Consider a switching system composed by M LTI modes subject to, possibly, heterogeneous constraints and levels of disturbance. Each mode is modelled
in a general state space representation following the structure presented in Section 2.2, thus the overall switching dynamics can be cast as follows
x(t + 1) = Aσ(t)x(t) + Bσ(t)u(t) + w(t) (4.1a)
x(t) ∈ Xσ(t) ⊂ Rnx (4.1b)
u(t) ∈ Uσ(t) ⊂ Rnu (4.1c)
w(t) ∈ Wσ(t) ⊂ Rnx. (4.1d)
At any particular time t, the mode that drives the system, alongside with the constraints that bound it and the disturbances that affect it, are entirely defined by the value of the switching signal σ(·). The latter is a assumed to be a piecewise constant function that, at each sampling time, takes values in the finite set M = {1, . . . , M }. Following the discussion in Chapter 2, it is assumed that, for all m ∈ M, Xm and Um are PC-sets and Wm is a C-set,
but furthermore, that all three constraint sets are polyhedrons. The latter assumption will become relevant for the verification of the different inclusion conditions related to the computation of MDTs for feasible switching. Finally, it is also required that Assumption 2.1 holds for every pair (Am, Bm) with
m ∈ M.
The switching instances are {t0, t1, . . . , tk, . . .} with t0 = 0 and tk ≥ tk−1+ 1,
which results in σ(t) being constant in the interval [tk−1, tk) for all k ≥ 1.
This structure implies that the dwell-times have to be at least of length 1, and that the switches take place exactly at the sampling instances. The former is necessary otherwise several modes could become active at the same time instance, while the latter is crucial since (4.1a) is usually a discretised version of a continuous-time process, hence switching that does not match the sampling instances would effectively result in prediction errors throughout the sampling intervals. Finally, it is assumed that the values of the switching signal are known instantly at each time t. This is a standard assumption in the analysis of switching systems [114,122,124–126] and it introduces less conservatism than assuming a-priori knowledge of the switching sequence (such as in [125]).
The concept of mode-dependent dwell-time, as it will be considered in this chapter, is now defined.
Definition 4.1. The mode-dependent dwell-time (MDT) associated to mode m ∈ M, say τm, is the minimum amount of time during which the dynamics of
the switching system (4.1) remain fixed at mode m before leaping into another allowable mode. It follows that if mode m became active at time tk, that is
σ(tk) = m but σ(tk− 1) 6= m, then tk+1− tk≥ τm.
In what follows, minimum values for τm that guarantee feasible and stable
switching between the different modes of (4.1) will be computed. These lower bounds effectively constrain how arbitrary the switching sequence can be, how- ever there are other types of restrictions that can be placed on the switching sequence. A particular instance of the former is the case in which switching between certain modes is not allowed. In such cases σ(·) is referred to as a constrained switching signal (CSS), which can be precisely represented by a di- rected graph G (M, E ), where M is the set of nodes, and E = {(s, d) |s, d ∈ M} the set of edges that link the nodes together. Each edge represents an allowed switch and for each (s, d) ∈ E , s represents the source node and d the destina- tion node (s and d will also be referred to as neighbouring modes). Note that sources and destinations are not interchangeable, hence for any pair m, l ∈ M, (m, l) ∈ E does not imply (l, m) ∈ E . Notice also that for all m ∈ M it is
assumed that (m, m) ∈ E , otherwise the MDT for mode m would be fixed at τm = 1. It follows that at each time instant t
σ(t) ∈ Mσ(t−1) = {d ∈ M| (σ(t − 1), d) ∈ E } ⊆ M.
The focus is placed on the regulation problem, i.e. the design of a control law u(t) = κ(x(t)) that admissibly stabilizes the origin (or a neighbourhood of it) for the switching system (4.1) and a driving CSS. Initially, standard stabilizing and admissible (robust) MPC controllers are deployed independently for each mode. As discussed in Section 4.1, arbitrary switching among independently stabilizing controllers can result in an unstable switching closed-loop. Furthermore, the heterogeneity of the constraints may result in constraint violation at the moment of switching. To avoid this issues MDTs that allow for feasible and stabilizing switching between the independent MPCs are computed. Standard MPC controllers are designed for the disturbance free case, while tube MPC is employed for the disturbed case. The standard MPC implementation used to control undisturbed LTI systems with constraints can be seen as a special case of the tube-based MPC controller presented in Chapter 2. In view of this the tube-based approach is now briefly recast to account for the change in notation owing to the different modes.
