A nonlinear tracking model predictive control scheme for dynamic target signals
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A nonlinear tracking model predictive control scheme for
dynamic target signals ?
Johannes Köhler1 , Matthias A. Müller2 , Frank Allgöwer1
1
Institute for Systems Theory and Automatic Control, University of Stuttgart, 70550 Stuttgart, Germany
2
Institute of Automatic Control,Leibniz University Hannover, 30167 Hannover, Germany
Abstract
We present a nonlinear model predictive control (MPC) scheme for tracking of dynamic target signals. The scheme combines
stabilization and dynamic trajectory planning in one layer, thus ensuring constraint satisfaction irrespective of changes in the
dynamic target signal. For periodic target signals we ensure exponential stability of the optimal reachable periodic trajectory
using suitable terminal ingredients and a convexity condition for the underlying periodic optimal control problem. Furthermore,
we introduce an online optimization of the terminal set size to automate the trade-off between fast convergence and operation
close to the constraints. In addition, we show how stabilization and dynamic trajectory planning can be formulated as
partially decoupled optimization problems, which reduces the computational demand while ensuring recursive feasibility and
convergence. The main tool to enable the proposed design is a novel reference generic offline computation that provides
suitable terminal ingredients for tracking of dynamic reference trajectories. The practicality of this approach is demonstrated
on benchmark examples, which demonstrates superior performance compared to state of the art approaches.
Key words: model predictive control; control of constrained systems; output regulation; periodic references; reference
tracking; nonlinear systems; stability
1 Introduction In many applications, the control goal goes beyond the
stabilization of a pre-determined setpoint. These prac-
Model Predictive Control (MPC) is a well established tical challenges include tracking of changing reference
control method, that computes the control input by re- setpoints, stabilization of dynamic trajectories, output
peatedly solving an optimization problem online [29]. regulation and general economic optimal operation. De-
The main advantages of MPC are the ability to cope with signing MPC schemes that provide theoretical guaran-
nonlinear dynamics, hard state and input constraints, tees (recursive feasibility, stability and performance) for
and the inclusion of performance criteria. such control problems is the focus of much research.
Motivation: Most of the existing theoretical results for Related work: In [10,15] tracking of reachable dynamic
MPC consider the problem of stabilizing some given reference trajectories is studied and stability is ensured
steady-state [29]. Theoretical properties, such as recur- by using suitable terminal ingredients. In [14], a track-
sive feasibility and asymptotic stability can be ensured ing MPC scheme without terminal ingredients for un-
by including suitable terminal ingredients (terminal set reachable target signals is studied and (practical) sta-
and terminal cost) in the optimization problem, which bility of the optimal reachable trajectory is established.
are computed offline, compare [5,24]. The MPC scheme is simple to implement, however, the
theoretical guarantees depend on a sufficiently large pre-
? The material in this paper was partially presented at the diction horizon that may be conservative for many ap-
6th IFAC Conference on Nonlinear Model Predictive Con-
plications.
trol, August 19—22, 2018, Madison, Wisconsin (USA).
Email addresses: A promising alternative to tackle the problem of un-
johannes.koehler@ist.uni-stuttgart.de (Johannes reachable target signals, is the simultaneous opti-
Köhler1 ), mueller@irt.uni-hannover.de (Matthias A. mization of an artificial reference, which is pursued
Müller2 ), frank.allgower@ist.uni-stuttgart.de (Frank in [18,19,21,9,26,11]. By using terminal constraints for
Allgöwer1 ). the artificial reference, these schemes provide a large
Preprint submitted to Automatica 14 April 2020region of attraction and ensure recursive feasibility in- sive exposition of the subject, unify the consideration
dependent of the (typically exogenous) target signal. of different terminal ingredients, include an online opti-
In particular, in [18] a setpoint tracking MPC scheme mization of the terminal set size, discuss how to partially
for linear systems has been introduced based on simul- decouple the reference update and add an example with
taneous optimization of an artificial steady state and a quantitative comparison to state of the art approaches.
tracking of this steady state. Compared to a standard
MPC formulation, this scheme ensures recursive feasi- Outline: Sec. 2 discusses preliminaries regarding track-
bility independent of the target signal and provides a ing MPC. Sec. 3 contains the proposed MPC scheme, in-
large region of attraction. In [31], the complexity of the cluding theoretical analysis and extensions. Sec. 4 shows
polyhedral invariant set for tracking from [18] has been the applicability and advantages of the proposed method
reduced by considering an online optimization of the using numerical examples. Sec. 5 concludes the paper.
terminal set size. In [21], for linear systems the approach
in [18] has been extended to periodic target signals by
using an artificial periodic trajectory and a terminal Notation: The quadratic norm with respect to a pos-
equality constraint. Similarly, in [20] for linear systems itive definite matrix Q = Q> is denoted by kxk2Q =
the economically optimal periodic trajectory is stabi- x> Qx. By K∞ we denote the class of functions α :
lized using a tracking formulation. The method in [18] R≥0 → R≥0 , which are continuous, strictly increasing,
has been extended to setpoint tracking for nonlinear unbounded and satisfy α(0) = 0. The identity matrix is
systems in [19]. Economic MPC schemes based on ar- In ∈ Rn×n . The interior of a set X is denoted by int(X ).
tificial steady states have been considered in [9,26,11],
for both linear and nonlinear systems.
