###### Proof.

######  Configuration-Constrained Tube MPC for Tracking

###### Abstract

This paper proposes a novel tube-based Model Predictive Control (MPC)
framework for tracking varying setpoint references with linear systems subject
to additive and multiplicative uncertainties. The MPC controllers designed
using this framework exhibit recursively feasible for changing references, and
robust asymptotic stability for piecewise constant references. The framework
leverages configuration-constrained polytopes to parameterize the tubes,
offering flexibility to optimize their shape. The efficacy of the approach is
demonstrated through two numerical examples. The first example illustrates the
theoretical results, and the second uses the framework to design a lane-change
controller for an autonomous vehicle.

##  1 Introduction

The inherent robustness of Model Predictive Control (MPC) derives from solving
optimal control problems in a receding horizon manner based on state feedback,
rendering it effective in managing complex control tasks. However, in the
presence of significant modelling errors or large disturbances, this may prove
inadequate [20], due to the mismatch between predictions and the actual
behavior of the controlled process. Tube-based MPC (TMPC) has emerged as a
suitable candidate to take an uncertainty model into account. The principle
behind TMPC is simple: instead of optimizing over vector-valued system
trajectories, one optimizes over Robust Forward Invariant Tubes (RFITs) —
together with their associated feedback policies — that are set-valued
sequences enclosing all realizations of the uncertain system. Particular set
and control parameterizations lead to different TMPC methods, for example,
rigid- [12, 16], homothetic [9, 18], elastic [19, 3], configuration-
constrained tube MPC [23] as well as other methods based on ellipsoidal [24]
or more general sets [8].

In most practical applications, the control system is used to make the outputs
track given set points — typically determined by an external module, such as a
steady-state optimizer [14] — that may change over time. For example, in
autonomous driving problems, the desired lateral displacement changes as the
vehicle switches lanes. The extension of TMPC to reference tracking problems
then represents a significant milestone in robust MPC design, addressing
challenges arising from uncertain environments and the change in operating set
points. Early TMPC approaches for reference tracking were based on a rigid
tube formulation and were limited to linear systems with additive disturbances
[10]. Subsequent works [6, 15] broadened this concept to include
multiplicative uncertainty. All these approaches have in common that they
enforce asymptotic stability by using parametric terminal sets, which requires
solving optimization problems with additional variables.

#### Contribution

This paper introduces a variant of Configuration-Constrained Tube MPC (CCTMPC)
for varying setpoint reference tracking, based on linear prediction models
subject to both additive and multiplicative uncertainty. In contrast to the
CCTMPC scheme of [23] which relies on an economic MPC formulation, our
approach is developed using a tracking MPC formulation. This enables the
synthesis of reference tracking controllers that hinge on the solution of a
single quadratic program to guarantee asymptotic stability for piecewise
constant references. This extends the applicability of CCTMPC beyond
regulation problems.

The use of configuration-constrained polytopes for parameterizing the RFIT
results in a controller that exhibits superior properties compared to [10, 6].
Notably, the need for a robust affine feedback law is eliminated, and the
resulting RFITs exhibit an improved representational capability, leading to
less conservativeness. A particularly relevant feature for the reference
tracking problem is the unrestricted online optimization of the shape of the
optimal steady-state set around the desired reference. The ability of our
approach to handle multiplicative uncertainty, coupled with the removal of the
requirement for precomputing a terminal set, renders it apt for designing
tracking controllers for practical systems that can be represented by linear
parameter-varying (LPV) models [7]. Furthermore, the scheme can seamlessly
incorporate model refinement while preserving asymptotic stability, making it
well-suited for the development of adaptive MPC schemes [2].

The paper is organized as follows: Section 2 introduces the problem
formulation. Section 3 presents the tracking CCTMPC formulation as well as a
novel terminal cost and related theoretical guarantees regarding recursive
feasibility in the presence of reference changes, robustness, and stability,
as summarized in Theorem 1 and Corollary 1. Numerical examples are presented
in Section 4. Section 5 concludes the article.

Notation: The symbol $\mathbb{I}$ represents the identity matrix. For vectors
$a$ and $b$, the symbol $(a,b)$ denotes $[a^{\top}\ b^{\top}]^{\top}$. A
block-diagonal matrix formed by matrices
$Q_{1},\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern 0.09995pt,Q_{n}$
is denoted by
$\mathrm{blkd}(Q_{1},\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,Q_{n})$. The symbol $\operatorname{convh}$ denotes the convex hull,
and we define $\left\lVert z\right\rVert_{Q}^{2}:=z^{\top}Qz$.

##  2 Tube Model Predictive Control

This section introduces the necessary notation for our reference-tracking Tube
MPC formulation. Here, we focus on Configuration-Constrained polytopic tube
parameterization, as originally introduced in [23].

###  2.1 Uncertain Linear Systems

This paper considers uncertain systems of the form

$\displaystyle x^{+}=Ax+Bu+w\quad\text{with}\quad z=Cx+Du,$ (1)

where $x\in\mathbb{R}^{n_{x}}$, $u\in\mathbb{R}^{n_{u}}$, and
$z\in\mathbb{R}^{n_{z}}$ denote the state, input and output respectively, and
$w\in\mathcal{W}$ an additive disturbance. The set $\mathcal{W}$ is compact
and convex. While $C$ and $D$ are given matrices, $A$ and $B$ are only known
to satisfy
$\displaystyle(A,B)\in\Delta:=\operatorname{convh}(\\{(A_{1},B_{1}),\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,(A_{p},B_{p})\\})$ . The system is subject to closed and convex
constraints $x\in\mathcal{X}$ and $u\in\mathcal{U}$. Our aim is to design an
MPC scheme that robustly steers the output $z$ towards a piecewise constant
reference $r$ that may be updated online.

