Equi-normalization and exact scaling dynamics in homothetic tube model predictive control

Abstract

This paper develops an equi-normalization process in order to verify the existence of, and characterize, the minimal robust positively invariant set generated by a number of a-priori, but suitably, selected linear inequalities. It also develops the exact scaling dynamics associated with this set. The paper then proposes an improved homothetic tube model predictive control synthesis for linear systems subject to additive disturbances and polytopic constraints.

Introduction

Model predictive control (MPC) need not be robust [1], [2], [3], while min–max feedback MPC [4] is computationally intractable, hence the need for robust MPC synthesis that strikes a balance between optimality and computability. Tube MPC (TMPC) [2], [3], [5], [6] forms a rigorous set theoretic approach to robust MPC synthesis, which results in computationally efficient handling of constraints and uncertainty. Early TMPC [3], [5] used “rigid” tubes with fixed cross-section shape sets whose centers were optimized on-line. The deployment of the fixed tube cross-section shape sets results in a degree of conservatism. Latter “homothetic” tubes allowed for more flexibility: the cross-sections of homothetic state and control tubes are parameterized in terms of the sequences of associated centers and scalings [7]. These sequences form decision variables for the on-line optimization, and are subject to constraints that enforce robust constraint satisfaction. Global conditions describing the tube dynamics and ensuring robust constraint satisfaction were given in [7]. The dynamics of the local homothetic state and control tubes is characterized by the dynamics of the local state and control tubes centers and scalings. These dynamics plays a crucial role for derivation of the stabilizing conditions through the use of a terminal constraint set and a local Lyapunov function. In [7], an affine relationship between the tube scalings at consecutive times was used. The affine scalings dynamics represents an upper approximation of the exact dynamics and, as such, it induces a degree of conservatism.

This article introduces an equi-normalization process, derives a description of the exact scaling dynamics and achieves a significant constraint relaxation. Using the proposed equi-normalization we establish the existence of, and characterize, the minimal robust positively invariant (RPI) set generated by a number of a-priori, but suitably, selected linear inequalities. The function inducing the exact scaling dynamics is shown to be a piecewise affine, convex, continuous, non-negative and monotonically non-decreasing function. Remarkably, this function is composed of, at most, two affine functions irrespective of the system dimension and the number of utilized linear inequalities. In particular, this function can be expressed as the maximum of two affine functions and yields the exact scaling dynamics (in contrast to the affine upper approximate scaling dynamics of [7]). The equi-normalized minimal RPI (mRPI) set together with the exact scaling function is used to analyze the dynamics of the local state and control tubes centers $(z,Kz)$ and scalings $\alpha$. (The latter dynamics is determined completely by the $(z,\alpha )$-dynamics.) This analysis enables us to ensure the maximality (w.r.t. both the volume and set inclusion) of the constraint set on the local state and control tubes centers $(z,Kz)$ and scalings $\alpha$. (These constraints are characterized fully by constraints on the $(z,\alpha )$-variable.) The equi-normalized mRPI set induces the minimal (w.r.t. set inclusion) homothetic tubes cross-section shape sets, while the exact scaling function induces the tightest (w.r.t. set inclusion) $(z,\alpha )$-dynamics. In turn, this yields a significantly improved terminal constraint set, and relaxed conditions for the applicability of both the homothetic tube optimal control (HTOC) and homothetic tube MPC (HTMPC). It is also shown that the equi-normalized mRPI set, the exact scaling dynamics and the enlarged terminal constraint set can be incorporated into HTMPC through a generalization of the cost and terminal cost functions and a local Lyapunov condition. This leads to an improved HTMPC synthesis, which is applicable under relaxed conditions compared to the existing approaches [5], [7].

Paper structure: Preliminaries and brief overview of the existing TMPC syntheses are given in Section 2, while Section 3 introduces equi-normalization, and derives the exact tube scaling dynamics and improved terminal constraint set. The improved HTOC and HTMPC are outlined in Section 4. Comparative remarks are presented in Section 5, while conclusions are drawn in Section 6.

Nomenclature and typographical conventions: The sets of non-negative and positive integers and non-negative reals are denoted by ${\mathbb{N}}_{\ge 0}$, ${\mathbb{N}}_{\ge 1}$, and ${\mathbb{R}}_{\ge 0}$, respectively. Given $a,b\in {\mathbb{N}}_{\ge 0}$ with $a<b$ we denote ${\mathbb{N}}_{[a:b]}≔\{a,a+1,\dots ,b-1,b\}$; we write ${\mathbb{N}}_b$ for ${\mathbb{N}}_{[0:b]}$. For $a,b\in {\mathbb{R}}^n$, inequalities such as $a\le b$ apply component-wise. For $M\in {\mathbb{R}}^{n\times n}$, $\rho \left(M\right)$ denotes its spectral radius. Given $X,Y\subset {\mathbb{R}}^n$ and $x\in {\mathbb{R}}^n$, the Minkowski set addition is defined by $X\oplus Y≔\{x+y:x\in X,y\in Y\}$, we write $x\oplus X$ instead of $\left\{x\right\}\oplus X$. Given a set $X$ and a real matrix $M$ of compatible dimensions, the image of $X$ under $M$ is denoted by $MX≔\{Mx:x\in X\}$. A function $f(\cdot ):{\mathbb{R}}^n\to {\mathbb{R}}^m$ is said to be positively homogeneous of the first degree if $f\left(\alpha x\right)=\alpha f\left(x\right)$ for all $\alpha \in {\mathbb{R}}_{\ge 0}$ and all $x\in {\mathbb{R}}^n$. If $f(\cdot )$ is a set-valued function from, say, $X$ into $U$, then its graph is the set $graph\left(f\right)≔\{(x,y):x\in X,y\in f\left(x\right)\}$. A set $X\subset {\mathbb{R}}^n$ is a $C$-set if it is compact, convex, and contains the origin; if in addition the origin is in the interior of $X$, then $X$ is said to be a proper $C$-set, or just a $PC$-set. A polyhedron is the (convex) intersection of a finite number of open and/or closed half-spaces and a polytope is a closed and bounded polyhedron. We say that a set $X$ is a ($C$-) $PC$-polytopic set if it is a polytope which is a ($C$-) $PC$-set. $\mathrm{interior}\left(X\right)$ denotes the interior of a set $X$ whereas $convh\left(X\right)$ denotes its convex hull. Sets $X\subset {\mathbb{R}}^n$ and $Y\subset {\mathbb{R}}^n$ are homothetic if and only if $Y=z\oplus \alpha X$ for some $z\in {\mathbb{R}}^n$ and $\alpha \in {\mathbb{R}}_{\ge 0}$. Given a non-empty closed convex set $X\subset {\mathbb{R}}^n$ the associated support function is given, for all $y\in {\mathbb{R}}^n$, by $support(X,y)≔{sup}_x\{y^{T}x:x\in X\}$. We distinguish the row and column vectors only when needed, we use the same symbol for a variable $x$ and its vectorized form, and we provide brief proofs of less obvious statements in the Appendix.