2 Tracking MPC for reachable references
Contribution: The goal of this paper is to generalize and
unify the methodologies from [18,19,21] to design non-
linear MPC schemes that exponentially stabilize the op- We consider the following nonlinear discrete-time sys-
timal reachable periodic trajectory given a possibly un- tem xt+1 = f (xt , ut ) with state x ∈ Rn , control input
reachable periodic output target signal, which is a funda- u ∈ Rm , and time t ∈ N. We impose point-wise in time
mental step towards practical nonlinear MPC schemes. constraints on the state and input (xt , ut ) ∈ Z, with
We first generalize the conditions on the terminal ingre- some compact set Z. As a preliminary problem, we con-
dients, such that we can present a unified theorem that sider tracking of a given reachable reference trajectory
incompases terminal equality constraints (TEC) (under rt = (xrt , urt ) ∈ Rn+m , analogous to [15].
suitable controllabiltiy conditions) and suitable termi-
nal cost and terminal set (which can be designed using Assumption 1 The reference trajectory r satisfies rt ∈
the reference generic offline computation in [15]). Then Zr , for all t ≥ 0, with some set Zr ⊆ int(Z). Further-
we design a nonlinear tracking MPC scheme for unreach- more, the evolution of the reference trajectory is restricted
able periodic target signals. We provide a novel proof by rt+1 ∈ R(rt ), with R(r) = {(xr+ , ur+ ) ∈ Zr | xr+ =
to show that the optimal reachable periodic trajectory f (xr , ur )}.
is exponentially stable for the resulting closed-loop sys-
tem, if the set of feasible periodic output trajectories
is convex. Furthermore, we extend this method to al- This assumption characterizes that the reference tra-
low for an online optimization of the terminal set size, jectory r is reachable, i.e., follows the dynamics f and
which significantly improves the performance, similar (strictly) satisfies the constraints Z for all times. Denote
to [31]. This extension requires one additional scalar op- the tracking error by et := xt − xrt . The control goal is
timization variable and automates the trade-off between stability of the tracking error et = 0 and constraint sat-
fast convergence and operation close to the constraints, isfaction (xt , ut ) ∈ Z, ∀t ≥ 0. We define the quadratic
which typically needs to be decided offline. In addition, reference tracking stage cost
we provide a novel algorithm that partially decouples
the reference trajectory updates and the computation of `(x, u, r) = kx − xr k2Q + ku − ur k2R ,
the closed-loop input. Finally, we demonstrate the ap-
plicability and practicality of the proposed methodology with positive definite weighting matrices Q, R. Denote
using two nonlinear benchmark examples. Furthermore, the reference r over the prediction horizon N by r·|t
we showcase superior performance compared to state of with rk|t = rt+k , k = 0, . . . , N . Given a predicted state
the art approaches in a quantitative comparison by in-
cluding the proposed terminal ingredients and optimiz- and input sequence x·|t ∈ Rn×(N +1) , u·|t ∈ Rm×N the
ing terminal set size online. tracking cost with respect to the reference r·|t is given by
A preliminary version of the proposed approach can be N
X −1
found in the conference proceedings [13]. Compared to JN (x·|t , u·|t , r·|t ) := `(xk|t , uk|t , rk|t ) + Vf (xN |t , rN |t ),
the conference version, we provide a more comprehen- k=0
2with the terminal cost Vf . The MPC scheme is based on sible and the tracking error e = 0 is uniformly exponen-
the following (standard) MPC optimization problem tially stable for the resulting closed-loop system.
Vt := min JN (x·|t , u·|t , r·|t ) (1a)
u·|t PROOF. This theorem is a straight forward exten-
s.t. xk+1|t = f (xk|t , uk|t ), (1b) sion of standard MPC results [29], compare also [15,10].
(xk|t , uk|t ) ∈ Z, k = 0, . . . , N − 1, (1c) Given the optimal solution u∗·|t , the candidate sequence
x0|t = xt , xN |t ∈ Xf (rN |t ), (1d) (
u∗k+1|t 0≤k ≤N −2
uk|t+1 = ,
where Xf (r) ⊂ Rn denotes the terminal set. The solution kf (x∗N |t , rN |t ) k = N − 1
to this optimization problem are the value function Vt
and the optimal state and input trajectories x∗·|t , u∗·|t . is a feasible solution to (1) and implies
For simplicity, we assume throughout the paper that
the optimization problems admit a unique minimizer 1 . Vt+1 − Vt ≤ −`(xt , ut , rt ) ≤ −ket k2Q . (4)
In closed-loop operation we apply the first part of the
optimized input trajectory to the system, leading to the Compactness of Z in combination with the local
closed-loop system xt+1 = f (xt , u∗0|t ) = x∗1|t , t ≥ 0. As quadratic upper bound (3) imply ket k2Q ≤ Vt ≤ cv ket k2Q ,
discussed in the introduction, we need suitable terminal for some cv ≥ cu ≥ 1 and all xt such that Problem (1) is
ingredients to ensure stability and recursive feasibility feasible, compare [29, Prop. 2.16]. Uniform exponential
for the closed loop. stability follows from standard Lyapunov arguments
based on the value function V .
Assumption 2 There exist a terminal controller kf :
Rn × Zr → Rm , a terminal cost Vf : Rn × Zr and a ter-
minal set Xf (r) ⊂ Rn , such that the following properties Thus, the closed-loop tracking MPC scheme given by (1)
hold for any r ∈ Zr , any x ∈ Xf (r) and any r+ ∈ R(r) has all the (standard) desirable properties, in case that a
reachable reference trajectory shall be tracked. The main
Vf (x+ , r+ ) ≤Vf (x, r) − `(x, kf (x, r), r), (2a) contribution of this work (see Sec. 3) is the development
(x, kf (x, r)) ∈Z, (2b) of MPC schemes for more general tracking problems, in-
cluding cases where an unreachable dynamic target sig-
x+ ∈Xf (r+ ), (2c) nal is given. We note that the tracking MPC scheme (1)
(and correspondingly also the proposed scheme in Sec-
with x+ = f (x, kf (x, r)). Furthermore, there exist con- tion 3) can be easily modified to ensure robust reference
stants cu , > 0, such that for any reference r·|t satisfying tracking, compare [16], [15, Thm. 2], and the discussion
Ass. 1, and any xt ∈ Rn with ket kQ ≤ , Problem (1) is in Remark 9.
feasible and the value function satisfies
In the following, we briefly detail how Assumption 2 can
Vt ≤ cu ket k2Q . (3) be satisfied using either a terminal equality constraint
or a suitable terminal cost.