###  2.2 Robust Forward Invariant Tubes

Tube MPC methods are based on the construction of Robust Forward Invariant
Tubes (RFITs). Let

denote the set of one-step forward reachable sets from a given set
$X\subseteq\mathbb{R}^{n_{x}}$. A sequence
$(X_{t}\subseteq\mathbb{R}^{n_{x}})_{t\in\mathbb{N}}$ of sets forms an RFIT if
for all $t\in\mathbb{N}$, $X_{t+1}\in\mathcal{F}(X_{t})$ holds. If
$X_{s}\in\mathcal{F}(X_{s})$, then the set $X_{s}$ is robust control invariant
(RCI) since $X_{s}$ contains its forward reachable set [1].

###  2.3 Configuration-constrained polytopic tubes

Our focus is on polytopic RFITs defined as

$\displaystyle X(y_{t}):=\\{x\in\mathbb{R}^{n_{x}}\,|\,Fx\leq y_{t}\\}\;.$ (2)

Here, $F\in\mathbb{R}^{m\times n_{x}}$ is a given matrix whose rows encode the
facet normals of $X(y_{t})$. We enforce configuration-constraints
$y_{t}\in\mathcal{E}:=\left\\{y\in\mathbb{R}^{m}\mid Ey\leq 0\right\\}$ for a
given matrix $E$. Such constraints fix the facial configuration of the
parametric polytopes $X(y_{t})$ enabling a joint hyperplane-vertex
parameterization of the form

$\displaystyle
X(y)=\operatorname{convh}(\\{V_{j}y\in\mathbb{R}^{n_{x}}\,|\,j\in\\{1,\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,v\\}\\});$ (3)

see [23, Section 3.5] for a construction of matrices $E$ and
$V_{j}\in\mathbb{R}^{n_{x}\times m}$. We define
$U_{j}=e_{j}\otimes\mathbb{I}_{v}$, stacking the vertex control inputs as
$u_{t}:=(u_{1,t},\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,u_{v,t})$, such that $u_{j,t}=U_{j}u_{t}$. Let
$\mathbb{S}\subseteq\mathbb{R}^{m}\times\mathbb{R}^{vn_{u}}\times\mathbb{R}^{m}$
be defined as

(4)

with $d_{i}:=\max\\{Y_{i}w:w\in\mathcal{W}\\}$ for all
$i\in\\{1,\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,m\\}$.

######  Proposition 1.

Let $(y_{t}\in\mathbb{R}^{m})_{t\in\mathbb{N}}$ be such that for all
$t\in\mathbb{N}$, there exists some $u_{t}\in\mathbb{R}^{vn_{u}}$ such that
$(y_{t},u_{t},y_{t+1})\in\mathbb{S}$. Then, $(X(y_{t}))_{t\in\mathbb{N}}$ is
an RFIT.

Proof: The result follows from [23, Corollary 4]. ∎

######  Remark 1.

The method from [11] identifies m-dimensional faces of the polytope
$\big{\\{}(x,y)\mid Fx-y\leq 0\big{\\}}$ whose projection onto the
y-coordinate space characterizes regions where the vertices of $X(y)$ depend
linearly on y. Exploring its relationship with configuration constraints is a
future research direction.

###  2.4 Optimal robust control invariant set

We design a tube MPC scheme that computes RFITs which converge to an optimal
RCI set for a given reference. By the definition of $\mathbb{S}$, $X(y)$ is an
RCI set whenever $(y,u,y)\in\mathbb{S}$ for some $u\in\mathbb{R}^{vn_{u}}$.
For a given reference $r$, we define an optimal RCI set as one whose
parameters $(y,u)$ minimize a given cost $\ell(y,u,r)$ over $\mathbb{S}$. As
an example of such a cost, consider the function

$\ell_{1}(y,u)=\sum^{v}_{j=1}\left\|\left((\overline{V}-V_{j})y,\
(\overline{U}-U_{j})u\right)\right\|^{2}_{Q_{\mathrm{v}}},$

which penalizes the distance of the vertices of $X(y)$ and the vertex control
inputs to their centers,

$\overline{V}y=\frac{1}{v}\sum^{v}_{j=1}V_{j}y\quad\text{and}\quad\overline{U}u=\frac{1}{v}\sum^{v}_{j=1}U_{j}u\;.$
(5)

Now, let $\overline{A}$ and $\overline{B}$ be the average of matrices $A_{i}$
and $B_{i}$ over $i=1,\cdots,p$ respectively. Assuming stabilizability of the
system $x^{+}=\overline{A}x+\overline{B}u$, let the columns of
$M\in\mathbb{R}^{(n_{x}+n_{u})\times n_{u}}$ be a basis for the nullspace of
the matrix $[\overline{A}-\mathbb{I}\ \overline{B}]$. Recalling that every
steady state of this system can be expressed as
$(x_{\mathrm{s}\mathrm{s}},u_{\mathrm{s}\mathrm{s}})=M\theta$, the function