4.2.1
Single mode tube-based MPC
At each time instant, the optimal control problem solved by the m-TMPC controller is PNm(x(t)) : min ¯ u,¯x0 JNm( ¯u, ¯x0) (4.2a) s.t. (for k = 0, . . . , Nm− 1) x(t) − ¯x0 ∈ Sm (4.2b) ¯ xk+1 = Amx¯k+ Bmu¯k (4.2c) ¯ xk ∈ ¯Xm ⊆ Xm Sm (4.2d) ¯ uk ∈ ¯Um ⊆ Um KmSm (4.2e) ¯ xNm ∈ ¯Xf,m ⊆ ¯Xm, (4.2f)
where again (¯xk, ¯uk) are the nominal predictions, updated at each time instant
to account for the newly measured true state, Nm is the prediction horizon
employed by mode m, and ¯u = {¯u0, . . . , ¯uNm−1} is the input sequence to be
optimized. The sets Sm and ¯Xf,m are respectively an RPI and a PI set for the
uncertain and nominal dynamics (4.2c) of mode m for a given stabilizing Km
according to Definition 2.2.
The cost function is, again, designed to approximate the infinite horizon LQR cost JNm( ¯u, ¯x0) = Nm−1 X k=0 ||¯xk||2Qm + ||¯uk|| 2 Rm + ||¯xN|| 2 Pm,
with Qm, Rm > 0 and ¯A>mPmA¯m + Qm + Km>RmKm − Pm = 0, where ¯Am =
(Am+ BmKm). Note that the matrix inequality in Proposition 2.1–(c) is now
replaced by an equality, hence the unconstrained infinite horizon LQR cost is not only approximated but exactly met. This is done to guarantee that the MPC control law is linear and time invariant when inside the terminal set ¯Xf,m,
a feature that is needed for subsequent developments. Define, as in (2.9), ¯
u∗(x(t)), ¯x∗0(x(t)) = arg PNm(x(t))
VNm(x(t)) = JNm u¯
∗
(x(t)), ¯x∗0(x(t)),
set the nominal input to the associated receding horizon control law ¯u(t) = ¯
κm(x(t)) = ¯u?0(x(t)) and let the nominal trajectories be updated with ¯x(t) =
¯ x?
feasible when constraint (4.2b) is replaced by ¯x0 = x(t), then Proposition 2.1
can be recast as follows.
Proposition 4.1. If (a) Assumption 2.1 holds with a certain Km, (b) the sets
Sm and ¯Xf,m are, correspondingly, admissible RPI and PI sets for ¯Am with
respect to constraints (4.1b) and (4.1c), disturbance set Wm and tightened
constraint (4.2d), (c) the sets Sm and ¯Xf,m are PC-polyhedrons, (d) the loop
is closed with u(t) = κm(x(t)) = ¯κm(x(t)) + Km(x(t) − ¯x∗0(x(t))), then (1)
the optimization problem (4.2) is recursively feasible with feasibility region XNm = Sm ⊕ ¯XNm, (2) the sets ¯XNm and ¯XNm−1 are PC-polyhedrons and
invariant under ¯u∗0(x(t)), (3) state and input constraints are met at all times despite the disturbances, and (4) there exist constant scalars bm, dm, fm > 0
such that for all x ∈ XNm and w ∈ Wm it holds that:
bm|¯x∗0(x)| 2 2 ≤VNm(x) ≤ dm|¯x ∗ 0(x)| 2 2 (4.3a) VNm(Amx + Bmκm(x) + w) −VNm(x) ≤ −fm|¯x ∗ 0(x)|22. (4.3b)
Analogously, a corollary is provided to explicitly establish the exponential stability result arising from Proposition 4.1.
Corollary 4.1. The system of inequalities (4.3) implies that there exist con- stant scalars cm > 0 and λm ∈ (0, 1) such that for all x(0) ∈ Sm⊕ ¯XNm, it holds
that
|¯x(t)|2 ≤ cmλtm|¯x(0)|2. (4.4)
Therefore the origin is exponentially stable for the nominal trajectories of mode m when in closed-loop with ¯u∗0(x(t)).
Proofs for Proposition 4.1 and Corollary 4.1 are again omitted and can be found in [1, 2]. A proof for the additional claim about the polytopic shape of the feasibility regions can be found in [99, 137]. In the subsequent developments specific values of the exponential stability constants are considered, in particular cm =
pd
m/bm and λm = p1 −fm/dm. Furthermore, although not stated, the
linear gain employed to compute the terminal cost matrix Pm needs not to
be equal to the tube gain employed in the composite control law κm(·). Both
are identified as Km to simplify notation, however they could be designed
independently to pursue different objectives.
In what follows MDTs are computed such that the switching control law
(robustly) stabilizes the origin for the constrained switching system (4.1). This selection of control law implies that at any given time instant, the control action is entirely governed by the MPC designed for the currently active mode, instead of a governing MPC that spans all modes such as in [122, 124].