The first set of conditions (2) is standard in (reference Terminal equality constraint - controllability
tracking) MPC, compare for example [5,29,10,15]. Con-
dition (3) ensures that the value function admits a local The following proposition shows that a simple terminal
quadratic upper bound. 2 In Prop. 4 and Lemma 5 be- equality constraint (TEC) with Xf (r) = xr , Vf = 0 sat-
low, we provide sufficient conditions for Assumption 2 isfies Ass. 2, if the system is locally uniformly exponen-
using controllability and stabilizability conditions. tially finite time controllable.
Theorem 3 Let Ass. 1–2 hold. Assume that Problem (1) Proposition 4 Suppose there exist constants N0 ∈ N,
is feasible at t = 0. Then Problem (1) is recursively fea- cu , TEC > 0, such that for any reference trajectory
r·|t satisfying Ass. 1, and any state xt ∈ Rn satisfying
1
Existence of a minimizer is, e.g., guaranteed if f, `, Vf are ket kQ ≤ TEC , there exists an input trajectory u·|t ∈
continuous and the constraint set is compact. If the mini- Rm×N0 such that
mizer is not unique, an arbitrary minimizer can be chosen.
2
The weak controllability condition [29, Ass. 2.17], which xk+1|t = f (xk|t , uk|t ), xN0 |t = xr,N0 |t ,
does not require feasibility of Problem (1) locally around
the reference trajectory r, would be sufficient to prove The- JN0 (x·|t , u·|t , r·|t ) ≤ cu ket k2Q .
orem 3. Assumption 2, similar to condition [19, Ass. 4] used
for setpoint stabilization, is stronger and will be used in the Then for any N ≥ N0 , Ass. 2 holds with Xf (r) = xr ,
proof of Theorem 8 in Section 3. Vf (x, r) = 0, kf (x, r) = ur and some constant > 0.
3PROOF. Satisfaction of (2) is standard. Consider set Xf the terminal controller kf is a feasible solution
Problem (1) at time t with initial condition satisfying to (1) and thus using (2a) the terminal cost Vf is an up-
ket kQ ≤ ≤ TEC , some prediction horizon N ≥ N0 , per bound on the value function V , which ensures (3).
and the input u·|t appended by uk|t = urk|t , k ≥ N0 . This
candidate input u satisfies JN (x·|t , u·|t , r·|t ) ≤ cu ket k2Q
and kxk|t − xrk|t k2Q + kuk|t − urk|t k2R ≤ cu 2 . Given that Proposition 3 in [15] provides a semidefinite program
rk|t ∈ Zr ⊆ int(Z), there exists a small enough con- (SDP) to compute matrices Pf , Kf satisfying (6) using
stant ∈ (0, TEC ], such that (xk|t , uk|t ) ∈ Z. Thus, the a parametrization of the form Pf = X −1 , Kf = Y Pf ,
θj : Z → R,
candidate solution is a feasible solution to (1) and the
bound (3) holds. X X
X(r) = X0 + θj (r)Xj , Y (r) = Y0 + θj (r)Yj ,
j j
The considered controllability condition holds, for exam-
ple, if the dynamic f is continuously differentiable and using methods for (quasi) linear parameter varying
the linearized dynamic f around any reachable trajec- (LPV) systems and gain scheduling [30]. This reference
tory r is N0 -step uniformly controllable. generic offline design is only done once for a given sys-
tem f and class of reference trajectories (Ass. 1) and
Terminal cost - reference generic offline computations the resulting terminal ingredients can be applied if the
reference r is optimized online (cf. Sec. 3). Details on
the computation with numerical examples can be found
In the following, we discuss how to compute a terminal
in [15].
cost Vf and a terminal set Xf satisfying Ass. 2. Denote
the Jacobian of f evaluated around an arbitrary point
r ∈ Zr by Discussion: The main advantage of using a terminal
equality constraint (TEC, Prop. 4) is the fact that no of-
∂f
∂f
fline design is needed. One of the main benefits of using
A(r) = , B(r) = . (5) a terminal cost/set (QINF, Lemma 5) is that the desired
∂x (x,u)=r ∂u (x,u)=r properties also hold for an arbitrarily small prediction
horizon N . Furthermore, the values of cu , can be com-
puted explicitly and are typically significantly less con-
Lemma 5 Suppose that f is twice continuously differen- servative than the ones obtained when using a terminal
tiable. Assume that there exist a matrix Kf (r) ∈ Rm×n equality constraint (TEC, Prop 4), which impacts the
and a positive definite matrix Pf (r) ∈ Rn×n continuous closed-loop convergence rate. This impact on closed-loop
in r, such that for any r ∈ Zr , r+ ∈ R(r), the following performance is quantitatively investigated with numer-
matrix inequality is satisfied ical examples in Section 4.
(A(r) + B(r)Kf (r))> Pf (r+ )(A(r) + B(r)Kf (r)) 3 Nonlinear tracking MPC for dynamic target
≤Pf (r) − (Q + Kf (r)> RKf (r)) − ˜In (6) signals
with some constant ˜ > 0. Then there exists a sufficiently In the following, we design a nonlinear MPC scheme for
small α, such that Ass. 2 is satisfied for any N ≥ 0 with tracking of exogenous (unreachable) periodic target sig-
nals using the terminal ingredients in Assumption 2. For
Vf (x, r) = kx − xr k2Pf (r) , cu = sup λmax (Pf (r), Q), the resulting closed-loop system, Theorem 8 establishes
r∈Zr
p exponential stability of the optimal reachable periodic
Xf (r) = {x ∈ Rn | Vf (x, r) ≤ α}, = α/cu , trajectory. In Section 3.3 we discuss how online optimiza-
kf (x, r) = ur + Kf (r) · (x − xr ), tion of the terminal set size α can speed up convergence,
while at the same time allowing optimal operation (ar-
bitrarily) close to the constraints. Algorithm 1 in Sec-
where λmax (P, Q) denotes the maximal generalized eigen-
tion 3.4 provides a simple means to reduce the compu-
value solving (P − λQ)v = 0, for some v ∈ Rn .
tational demand by partially decoupling the trajectory
planning and the reference tracking problem.