$\displaystyle\ell_{2}(y,u,r,\theta):=\left\|\begin{bmatrix}\overline{V}&0\\\
0&\overline{U}\end{bmatrix}\begin{bmatrix}y\\\
u\end{bmatrix}-M\theta\right\|_{Q_{\mathrm{c}}}^{2}+\left\|r-\begin{bmatrix}C\
\ D\end{bmatrix}M\theta\right\|_{Q_{\mathrm{r}}}^{2}$

penalizes the distance of the center $(\overline{V}y,\overline{U}u)$ to the
steady state $M\theta$ and the distance of its output to the reference $r$. We
can now define a function $\ell$ given by

$\ell(y,u,r):=\min_{\theta}\left(\ell_{1}(y,u,r)+\ell_{2}(y,u,r,\theta)\right),$

with $Q_{\mathrm{v}}$, $Q_{\mathrm{c}}$, and $Q_{\mathrm{r}}$ being positive
semidefinite matrices of appropriate dimensions such that $\ell$ is a convex
quadratic function. In conclusion, for a given reference $r$, the set
$X(y_{\mathrm{o}})$ is an optimal RCI set with vertex control inputs
$u_{\mathrm{o}}$, if $(y_{\rm o},u_{\rm o})$ is a minimizer of

$\displaystyle\mathcal{C}_{\mathrm{o}}:=\min_{y,u}\ \ell(y,u,r)\ \
\text{s.t.}\ \ (y,u,y)\in\mathbb{S}.$ (6)

######  Assumption 1.

There exists a pair $(y,u)$ satisfying $(y,u,y)\in\mathbb{S}$, such that
$X(y)$ is an RCI set.

Problem Statement: Design a CCTMPC scheme for reference tracking that
guarantees $(i)$ recursive feasibility for changing references; and $(ii)$
robustness, stability, and convergence to the optimal RCI set of the current
reference for piecewise constant references.

###  2.5 Tube MPC with Stabilizing Initial and Terminal costs

For a fixed reference $r$, one can design a CCTMPC scheme based on the
optimization problem

$\displaystyle\min_{\mathbf{y},\mathbf{u}}\
L_{0}(y_{0},r)+\sum_{k=0}^{N-1}\ell(y_{k},u_{k},r)+L_{N}(y_{N},r)$
$\displaystyle\ \text{s.t.}\ \ Fx\leq y_{0},\quad y_{N}\in\mathbb{X}_{N},$ (7)
$\displaystyle\qquad(y_{k},u_{k},y_{k+1})\in\mathbb{S},\ \ \forall\
k\in\\{0,\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,N-1\\}.$

We use the bold symbols $\mathbf{y}$ and $\mathbf{u}$ to denote the vectors
stacking the tube parameters
$y_{0},\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern 0.09995pt,y_{N}$
and vertex inputs $u_{0},\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,u_{N}$. The state and reference measurements are $x$ and $r$, the
initial and terminal costs $L_{0}$ and $L_{N}$, and the terminal region
$\mathbb{X}_{N}$. In [23], a construction of $L_{0}$, $L_{N}$ and
$\mathbb{X}_{N}$ to guarantee asymptotic stability is presented. In general
(7) is an economic formulation as the cost, while convex, is not positive
definite. While this formulation yields a stabilizing controller for a fixed
reference $r$, adapting it to track changing references is challenging since
$L_{0},L_{N}$ and $\mathbb{X}_{N}$ are constructed using the solution of
Problem (6) reformulated as

$\displaystyle\min_{y,u,y^{+}}\ \ell(y,u,r)\ \ \text{s.t.}\ \
(y,u,y^{+})\in\mathbb{S},\ y=y^{+}|\lambda.$ (8)

These ingredients depend on the primal-dual solution $(y_{\rm o},u_{\rm
o},y_{\rm o},\lambda_{\rm o})$, which in turn depends on the reference $r$.
Therefore, to address the challenges highlighted in the Problem Statement, a
control scheme would require one module for solving Problem (8) to formulate
$L_{0}$, $L_{N}$ and $\mathbb{X}_{N}$ whenever a change in the reference is
detected, and another for solving Problem (7). While such a scheme could
operate the system safely, its cumbersome implementation can be unwieldy in
practice. We address this using a reference tracking CCTMPC formulation that
eliminates the need for reformulating the underlying problem when a reference
change occurs.

######  Remark 2.

Elastic Tube-MPC (ETMPC) [19] formulations can also be used to construct RFITs
by optimizing $y_{t}$. However, such RFITs can be more conservative than those
computed using CCTMPC, since CCTMPC uses vertex control laws to induce
invariance instead of affine control laws. See [23] for further details.

######  Remark 3.

The methods in [5, 13] etc., can be used to compute matrix $F$ of desired
complexity verifying Assumption 1.

##  3 Tube MPC for reference tracking

We now present a CCTMPC formulation that directly tackles the challenges posed
in the Problem statement.

