\geq}$
$\displaystyle\min_{{\theta}\in\mathbb{B}^{p}}\Big{\\{}\bigl{\|}\text{vec}\Big{(}\sum_{i=1}^{p}\begin{bmatrix}\epsilon^{\prime}_{\mathcal{B}}A_{K,i}&\sqrt{N_{u}}B_{i}\end{bmatrix}[{\theta}]_{i}\Big{)}\bigr{\|}^{2}\epsilon_{s}\Big{\\}}$
$\displaystyle\overset{(\text{I4})}{\geq}$
$\displaystyle\epsilon_{s}\sigma^{2}$

where
$\epsilon^{\prime}_{\mathcal{B}}=\sqrt{(N_{u}-n_{x})\epsilon_{\mathcal{B}}}$
and $\sigma$ is the minimum singular value of the matrix whose $i$th column is
$[\epsilon^{\prime}_{\mathcal{B}}\operatorname{vec}(A_{K,i})^{\top}\
\sqrt{N_{u}}\operatorname{vec}(B_{i})^{\top}]^{\top}$ for
$i\in\mathbb{N}_{[1,p]}$. Here equality (E1) holds because $x_{k|t}$ and the
injected noise $s_{k|t}$ are independent and $\mathbb{E}[s_{k|t}]=0$,
inequality (I2) holds by substitution, inequality (I3) follows from
Assumptions 2 and 6, and $\sigma>0$ by Assumption 2.

Applying (33) for $\kappa=N,N-1,\ldots,N-N_{pe}+1$, and noting that
$\epsilon_{t}={\beta}_{N-N_{pe}|t}$, we therefore have

$\mathbb{E}_{t}\bigl{[}\epsilon_{t+{N_{pe}}}\bigr{]}\geq\mathbb{E}_{t}\bigl{[}\beta_{N|t}\bigr{]}\geq\epsilon_{\Phi}$

with probability $\smash{p_{s}^{N_{pe}}}>0$, where
$\epsilon_{\Phi}=\epsilon_{s}\sigma^{2}>0$. ∎

If Assumptions 1-4 and 6 hold, then Lemma 1 and Theorem 9 imply that the
parameter set estimate $\Theta_{t}$ converges under Algorithm 1 to
$\\{\theta^{\ast}\\}$ with probability 1.

##  4 Numerical example

In this section, we use the example in Lorenzen et al. (2019) to illustrate
that the proposed adaptive MPC algorithms can produce persistently exciting
input signals while satisfying the system input and state constraints, and
providing performance close to the optimal performance for MPC without PE
constraints.

Consider the second-order uncertain linear system (Lorenzen et al., 2019)
defined by (1) with

$\displaystyle A_{0}=\begin{bmatrix}0.5&0.2\\\ -0.1&0.6\end{bmatrix},\
A_{1}=\begin{bmatrix}0.042&0\\\ 0.072&0.03\end{bmatrix},$ $\displaystyle
A_{2}=\begin{bmatrix}0.015&0.019\\\ 0.009&0.035\end{bmatrix},\
A_{3}=\begin{bmatrix}0&0\\\ 0&0\end{bmatrix},$ $\displaystyle
B_{0}=\begin{bmatrix}0\\\ 0.5\end{bmatrix},\ B_{1}=\begin{bmatrix}0\\\
0\end{bmatrix},\ B_{2}=\begin{bmatrix}0\\\ 0\end{bmatrix},\
B_{3}=\begin{bmatrix}0.040\\\ 0.054\end{bmatrix},$

$\theta^{*}=[0.8\ \ 0.2\ \ {-0.5}]^{\top\\!}$ and $K=[0.017\ \ {-0.41}]$. The
initial parameter set estimate is
$\Theta_{0}=\\{\theta:\|\theta\|_{\infty}\leq 1\\}$, and for all $t\geq 0$,
$\Theta_{t}$ is a hyperrectangle, with $\mathcal{M}_{\theta}=[{\mathcal{{I}}}\
\ {-{\mathcal{{I}}}}]^{\top}$. The bounded disturbance sequence
$\\{w_{0},w_{1},\ldots\\}$ follows a truncated normal distribution with zero
mean, covariance $\sigma_{w}=0.06$, and bounds
$w_{t}\in\mathcal{W}=\\{w:|w|\leq 0.05\\}$. The injected noise sequence
$\\{s_{0},s_{1},\ldots\\}$ follows a uniform distribution with zero mean and
$s_{t}\in\mathcal{S}=\\{s:|s|\leq 0.005\\}$. The state and input constraints
are $[x_{t}]_{2}\geq-0.3$ and $u_{t}\leq 1$. The MPC prediction horizon and PE
window length are $N=10$ and $N_{u}=3$ respectively. For these parameters we
obtain $\epsilon_{s}=8.33\times 10^{-6}$, $\sigma=1.27\times 10^{-3}$ and
$\epsilon_{\Phi}=1.35\times 10^{-11}$ in Theorem 9.