PROOF. Satisfaction of conditions (2) follows from [15,
Lemma 1] based on standard differentiability/Lipschitz 3.1 Nonlinear periodic tracking MPC
continuity arguments, compare also [5,29,10]. Further-
more, kx−xr k2Q ≤ implies Vf (x, r) ≤ cu kx−xr k2Q ≤ α In [18,19,21], tracking MPC schemes based on simulta-
and hence x ∈ Xf (r). Using standard arguments (com- neous optimization of an artificial reference have been
pare for example [29, Prop. 2.35]), within the terminal introduced. Compared to a standard reference tracking
4MPC formulation such as (1), these schemes ensure re- using JT . As we will see later in the theoretical analysis
cursive feasibility independent of the (potentially un- (Thm. 8) and the numerical examples (Sec. 4), this for-
reachable) target signal and provide a large region of at- mulation ensures that the closed loop smoothly tracks
traction. In the following, we extend these methods to the optimal reachable periodic trajectory xT ∗ .
nonlinear periodic reference tracking using general ter-
minal ingredients (Ass. 2, using a terminal equality con- 3.2 Theoretical analysis
straint (TEC, Prop. 4) or a terminal cost/set (QINF,
Lemma 5)).
In the following, we derive the theoretical properties of
r r r the closed-loop system based on (8). The following con-
We consider a nonlinear output function y = h(x , u ) ∈ dition is used to ensure exponential stability of the opti-
Rp and assume that at time t an exogenous T -periodic mal trajectory xT ∗ , similar to [21, Ass. 2],[19, Ass. 1-2].
e
target signal y·|t ∈ Rp×T is given. For some T -periodic
reference trajectory r·|t = (xr·|t , ur·|t ) ∈ R(n+m)×T , the Assumption 6 There exist (unique) locally Lipschitz
tracking cost w.r.t. this target signal y e is defined as continuous functions gx : Rp×T → Rn×T , gu : Rp×T →
Rm×T such that gx (y·|t r
) = xr·|t , gu (y·|t
r
) = ur·|t , for any
T
X −1 feasible solution to (7). The set of feasible solutions to (7)
e
JT (r·|t , y·|t ) := k h(xrj|t , urj|t ) −yj|t
e 2 r
kS = ky·|t e 2
− y·|t kS , is convex 3 in y·|t
r
.
j=0 | {z }
r
=yj|t
This assumption implies that (7) is a strictly convex
problem and the minimizer rT ∗ is unique. Thus, for any
with some positive definite weighting matrix S ∈ Rp×p . y r 6= y T ∗ it is possible to incrementally change y r , such
The objective is to stabilize the reachable (Ass. 1) T - that it remains feasible and the cost JT decreases. Fur-
T∗
periodic reference trajectory r·|t = (xT·|t∗ , uT·|t∗ ) that min- thermore, due to convexity the directional derivative of
imizes the distance to the signal y e , which is defined as JT at y T ∗ in any feasible direction is non-negative, i.e.,
the minimizer to the following optimization problem for any reference r·|t that satisfies the constraints in (7),
r
the corresponding output y·|t satisfies
e e
VT (y·|t ) = min JT (r·|t , y·|t ) (7)
r·|t
r T∗ >
s.t. rj+1|t ∈ R(rj|t ) ⊆ Zr , r0|t = rT |t , j = 0, . . . , T − 1. (y·|t − y·|t ) ∇yr JT |yr =yT ∗ ≥ 0, (10)
T∗ which can be equivalently written as
The corresponding output reference is denoted by y·|t ,
T∗ T∗
with yk|t = h(rk|t ). In order to find and stabilize this pe- e
JT (r·|t , y·|t e
) ≥ VT (y·|t T∗
) + ky·|t r 2
− y·|t kS . (11)
riodic trajectory, we consider the following optimization
problem, similar to [21]
e e
WT (xt , y·|t ) = min JN (x·|t , u·|t , r·|t ) + JT (r·|t , y·|t ) Remark 7 Similar to [19, Remark 1], existence of gx ,
u·|t ,r·|t
gu can be ensured based on the implicit function theorem,
s.t. xk+1|t = f (xk|t , uk|t ), x0|t = xt , (8a) if a rank condition on the linearization of a suitably de-
(xk|t , uk|t ) ∈ Z, xN |t ∈ Xf (rN |t ), (8b) fined T -step system is satisfied and f , h are continuously
rj+1|t ∈ R(rj|t ) ⊆ Zr , rl+T |t = rl|t , (8c) differentiable.
For a linear output h(x, u) = Cx + Du, the convexity
j = 0, . . . , T − 1, k = 0, . . . , N − 1, assumption is ensured if the constraint set in (7) (de-
l = 0, . . . , max{0, N − T }. scribing reachable periodic orbits) is convex. For T = 1,
this reduces to convexity of the steady-state manifold Zr ,
The optimal state and input trajectory is given by u∗·|t , which is often easy to verify. Furthermore, even in case
x∗·|t , with the artificial reference r·|t
∗
= (xr∗ r∗ the set of reachable periodic trajectories is non-convex,
·|t , u·|t ) and the
output yk|t r∗ ∗
= h(rk|t ). In closed-loop operation we apply it may be possible to choose a suitable nonlinear output
h, such that the convexity condition (Ass. 6) is satisfied,
the first part of the optimized input trajectory to the compare [7].
system, leading to the following closed-loop system If the convexity condition in Assumption 6 is not satis-
fied, the MPC scheme will not necessarily stabilize the
xt+1 = f (xt , u∗0|t ) = x∗1|t , t ≥ 0. (9) optimal reachable trajectory xT ∗ , but could instead stabi-
lize a suboptimal periodic trajectory, similar to [26]. The
The rational behind this optimization problem is to pe-
3
nalize the (standard) tracking cost JN w.r.t. some arti- Given two feasible solutions r1 , r2 with corresponding
ficial periodic reference r together with the distance of outputs y1r , y2r , the reference r·|t = (gx (y·|t
r r
), gu (y·|t )) is a
r r r
the output of this artificial reference to the target signal feasible solution to (7) with y·|t = βy1 + (1 − β)y2 , β ∈ [0, 1].