###  3.1 Reference-tracking CCTMPC

Problem (7) necessitates the use of a rotated stage cost due to presence of
economic stage cost terms in the objective. Alternatively, we adopt a convex
tracking cost $s(y-y_{\mathrm{s}},u-u_{\mathrm{s}})$ that vanishes at the
origin and remains positive for all other points. For instance, we can define
$s(z,v)=\left\lVert(z,v)\right\rVert_{Q}^{2}$ using a positive definite matrix
$Q$. Next, we formulate the optimization problem

$\displaystyle\min_{\mathbf{y},\mathbf{u},y_{\mathrm{s}},u_{\mathrm{s}}}\
\scalebox{0.93}{\mbox{$\displaystyle\ell(y_{\mathrm{s}},u_{\mathrm{s}},r)+\sum_{k=0}^{N-1}s(y_{k},y_{\rm
s},u_{k},u_{\rm s})+\mathrm{M}(y_{N},y_{\mathrm{s}},u_{\mathrm{s}})$}}$
$\displaystyle\hskip 3.0pt\ \text{s.t.}\ \ F{x}\leq y_{0},\
(y_{\mathrm{s}},u_{\mathrm{s}},y_{\mathrm{s}})\in\mathbb{S},\
y_{N}\in\mathbb{T}(y_{\mathrm{s}},u_{\mathrm{s}}),\vspace{3pt}$ (9)
$\displaystyle\hskip 6.0pt\qquad(y_{k},u_{k},y_{k+1})\in\mathbb{S},\ \
\forall\ k\in\\{0,\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,N-1\\},$

where the variables $(y_{\mathrm{s}},u_{\mathrm{s}})$ are enforced to
represent an RCI set $X(y_{\mathrm{s}})$, and the tracking cost penalizes the
deviation between $(y_{k},u_{k})$ and $(y_{\mathrm{s}},u_{\mathrm{s}})$. We
now design a terminal cost $\mathrm{M}(y,y_{\mathrm{s}},u_{\mathrm{s}})$ and
set $\mathbb{T}(y_{\mathrm{s}},u_{\mathrm{s}})$ that guarantee recursive
feasibility and asymptotic stability.

###  3.2 Terminal ingredients design

As a preparation for designing the terminal cost, we define the cost-to-travel
function [22]

$\displaystyle\mathrm{V}(y,y^{+},y_{\mathrm{s}},u_{\mathrm{s}}):=$
$\displaystyle\min_{u}\ \
\left\lVert\begin{pmatrix}y-y_{\mathrm{s}},u-u_{\mathrm{s}}\end{pmatrix}\right\rVert_{Q}^{2}$
(10) $\displaystyle\ \ \text{s.t.}\ \ \ (y,u,y^{+})\in\mathbb{S},$

for fixed $(y_{\mathrm{s}},u_{\mathrm{s}})$. We define
$\mathrm{V}(y,y^{+},y_{\mathrm{s}},u_{\mathrm{s}})=\infty$ if (10) is
infeasible. Since the definition of $V$ is based on a tracking term, we can
enforce asymptotic stability in the sequel without introducing an initial
cost. Denoting

Problem (9) can be equivalently written as

$\displaystyle\min_{\mathbf{y},y_{\mathrm{s}},u_{\mathrm{s}}}\
\scalebox{0.99}{\mbox{$\displaystyle\ell(y_{\mathrm{s}},u_{\mathrm{s}},\hat{r})+\mathcal{V}_{N}(\mathbf{y},y_{\mathrm{s}},u_{\mathrm{s}})+\mathrm{M}(y_{N},y_{\mathrm{s}},u_{\mathrm{s}})$}}$
$\displaystyle\text{s.t.}\ \ F\hat{x}\leq y_{0},\ \
y_{N}\in\mathbb{T}(y_{\mathrm{s}},u_{\mathrm{s}}),(y_{\mathrm{s}},u_{\mathrm{s}},y_{\mathrm{s}})\in\mathbb{S}$

For some $\gamma\in[0,1)$, we define the terminal cost as

$\displaystyle\mathrm{M}(y,y_{\mathrm{s}},u_{\mathrm{s}}):=$
$\displaystyle\min_{u}\
\left\lVert\begin{pmatrix}y-y_{\mathrm{s}},u-u_{\mathrm{s}}\end{pmatrix}\right\rVert_{P}^{2}$
(11l) $\displaystyle\hskip 5.0pt\text{s.t.}\ \ (y,u,\gamma
y+(1-\gamma)y_{\mathrm{s}})\in\mathbb{S},$

where $P$ is a positive definite matrix. We show that for any constant
$\gamma\in[0,1)$, $M(y,y_{\mathrm{s}},u_{\mathrm{s}})$ satisfies the Lyapunov
descent condition inside the implicit terminal set
$\mathbb{T}(y_{\mathrm{s}},u_{\mathrm{s}}):=\big{\\{}y\ \mid\
\mathrm{M}(y,y_{\mathrm{s}},u_{\mathrm{s}})<\infty\big{\\}}.$

######  Theorem 1.

If Assumption 1 holds, then for any $(y_{\mathrm{s}},u_{\mathrm{s}})$
satisfying
$\displaystyle(y_{\mathrm{s}},u_{\mathrm{s}},y_{\mathrm{s}})\in\mathbb{S}$,
$\mathrm{M}(y,y_{\mathrm{s}},u_{\mathrm{s}})$ is positive definite at
${y=y_{\mathrm{s}}}$. Moreover, if the matrix inequality $Q+\gamma^{2}P\preceq
P$ holds, then for any $\bar{y}\in\mathbb{T}(y_{\mathrm{s}},u_{\mathrm{s}})$,
functions $M$ and $V$ satisfy the Lyapunov descent condition