For comparison with Algorithm 1 we define a robust adaptive MPC law (Algorithm
2) with online parameter set update but without PE constraints in the MPC
optimization.

Algorithm 2 Robust Adaptive MPC

Modify Algorithm 1 by omitting the constraints (23) and (24) from problem
$\mathcal{P}_{>0}$ and omitting step (e).

All simulations were performed in Matlab on a 2.9 GHz Intel Core i7-10700F
processor, and the online MPC optimization was formulated using Yalmip
(Löfberg, 2019). Mosek (MOSEK ApS, 2019) and Gurobi (Gurobi Optimization, LLC,
2021) were used to solve SDP and QP problems respectively. For the purposes of
comparison, identical disturbance sequences and the same parameter update laws
were employed in each simulation.

Table 1 compares Algorithms 1, 2, and the linear feedback law with injected
noise ($u_{t}=Kx_{t}+s_{t}$), in terms of the parameter set volume (denoted
$\lvert\Theta_{t}\rvert$), empirical mean closed loop PE coefficient
($\hat{\epsilon}$), and average solver run-time. Here the observed average
closed loop PE coefficient is defined as

$\hat{\epsilon}=\frac{1}{N_{av}N_{w}}\sum_{i=1}^{N_{w}}\sum_{k=0}^{N_{av}-1}\epsilon_{k}$

with $N_{av}=500$ time steps and $N_{w}=30$ disturbance sequences. Algorithm 1
outperforms Algorithm 2 in terms of the convergence rate of its estimated
parameter set, which is consistent with the higher value of $\hat{\epsilon}$
observed for Alg. 1. The linear feedback law provides a value of
$\hat{\epsilon}$ similar to Alg. 1, however this control law does not
guarantee constraint satisfaction because the initial state is outside the
terminal region.

Table 1: Parameter set volume, PE coefficient, and computation

$Kx_{t}+s_{t}$ Alg. 1 Alg. 2 $\lvert\Theta_{100}\rvert/\lvert\Theta_{0}\rvert$
$(\%)$ 22.82 20.83 22.18 $\lvert\Theta_{200}\rvert/\lvert\Theta_{0}\rvert$
$(\%)$ 21.47 19.93 21.00 $\lvert\Theta_{500}\rvert/\lvert\Theta_{0}\rvert$
$(\%)$ 17.24 15.54 17.54 $\hat{\epsilon}$ ($\times 10^{-7}$) $1.042$ $1.044$
$0.547$ Step (c) solver time (s) - 0.080 - Step (d) solver time (s) - 0.027
0.075 Step (e) solver time (s) - 3.82 -

The average computation time for the QP in step (d) of Alg. 2 exceeds that of
the SDP in step (d) of Alg. 1. However, Alg. 1 requires an additional $0.08$ s
per time step to compute
$\smash{\hat{\boldsymbol{\beta}}\mbox{}_{t}^{\prime}}$ in step (c) and $3.82$
s to perform step (e) with $N_{s}=30$. Note that
$\smash{\beta_{\kappa|t}<\hat{\beta}_{\kappa|t}^{s}}$ occurred in step (e) of
Alg. 1 at only $20\%$ of time-steps. This is consistent with the observation
that, due to the presence of the constraints (23) and (24) in problem
$\mathcal{P}_{>0}$, Alg. 1 provides a higher value of $\hat{\epsilon}$ and
faster parameter convergence than Alg. 2 even if the time-consuming step (e)
is omitted from Alg. 1, although the guarantee of Theorem 9 then no longer
applies.

Figure 1: Parameter convergence: Alg. 1, 2 and $u_{t}=Kx_{t}+s_{t}$

Figure 1 compares the volumes of estimated parameter sets for Algorithms 1, 2
and linear feedback with injected noise. For Alg. 1 (blue) with PE
constraints, the volume decreases faster than for Alg. 2 (red) without PE
constraints or linear feedback with injected noise (green). Figure 2 shows the
closed loop value of $\epsilon_{t}$ obtained using Algorithms 1, 2 and linear
feedback with injected noise. As expected from the empirical mean
$\hat{\epsilon}$ (Table 1), Alg. 1 is observed to inject more excitation into
the system than Alg. 2, and this explains its faster convergence rate. The
smaller parameter set volume brings 2.2% improvement in closed loop cost for
Alg. 1 over Alg. 2 with identical initial conditions and disturbance
sequences.

Figure 2: PE coefficient $\epsilon_{t}$: Alg. 1, 2 and $u_{t}=Kx_{t}+s_{t}$

##  5 Conclusions

The proposed robust adaptive MPC algorithm exploits the persistence of
excitation properties of linear feedback laws with injected noise while
ensuring satisfaction of constraints on states and control inputs, and optimal
tracking/regulation performance. This is achieved by incorporating recursively
feasible constraints into the online MPC optimization which, when combined
with a posterior sample-based check, ensures persistence of excitation.
Guarantees of input-to-state practical stability and asymptotic parameter set
convergence are provided.

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