5main alternative to using a tracking scheme with simulta- with some constant Lg . Thus, strict convexity (com-
neous optimization of the artificial trajectory such as (8), pare (11)) implies
would be to directly solve (7) and then apply a tracking
MPC for this reachable reference trajectory, compare Sec- ∗
JT (r·|t ∗T
, t) − VT ≥ ky·|t r∗ 2
− y·|t kS
tion 2. If (7) is solved with a standard convex solver, the
solver will most likely end in the same local minimum as ≥1/L2g kxT·|t∗ − xr∗ 2 2 T∗ r∗ 2
·|t kQ ≥ 1/Lg kx0|t − x0|t kQ .
the closed-loop MPC scheme. Thus, even if the convexity
condition is not satisfied, the proposed scheme (8) is still Correspondingly, using the fact that a2 +b2 ≥ 1/2(a+b)2
a good choice. for all a, b ∈ R yields the lower bound
The following theorem establishes exponential stability Wt ≥ kxt − xr∗ 2 ∗
0|t kQ + JT (r·|t , t) − VT
of the optimal reachable trajectory xT ∗ given suitable ∗T
terminal ingredients (Ass. 2) and the convexity condi- ≥kxt − xr∗ 2 2 r∗ 2 T 2
0|t kQ + 1/Lg kx0|t − x0|t kQ ≥ αW ket kQ ,
tion on the set of feasible periodic orbits (Ass. 6), which
is the main result of this paper. This result generalizes with αW = 21 min{1, 1/L2g }. In case keTt k2Q ≤ 2 , In-
and unifies the results in [19,21], by considering non- equality (3) in Assumption 2 ensures that r·|t = r·|t T∗
is
linear dynamics, periodic reference trajectories, estab- T 2
lishing exponential stability, and unifying the considera- a feasible solution to (8), which satisfies Wt ≤ cu ket kQ .
tion of different terminal ingredients (Ass. 2, Lemma 5, As in Theorem 3, compact constraints together with this
Prop. 4). local upper bound imply the imply the upper bound
in (12b), compare [29, Prop. 2.16]. Inequalities (12) im-
Theorem 8 Let Assumptions 2 and 6 hold. Assume that ply (uniform) stability of xT ∗ for the closed-loop system,
h is bounded on Zr and that the Problem (8) is feasible at but not necessarily asymptotic or exponential stability.
t = 0. Then the Problem (8) is recursively feasible for the Part III: Exponential stability - case distinction:
resulting closed-loop system (9), for arbitrary target sig- Case 1: Consider
nals y e . Furthermore, for a T -periodic target signal y e ,
T∗ 2
the optimal reachable trajectory xT ∗ is (uniformly) expo- kxt − xr∗ 2 r∗
0|t kQ ≥ γky·|t − y·|t kS , (14)
nentially stable for the resulting closed-loop system (9).
with a later specified positive constant γ. Then, (12a)
and (13) imply
PROOF. Part I: Recursive Feasibility: It suffices to
note that feasibility of (8) does not depend on the target (12a)
signal y e . Correspondingly, the input sequence u·|t+1 in Wt+1 − Wt ≤ −kxt − xr∗ 2
0|t kQ
∗
Theorem 3 with the shifted reference rk|t+1 = rk+1|t is (14)
T∗ 2
a feasible solution to (8). ≤ − 1/2(kxt − xr∗ 2 r∗
0|t kQ + γky·|t − y·|t kS )
Part II: Stability: Consider a periodic target signal (13)
e e e e T∗ 2
yk|t = yk−1|t+1 , which is denoted by yt+k := yk|t . Thus, ≤ − 1/2(kxt − xr∗ 2 2 r∗
0|t kQ + γ/Lg kx0|t − x0|t kQ )
∗
the minimizer of (7) is a periodic trajectory, i.e., xTk+t := ≤ − 1/4 min 1, γ/L2g kxt − xTt ∗ k2Q .
T∗ T∗ e
xk|t = xk−1|t+1 , and we write JT (r, t) := JT (r, y·|t ),
e
VT := VT (y·|t ), with JT (periodically) time-varying in Case 2: Assume
the second argument and VT constant in time. Define the
T∗ 2
candidate Lyapunov function Wt := WT (xt , y·|t e
) − VT kxt − xr∗ 2 r∗
0|t kQ ≤ γky·|t − y·|t kS . (15)
T T∗
and the error et := xt − xt . In the following, we show
that there exists a positive constant αW , such that Boundedness of h on Zr implies that there exists a con-
r r 2
stant ymax , such that ky·|t − ỹ·|t kS ≤ ymax , for any tra-
Wt+1 ≤ Wt − kxt − xr∗ 2
0|t kQ , (12a) r r
jectories y , ỹ that satisfy the constraints in (7). This
αW keTt k2Q ≤ Wt ≤ cv keTt k2Q , (12b) implies
(15)
holds for all xt such that Problem (8) is feasi- kxt − xr∗ 2 r∗ T∗ 2
0|t kQ ≤ γky·|t − y·|t kS ≤ γymax .