$\displaystyle\min_{y^{+}}\\{\mathrm{M}({y}^{+},y_{\mathrm{s}},u_{\mathrm{s}})+\mathrm{V}(\bar{y},{y}^{+},y_{\mathrm{s}},u_{\mathrm{s}})\\}\leq\mathrm{M}(\bar{y},y_{\mathrm{s}},u_{\mathrm{s}}).$
(11m)

Proof: Under Assumption 1, there exist $(y_{\mathrm{s}},u_{\mathrm{s}})$
satisfying $(y_{\mathrm{s}},u_{\mathrm{s}},y_{\mathrm{s}})\in\mathbb{S}$.
Then, positive definiteness of $M$ holds since the objective of Problem (11l)
is positive definite at $(\mathbf{0},\mathbf{0})$. Regarding the descent
condition, denoting the minimizer of (11l) at $y=\bar{y}$ by $\bar{u}$, we
have

$\displaystyle\mathrm{M}(\bar{y},y_{\mathrm{s}},u_{\mathrm{s}})=\left\lVert(\bar{y}-y_{\mathrm{s}},\bar{u}-u_{\mathrm{s}})\right\rVert_{P}^{2}.$
(11n)

The points $\bar{y}^{+}=\gamma\bar{y}+(1-\gamma)y_{\mathrm{s}}$ and
$u=\bar{u}$ are feasible for Problem (10) at $y^{+}=\bar{y}^{+}$. Hence,

$\displaystyle\mathrm{V}(\bar{y},\bar{y}^{+},y_{\mathrm{s}},u_{\mathrm{s}})\leq\left\lVert(\bar{y}-y_{\mathrm{s}},\bar{u}-u_{\mathrm{s}})\right\rVert_{Q}^{2}.$
(11o)

If $Q+\gamma^{2}P\preceq P$ holds, then (11n) and (11o) along with

$\displaystyle\mathrm{M}(\bar{y}^{+},y_{\mathrm{s}},u_{\mathrm{s}})\leq\gamma^{2}\left\lVert(\bar{y}-y_{\mathrm{s}},\bar{u}-u_{\mathrm{s}})\right\rVert_{P}^{2}$
(11p)

imply (11m). To show that (11p) holds, observe that

$\displaystyle F(A_{i}V_{j}y_{\mathrm{s}}+B_{i}U_{j}u_{\mathrm{s}})+d$
$\displaystyle\leq y_{\mathrm{s}},$ (11q) $\displaystyle
F(A_{i}V_{j}\bar{y}+B_{i}U_{j}\bar{u})+d$
$\displaystyle\leq\gamma\bar{y}+(1-\gamma)y_{\mathrm{s}}$

follow from (4). With $\bar{u}^{+}:=\gamma\bar{u}+(1-\gamma)u_{\mathrm{s}}$,
(11q) implies

$\displaystyle
F(A_{i}V_{j}\bar{y}^{+}+B_{i}U_{j}\bar{u}^{+})+d\leq\gamma\bar{y}^{+}+(1-\gamma)y_{\mathrm{s}}.$
(11r)

Since $u=\bar{u}^{+}$ is feasible for (11l) with $y=\bar{y}^{+}$, (11r)
implies
$\mathrm{M}(\bar{y}^{+},y_{\mathrm{s}},u_{\mathrm{s}})\leq\big{\lVert}(\bar{y}^{+}-y_{\mathrm{s}},\bar{u}^{+}-u_{\mathrm{s}})\big{\rVert}_{P}^{2}$
such that
$(\bar{y}^{+}-y_{\mathrm{s}},\bar{u}^{+}-u_{\mathrm{s}})=\gamma(\bar{y}-y_{\mathrm{s}},\bar{u}-u_{\mathrm{s}})$
implies (11p). ∎

In summary, Problem (9) is implemented as

$\displaystyle\min_{\mathbf{y},\mathbf{u},y_{\mathrm{s}},u_{\mathrm{s}}}\scalebox{0.91}{\mbox{$\displaystyle{\ell}(y_{\mathrm{s}},u_{\mathrm{s}},r)+\sum_{k=0}^{N-1}\left\lVert\begin{bmatrix}y_{k}-y_{\mathrm{s}}\\\
u_{k}-u_{\mathrm{s}}\end{bmatrix}\right\rVert_{Q}^{2}+\left\lVert\begin{bmatrix}y_{N}-y_{\mathrm{s}}\\\
u_{N}-u_{\mathrm{s}}\end{bmatrix}\right\rVert_{P}^{2}$}}$ $\displaystyle\ \
\text{s.t.}\ \ Fx\leq y_{0},\ \
(y_{\mathrm{s}},u_{\mathrm{s}},y_{\mathrm{s}})\in\mathbb{S},$
$\displaystyle\qquad\ (y_{k},u_{k},y_{k+1})\in\mathbb{S},\ \forall\
k\in\\{0,\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,N-1\\},$ $\displaystyle\qquad\ (y_{N},u_{N},\gamma
y_{N}+(1-\gamma)y_{\mathrm{s}})\in\mathbb{S}.$ (11s)

###  3.3 Closed-Loop Control Law

Denoting the parametric minimizers of (11s) as $\mathbf{y}^{*}(x,r)$,
$\mathbf{u}^{*}(x,r)$, $y_{\mathrm{s}}^{*}(x,r)$ and
$u_{\mathrm{s}}^{*}(x,r)$, the control law is constructed as a convex
combination of the vertex control inputs as

$\displaystyle\mu({x},r):=\sum_{j=1}^{v}\lambda_{j}(x,r)u^{*}_{0,j}(x,r),$
(11t)

where $\lambda(x,r)\in\mathbb{R}^{v}$ solves the quadratic program

$\displaystyle\min_{\lambda\geq 0}\ \|\lambda\|_{2}^{2}\ \ \text{s.t.}\ \
x=\sum_{j=1}^{v}\lambda_{j}V_{j}y^{*}_{0}(x,r),\ \sum_{j=1}^{v}\lambda_{j}=1.$

Then, the closed loop system associated with dynamics (1) and reference
sequence $(r_{t})_{t\in\mathbb{N}}$ is
$x_{t+1}=A_{t}x_{t}+B_{t}\mu(x_{t},r_{t})+w_{t},$ with output
$z_{t}=Cx_{t}+D\mu(x_{t},r_{t})$. We now present recursive feasibility and
stability properties of the control scheme. We denote the optimal values of
(6) and (11s) by $\mathcal{C}_{\mathrm{o}}(r)$ and $\mathcal{C}^{*}(x,r)$, and
the minimizer of (6) by $(y_{\mathrm{o}}(r),u_{\mathrm{o}}(r))$ emphasizing
the dependence of the optimal RCI set parameters on the reference $r$.

######  Assumption 2.

The function $\ell$ defining the optimal RCI set in Problem (6) is strictly
convex.

######  Corollary 1.

Suppose Assumption 2 holds. If $Q\succ 0$, $P\succ 0$, $\gamma\in[0,1)$
satisfy ${Q+\gamma^{2}P\preceq P}$, then the following statements hold for any
reference $(r_{t})_{t\in\mathbb{N}}$, independent of $w_{t}\in\mathcal{W}$ and
$[A\ B]_{t}\in\Delta$, $t\in\mathbb{N}$:

$(i)$ For the control law (11t) with $(x,r)=(x_{t},r_{t})$, (11s) is feasible
for all $t\in\mathbb{N}$ and $r_{t}\in\mathbb{R}^{n_{z}}$ assuming feasibility
at $t=0$;

$(ii)$ For a constant reference $r_{t}=r$,
$\mathcal{C}^{*}(x_{t},r)\geq\mathcal{C}_{\mathrm{o}}(r)$ and
$\underset{t\to\infty}{\lim}\mathcal{C}^{*}(x_{t},r)=\mathcal{C}_{\mathrm{o}}(r)$.
Moreover,
$\displaystyle\mathcal{L}(\mathbf{y}^{*}(x,r),y_{\mathrm{s}}^{*}(x,r),u_{\mathrm{s}}^{*}(x,r),r):=\mathcal{C}^{*}(x,r)-\mathcal{C}_{\mathrm{o}}(r)$
is a strictly descending Lyapunov function, and the minimizers converge to the
optimal RCI parameters as

$\displaystyle\underset{t\to\infty}{\lim}(y_{\mathrm{s}}^{*}(x_{t},r),u_{\mathrm{s}}^{*}(x_{t},r))={(y_{\mathrm{o}}(r),u_{\mathrm{o}}(r))}$

for all $k\in\\{0,\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995pt,N\\}$.  Proof: $(i)$ Suppose Problem (11s) is feasible at some
$t\in\mathbb{N}$. Denoting $\tilde{y}:=\gamma
y_{N}^{*}(x_{t},r_{t})+(1-\gamma)y_{\mathrm{s}}^{*}(x_{t},r_{t})$ and
$\tilde{u}:=\gamma
u_{N}^{*}(x_{t},r_{t})+(1-\gamma)u_{\mathrm{s}}^{*}(x_{t},r_{t})$, Theorem 1
implies $\mathbf{y}=[y_{1}^{*}(x_{t},r_{t})\
\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern 0.09995pt\
y_{N}^{*}(x_{t},r_{t})\ \tilde{y}]$,
$y_{\mathrm{s}}=y_{\mathrm{s}}^{*}(x_{t},r_{t})$,
$\mathbf{u}=[u_{1}^{*}(x_{t},r_{t})\kern-0.50003pt.\kern-0.09995pt.\kern-0.09995pt.\kern
0.09995ptu_{N}^{*}(x_{t},r_{t})\ \tilde{u}]$,
$u_{\mathrm{s}}=u_{\mathrm{s}}^{*}(x_{t},r_{t})$ are feasible with $x=x_{t+1}$
and any $r=r_{t+1}$, since $r$ affects only the objective;

$(ii)$ If $y_{0}^{*}\neq y_{\mathrm{s}}$, we have from (10) that
$\mathrm{V}(y_{0}^{*}(x_{t},r),y_{1}^{*}(x_{t},r),y_{\mathrm{s}}(x_{t},r),u_{\mathrm{s}}(x_{t},r))>0$.
Since (11) and (11s) are equivalent, the value of (11) with the feasible
solution in ($i$) is strictly less than $\mathcal{C}^{*}(x_{t},r)$. This
implies that the tracking components of (11) are driven to $0$, such that
$\mathcal{C}^{*}(x_{t},r)$ approaches $\mathcal{C}_{\mathrm{o}}(r)$ from above
and $\mathcal{L}$ is a Lyapunov function. The minimizers converge since
Assumption 2 implies (6) is strictly convex.