ble. The shifted reference r·|k+1 in Part I satisfies
∗
JT (r·|t+1 , t + 1) = JT (r·|t , t). Thus, feasibility in com-
∗ For γ ≤ γ1 := 2 /ymax , we have kxt − xr∗ 2 2
0|t kQ ≤ . Thus,
bination with (4) implies Wt+1 − Wt ≤ −`(xt , ut , r0|t ),
Assumption 2 implies
which implies (12a). Lipschitz continuity of g implies
T∗ T∗ kx∗1|t − xr∗ 2 ∗ ∗ ∗
1|t kQ ≤ JN (x·|t , u·|t , r·|t ) (16)
kxr∗ r∗
0|t − x0|t kQ ≤ kx·|t − x·|t kQ (13)
(3) (15)
r∗ T∗ r∗ T∗ T∗ 2
=kgx (y·|t ) − gx (y·|t )kQ ≤ Lg ky·|t − y·|t kS , ≤ cu kxt − xr∗ 2 r∗
0|t kQ ≤ γcu ky·|t − y·|t kS ≤ γcu ymax .
6For γ ≤ γ2 := 2 /(4cu ymax ) ≤ γ1 this implies kxt+1 − continuity, compare (13). Correspondingly, we have
xr∗ r
1|t kQ ≤ /2. Consider yk|t+1 = h(rk|t+1 ), where r·|t+1
is the candidate reference from Part I of the proof. At Wt+1 − Wt
time t + 1, define an auxiliary reference ∗
≤ JN (x̂, û, r̂) + JT (r̂, t + 1) − JT (r·|t , t) − kxt − xr∗ 2
0|t kQ
(19),(20)
ŷ r := βy·|t+1
r T∗
+ (1 − β)y·|t+1 , β ∈ [0, 1], (17) ≤ 2cu kxt+1 − xr∗ 2 r∗ 2
1|t kQ − kxt − x0|t kQ
T∗ 2
r∗
− ((1 − β 2 ) − 2cu L2g (1 − β)2 ) ky·|t − y·|t kS
| {z }
with the corresponding state and input trajectory x̂r = =:c2 (β)
gx (ŷ r ), ûr = gu (ŷ r ), r̂ = (x̂r , ûr ). Convexity (Ass. 6) en- (13),(16)
T∗ 2
sures that the auxiliary reference r̂ is a feasible solution ≤ (2c2u γ − c2 (β)/2)ky·|t
r∗
− y·|t kS
to (7) at t + 1. This definition implies T∗ 2
− c2 (β)/(2L2g )kxr∗ r∗ 2
0|t − x0|t kQ − kxt − x0|t kQ
T∗ 2
≤(2c2u γ − c2 (β)/2)ky·|t
r∗
− y·|t kS
T∗
ŷ r − y·|t+1
r
= (1 − β)(y·|t+1 r
− y·|t+1 ). (18) − min{1/2, c2 (β)/(4L2g )}kxt − xT0|t∗ k2Q .
The cost JT satisfies Let β = β2 := arg maxβ∈[β1 ,1] c2 (β), with c2 (β2 ) > 0.
For γ ≤ γ3 := c2 (β2 )/(4c2u ), this implies
∗
JT (r̂, t + 1) − JT (r·|t , t) Wt+1 − Wt ≤ − min 1/2, c2 (β2 )/(4L2g ) kxt − xTt ∗ k2 .
=(ŷ r − y·|t+1
r
)> S(ŷ r + y·|t+1
r e
− 2y·|t+1 )
(17)
Combine: Combining these two cases yields
T∗
= (1 − β)(y·|t+1 r
− y·|t+1 )> S
r
((1 + β)y·|t+1 T∗
+ (1 − β)y·|t+1 e
− 2y·|t+1 ) Wt+1 ≤Wt − γT kxt − xTt ∗ k2 , (21)
T∗ 2 c2 (β2 ) 1 γ
= − (1 − β 2 )ky·|t
r∗
− y·|t kS γT := min , , , γ := min{γ1 , γ2 , γ3 }.
T∗ r
4L2g 4 4L2g
+ (1 − β)(y·|t+1 − y·|t+1 )∇yr JT (y r , y e )|yr =yT ∗
(10) Uniform exponential stability follows using inequali-
T∗ 2
≤ − (1 − β 2 )ky·|t
r∗
− y·|t kS . (19) ties (12b), (21) and Lyapunov arguments.
Lipschitz continuity (compare (13)) implies
Theorem 8 ensures exponential stability of the optimal
reachable trajectory xT ∗ by showing quadratic lower and
kxt+1 − x̂r0 kQ ≤ kxt+1 − xr∗ r∗ r
1|t kQ + kx1|t − x̂0 kQ upper bounds and an exponential decay of the Lyapunov
e
r
≤/2 + Lg ky·|t+1 − ŷ r kS function Wt := WT (xt , y·|t ) − VT . The exponential de-
cay of Wt is shown by utilizing two distinct candidate
(18) r T∗
= /2 + Lg (1 − β)ky·|t+1 − y·|t+1 kS . solutions, namely (x̂, û, r̂) and the standard candidate
solution from Theorem 3. In particular, we distinguish
whether the tracking error kxr∗ 2
0|t −xt kQ is large/small (γ)
√
For β ∈ [β1 , 1] with β1 := 1−/(2Lg ymax ), this implies compared to the output tracking cost JT = ky·|t r∗ e 2
−y·|t kS .
kxt+1 − x̂r0 kQ ≤ . Thus, Assumption 2 ensures that If the reference tracking error is large, then the stan-
there exists some state and input sequence (x̂, û), such dard candidate solution, e.g. used in Theorem 3, ensures
that (x̂, û, r̂) is a feasible solution to (8) at time t + 1 a sufficient exponential decrease in the Lyapunov func-
and the tracking cost satisfies tion Wt . On the other hand, if the reference tracking
error is small enough (γymax ), then the convexity con-
(3) dition (Ass. 6) ensures that the artificial reference r can
JN (x̂, û, r̂) ≤ cu kxt+1 − x̂r0 k2Q (20) be incrementally (β) moved towards the optimal reach-
r∗ 2 able reference rT ∗ , which decreases the output tracking
≤2cu (kxt+1 − x1|t kQ + kxr∗ r 2
1|t − x̂0 kQ ) cost JT . The local quadratic bound (3) (Ass. 2) on the
≤2cu kxt+1 − xr∗ 2 2 r
1|t kQ + 2cu Lg ky·|t+1 − ŷ r k2S , value function V ensures that the optimization problem
(18) is feasible with the incrementally moved reference r̂ and
T∗ 2
≤ 2cu kxt+1 − xr∗ 2 2 2 r∗
1|t kQ + 2cu Lg (1 − β) ky·|t − y·|t kS , that the increase in the tracking cost JN is quadratically
bounded. Finally, there exists a sufficiently small change
(β2 ), such that this auxiliary candidate solution (x̂, û, r̂)
where the second to last inequality follows from Lipschitz ensures an exponential decay in Wt .
7Remark 9 Similar to the derivations in [19,21], Theo- Remark 11 We note that this result guarantees expo-
rem 8 assumes no model mismatch, which is rarely the nential stability of the optimal reachability trajectory
case in practical applications. To ensure robust recur- xT ∗ using either terminal equality constraints (Prop. 4)
sive feasibility despite disturbances, the MPC problem (8) or a terminal cost (Lemma 5), respectively. However,
needs to be adjusted using constraint tightening tech- the quantitative bounds regarding convergence rate may
niques from robust MPC. A corresponding formulation differ significantly. In particular, including suitably de-
for nonlinear robust tracking MPC systems can be found signed terminal ingredients greatly improves the closed-
in the recent paper [27], which combines the formulation loop performance, which is demonstrated in the numeri-
presented in Section 3.3 with the nonlinear robust MPC cal examples in Section 4, compare also examples in [15].
formulation in [16].
3.3 Online optimized terminal set size
In addition to possible feasibility issues, model mismatch
typically also implies non-zero offset, even in case of con- In case we use a terminal cost and terminal set
stant references. For the special case of setpoint track- (Lemma 5), the fact that Zr is chosen in advance (as also
ing (T = 1), this issue is typically resolved using offset- done in [19,21]) can be disadvantageous. In particular,
free MPC formulations (cf. [25] and references therein), similar to setpoint stabilization in [5], the terminal set
which rely on a disturbance estimator for constant off- size α in Lemma 5 is the minimum of two values: α1 and
sets. In order to transfer this concept to T -periodic tra- α2 . The first (α1 ) is independent of Zr and needs to be
jectories, the dimension of the disturbance model must such that (2a) holds (cf. [15, Alg. 1]). The second (α2 )
be correspondingly increased. An alternative approach to ensures constraint satisfaction (2b) and depends on the
ensure offset-free tracking is to use a parameter estima- difference between Z and Zr . Thus, by choosing Zr we
tion scheme with an adaptive MPC formulation, under trade achievable terminal set size (α2 ) (and hence con-
appropriate assumptions on the model mismatch, com- vergence speed of the closed-loop state) against opera-
pare e.g. [4]. Extending the proposed approach to ensure tion close to the boundary of the constraint set Z. In the
offset free tracking despite deterministic model mismatch following, we show how α can be optimized online, in-
is an open problem. stead of using a preassigned reference constraint set Zr .
In [31], a similar dynamic scaling of the terminal set has
Remark 10 The proposed approach can be extended to been suggested for linear setpoint tracking with a poly-
stabilize the economically optimal reachable reference r, topic terminal set (albeit with a different motivation).
as an extension to the linear approach in [20], assuming The proposed formulation is geared towards the termi-
that (7) remains (strictly) convex (Ass. 6). The exten- nal ingredients from Lemma 5 and polytopic constraints
sion of the approaches in [26,9] for economically opti- of the form Z = {r ∈ Rn+m | Li r ≤ li , i = 1, . . . , nz }.
mal steady-state operation, to optimal dynamic operation To this end, we define the following functions
based on this result is part of current research. Recently,
−1/2
in [2] for the special case of linear systems, the analysis LP K,i (r) := kPf (r)[In , Kf> (r)]L>
i k, i = 1, . . . , nz .
in Theorem 8 was extended to stage costs ` with Q semi-
definite using suitable observability conditions, which is The modified optimization problem is given by
especially relevant for input-output models.
e
min JN (x·|t , u·|t , r·|t ) + JT (r·|t , y·|t )
u·|t ,r·|t ,αst
The consideration of nonperiodic dynamic target signals
in this framework is still an open topic. Setpoint sta- s.t. xk+1|t = f (xk|t , uk|t ), x0|t = xt , (22a)
bilization is a special case in Theorem 8 with T = 1, (xk|t , uk|t ) ∈ Z, k = 0, . . . , N − 1 (22b)
R(r) = r and the feasible steady-state manifold Zr , which √ √
has also been considered in [19]. Compared to [19], Theo- Vf (xN |t , rN |t ) ≤ (αts )2 , αmin ≤ αts ≤ α1 , (22c)
rem 8 ensures exponential stability with convergence rate xrj+1|t = f (xrj|t , urj|t ), rl+T |t = rl|t , (22d)
cv −γT
cv < 1 and Lyapunov function Wt , while [19] uses a Li rj|t + LP K,i (rj|t )αts ≤ li , i = 1, . . . , nz (22e)
proof of contradiction to establish convergence. Further- j = 0, . . . , T − 1, l = 0, . . . , max{0, N − T }, .