######  Remark 4.

The study of equivalence of the Economic MPC problem (7) and the Tracking MPC
problem (9) could be an interesting topic for future research.

##  4 Numerical Examples

We present two case studies to demonstrate the efficacy of the proposed CCTMPC
scheme. In both cases, the terminal cost in (11l) is computed with
$\gamma=0.95$.

###  4.1 Illustrative example

We consider the open-loop unstable LTI system

$\displaystyle x^{+}=\begin{bmatrix}1.1&1\\\
0&1\end{bmatrix}x+\begin{bmatrix}0.5\\\ 1\end{bmatrix}u+w,$ $\displaystyle
z=[1\ \ 0]x,$ (11u)

with constraints $\mathcal{X}=\\{x:(-5,-2)\leq x\leq(5,3)$},
$\mathcal{U}=\\{u:-1\leq u\leq 2\\}$ and $\mathcal{W}=\\{w:|w|\leq(0,0.5)\\}$.
To design a tracking CCTMPC controller for this system, we construct matrix
$F$ in (2) with $m=v=12$ as in [23, Remark 3], and define $\ell$ in (6) with
$Q_{\mathrm{v}}=\mathrm{blkd}(10\mathbb{I}_{2},1)$,
$Q_{\mathrm{c}}=\mathrm{blkd}(\mathbb{I}_{2},1)$ and $Q_{\mathrm{r}}=100$. We
use
$Q=\sum_{k=1}^{v}\mathbf{V}_{k}^{\top}Q_{\mathrm{v}}\mathbf{V}_{k}+10^{-3}\mathbb{I}_{32}$
and $P=(1-\gamma^{2})^{-1}Q$, where $\mathbf{V}_{k}:=[\overline{V}-V_{k}\
\overline{U}-U_{k}]$, such that $Q+\gamma^{2}P\preceq P$. In Figure 1, we plot
the closed-loop results using this controller for $N=5$. In Figure 3, the
Lyapunov function
$\mathcal{L}_{t}:=\mathcal{L}(\mathbf{y}^{*}(x_{t},r_{t}),y_{\mathrm{s}}^{*}(x_{t},r_{t}),u_{\mathrm{s}}^{*}(x_{t},r_{t}),r_{t})$
is shown.

Figure 1 compares the feasible regions of our CCTMPC, ETMPC [17] modified for
tracking as in [10], and tracking RTMPC [10] controllers. In order to
parameterize the RFITs, the ETMPC controller uses the set $X(y)$, and the
RTMPC uses the fixed RCI set $X(y_{\mathrm{o}}(0))$. ETMPC induces invariance
in the minimal parameterized RPI set with the LQR controller with
$Q=\mathbb{I}_{2}$ and $R=1$, and both schemes use the maximal positive
invariant set [1] corresponding to their respective systems as the terminal
set. CCTMPC exhibits a larger feasible region: its Hausdorff distance to the
maximal RCI set is $0.0179$, compared to $0.2816$ for ETMPC and $1.1056$ for
RTMPC. Hence, we can control the system from states close to the boundary of
the MRCI set with a short horizon of $N=5$ with our tracking CCTMPC
controller.

###  4.2 Lane change maneuver

We now focus on the lateral vehicle dynamics model [13, Section V-C],

(11v)

with $x=(e_{y},e_{\psi},\dot{y},\dot{\psi})$,
$u=(\delta_{\mathrm{s}},\mu_{\mathrm{b}})$ and $z=e_{y}$. $e_{y}$ [m] is the
lateral deviation, $e_{\psi}$ [rad] the angular deviation, $\dot{y}$ [m/s] the
lateral velocity, $\dot{\psi}$ [rad/s] the yaw rate, $\delta_{\mathrm{s}}$
[rad] the steering angle, and $\mu_{\mathrm{b}}$ [Nm] the braking yaw moment.
The disturbance is $w\in[0,100]$, and $v^{x}\in[40,65]/3.6$ [m/s] is the
longitudinal velocity.

Treating (11v) as an LPV system with parameters $(v^{x},1/v^{x})$ allows us to
capture its behavior as a linear system with multiplicative uncertainty. By
defining the parameter set
$\mathcal{P}(v^{x}_{t}):=\\{(v^{x},1/v^{x}):v^{x}\in[v^{x}_{t},65]/3.6\\}$, we
can derive the matrices $(A_{i},B_{i})$ that characterize the system’s
multiplicative uncertainty using the vertices of of
$\hat{\mathcal{P}}\supset\mathcal{P}(40)$, where

$\displaystyle\hat{\mathcal{P}}\coloneqq\mathrm{convh}\scalebox{0.8}{\mbox{$\displaystyle\left\\{\begin{bmatrix}11.111\\\
0.090\end{bmatrix},\begin{bmatrix}18.056\\\
0.055\end{bmatrix},\begin{bmatrix}11.111\\\
0.086\end{bmatrix},\begin{bmatrix}17.217\\\ 0.055\end{bmatrix}\right\\}$}}.$

System constraints are $\mathcal{U}=\\{u:|u|\leq(10\pi/180,2)\\}$ and
$\mathcal{X}=\\{x:|x|\leq(3,4,20\pi/180,3)\\}$. Matrix $F$ in (2) is computed
using the approach in [13, Section IV-D]; therefore, $F=\hat{S}T^{-1}$, where
$\hat{S}=\left[\mathbf{I}^{4}\ -\mathbf{I}^{4}\right]^{\top}$ and

$T=\scalebox{0.8}{\mbox{$\displaystyle\begin{bmatrix}[r]0.0712&-0.2667&0.2667&0.0712\\\
2.5077&0&0&-1.4923\\\ -0.0718&-0.0024&-0.1472&-0.0011\\\ 0&1.1729&1.8271&0\\\
\end{bmatrix}$}},$

resulting in $m=8$ and $v=16$. The cost $\ell$ is defined with
$Q_{\mathrm{v}}=\mathrm{blkd}(10^{-3}\mathbb{I}_{4},10^{-2}\mathbb{I}_{2})$,
$Q_{\mathrm{c}}=\mathrm{blkd}(10^{2}\mathbb{I}_{4},10^{-2}\mathbb{I}_{2})$ and
$Q_{\mathrm{r}}=10$, while we set
$Q=\mathrm{blkd}(10^{-3}\mathbb{I}_{8},10^{-2}\mathbb{I}_{32})$ and
$P=(1-\gamma^{2})^{-1}Q$. For simulation, we simplify the “merge into faster
lane” maneuver [21], where a vehicle shifts from a slow lane ($r=-1.5$,
minimum speed $40/3.6$ [m/s]) to a fast lane ($r=1.5$, minimum speed $60/3.6$
[m/s]), during which the parameter $v_{x}$ is non-decreasing, resulting in a
decreasing multiplicative uncertainty over time, as
$\mathcal{P}(v^{x}_{t_{2}})\subseteq\mathcal{P}(v^{x}_{t_{1}})$ for $t_{2}\geq
t_{1}$. The tracking CCTMPC scheme can seamlessly account for this uncertainty
reduction: by recomputing the matrices $(A_{i},B_{i})$ at each time step using
the vertices of $\mathcal{P}(v^{x}_{t})\subset\hat{\mathcal{P}}$, we still
ensure stability.

In Figure 2 we illustrate the closed-loop dynamics using the aforementioned
online model adaptation, where the CCTMPC controller is formulated with $N=3$.
To visualize the set dynamics, we project $X(y_{0}^{*}(x_{t},r_{t}))$ along
$e_{y}$ (blue tube) and $e_{\psi}$ (angle between green arrows) states, in
accordance with the spatial evolution of the system. We also show the
projection of $X(y^{*}_{\mathrm{s}}(x_{t},r_{t}))$ along $e_{y}$ (green tube)
as well as the projection of $X(y_{\mathrm{o}}(r_{t}))$ for $e_{y}$ (red
tube). The yellow arrow denotes $e_{\psi}(t)$ — sampled every 15 time steps —
which consistently resides within the corresponding polytope projection. By
explicitly accounting for reduction in multiplicative uncertainty upon moving
into the fast lane, we reduce the uncertainty in the tubes as can be observed
from a reduction in their width. The Lyapunov function $\mathcal{L}_{t}$ is
shown in Figure 3.

Figure 1: Closed-loop results for System (11u). Feasible regions of CCTMPC,
RTMPC from [10], and ETMPC from [17], are compared. The reference is $r_{t}=5$
for $t\in[0,14]$ and $r_{t}=-5$ for $t\geq 15$. Closed-loop tubes and system
trajectory are plotted for $x_{0}=(-3.12,2.95)$. Figure 2: Closed-loop spatial
evolution of (11v).

Figure 3: Lyapunov function $\mathcal{L}_{t}$ for system (11u) (left) and
(11v) (right) respectively. Changes in reference for the two systems are
marked by vertical dashed lines.

###  4.3 Computational aspects

The CCTMPC scheme has $(N+1)\ v[m(1+p)+n_{\mathcal{X}}+n_{\mathcal{U}}]+m$
inequality constraints and $(N+1)(m+vn_{u})$ optimization variables, where
$n_{\mathcal{X}}$ and $n_{\mathcal{U}}$ are the number of halfspaces defining
polytopes $\mathcal{X}$ and $\mathcal{U}$. As highlighted in [23, Remark 2],
the matrix $E$ is typically sparse and contains redundant rows, which can be
reduced using an LP solver. Trade-off between complexity of the RFIT (hence
reduction in conservativeness over prediction steps) and computational burden
must be taken into account in real world applications, which can be mitigated
by means of high-performance QP solvers [4].

##  5 Conclusions

This paper has introduced a novel CCTMPC formulation for tracking piecewise
constant reference signals. The MPC scheme is based on the implementation of a
single convex quadratic program that simultaneously computes an RFIT and a
target invariant set. Moreover, Theorem 1 and Corollary 1 have established the
fact that the RFIT converges to the optimal RCI set for each constant
reference, while maintaining recursive feasibility and stability guarantees in
the presence of changing references. Finally, our numerical results have not
only illustrated the practical applicability of the proposed CCTMPC scheme,
but also its benefits compared to existing rigid-tube MPC methods for
reference tracking.

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