more, the general assumptions on the terminal ingredi-
ents (Ass. 2) allow us to use the continuously parame- Compared to (8), the reference constraints (8c) and
terized and thus differentiable terminal cost Vf based on in particular the polytopic constraints Zr are replaced
Lemma 5 (cf. [15]), instead of partitioning Zr , as done by (22d) and (22e) and we√have one additional op-
in [19]. The practicality of this result is demonstrated in timization variable αs = α, where αmin > 0 is
the numerical examples in Section 4. In the setpoint sta- needed to avoid robustness issues and retain the closed-
bilization case (T = 1) with N = 0, the candidate solu- loop properties of the original scheme (Thm. 8). The
tions in Theorem 8 correspond to an inner-loop controller −1/2
kf (x, r) with a corresponding (explicit) reference gover- expression Pf (r) in LP K,i denotes any matrix
nor (cf. [12]) for r given by (17), compare the numerical square root of X(r) = Pf−1 (r), which can be com-
example in Section 4. puted using the command sqrtm(X) or chol(X) in
8Matlab. In the numerical examples (Sec. 4), we imple- the polytopic characterization is reduced by online opti-
ment (22e) usingP the symbolic Cholesky decomposition mizing a scaling of the polytopic terminal set. The pro-
of X(r) = X0 + i θi (r)Xi , which is suitable for auto- posed approach, especially the simplified formula (23),
matic differentiation used in CasADi [1]. can be viewed as an extension of this approach to nonlin-
ear systems with ellipsoidal terminal sets. For compari-
Proposition 12 Suppose there exist matrices Pf , Kf son, in [19] the reference constraint set Zr is partitioned
and a constant α > 0, such that condition (2a) holds and a fixed size αi is considered for each partition Yi ,
for all r, r+ ∈ Z, Vf (x, r) ≤ α1 with xr+ = f (xr , ur ), which is conservative compared to the online optimized
x+ = f (x, ur + Kf (r)(x − xr )). Assume further that h is value αs , compare Fig. 1 in Section 4.
bounded on Z and Ass. 6 holds with Zr in (7) replaced by
Remark 13 The proposed approach with online opti-
√ mization of αs also provides a simple means to avoid ter-
Z̃r = {r| Li r + LP K,i (r) αmin ≤ li , i = 1, . . . , nz }.
minal equality constraints as a valuable extension of the
Then the MPC scheme based on (22) satisfies the theo- linear periodic tracking MPC in [21]. In case of linear
retical properties in Theorem 8. Consider the scheme (8) system dynamics with a polytopic terminal set and online
with a given constraint set Zr and correspondingly com- optimized αs (cf. [31]), the overall optimization problem
puted α2 . If α2 ∈ (αmin , α1 ), the scheme (22) can stabilize is a quadratic program (QP) with one additional scalar
references r ∈ / Zr and has a larger region of attraction. variable αs and linear inequality constraints instead of
the terminal equality constraint.
PROOF. First, note that (22e) is equivalent to (2b), Remark 14 It is possible to further relax the tightened
compare e.g. [6, Equation (10)] based on the support reference constraints (22e) or (23), by taking into ac-
function. Thus, every feasible solution count the fact that the terminal set is contractive with
√ to (8) is also a some constant ρ ∈ (0, 1), i.e.,
feasible solution to (22) with αts = α. Recursive fea-
sibility follows with the same candidate input by using
s
αt+1 = αts∗ , where αts∗ denotes the solution to (22). Parts Vf (f (x, kf (x, r)), r+ ) ≤ ρ2 Vf (x, r), (24)
II and III of Theorem 8 remain true sincepInequality (3) + r+ r
∀ Vf (x, r) ≤ α1 , r, r ∈ Z, x = f (x , u ), r
holds for all references r·|t ∈ Z̃r with = αmin /cu > 0.
Satisfaction of Ass. 6 with Z̃r as defined above ensures s
compare
√ √ [15, Prop. 1]. In particular, by redefining α =
that the reference
√ ŷ r (17) satisfies the constraints
√ in (22) α − αmin we can replace the constraints (22c),(22e)
with α = αmin . Furthermore, αs > α2 provides a
s
by the following constraints
larger terminal set and thus enlarges
√ the set of feasible
initial conditions. For αs < α2 we can consider ref- √
Vf (xN |t , rN |t ) ≤ (αts +αmin )2 ,
erences r ∈ / Zr (close to the boundary of Z) and thus √
provide a feasible solution for initial conditions close to Lmax,i (αts ρmod(j+T −N,T ) + αmin ) ≤ li − Li rj|t ,
√ √
the constraints and stabilize references r ∈ / Zr . αts ∈ [0, α1 − αmin ],
where mod denotes the modulo operator. The theoretical
Proposition 12 essentially confirms that this modifica- properties in Theorem 8 and Proposition 12 remain valid
s
tion preserves the theoretical properties in Theorem 8. with the candidate solution αt+1 = ραts∗ , which satisfies
√
In summary, the optimization over αs provides an addi- αt+1 ≥ max{ρ αt , αmin }, with αt∗ = (αts∗ + αmin )2 .
2 ∗
tional degree of freedom which can significantly enlarge These relaxed constraints are especially useful in tran-
the terminal set and lead to faster convergence. For nu- sient operation with active constraints on the reference r
merical reasons, one can also replace the constraint (22e) and ρT 1.
with the more conservative constraint
3.4 Partially decoupled reference updates
Lmax,i αts + Li rj|t ≤ li , j = 0, . . . T − 1, (23)
Lmax,i := max LP K,i (r), i = 1, . . . , nz .
r∈Z In the following, we demonstrate that the joint stabiliza-
tion and trajectory planning (Sec. 3) can be partially de-
This formulation also retains the properties in Theo- coupled, which can significantly reduce the online com-
rem 8 and the constraint (23) is linear in the optimiza- putational demand.
tion variables αs , r. This constraint on the reference r is Motivation: The main premise of the proposed approach
similar to the constraint tightening in robust MPC with (Sec. 3) is that the operating conditions change on a
a variable tube size αs . In the linear polytopic setpoint time scale similar to the system dynamics, which in turn
tracking case, a polyhedral invariant set for tracking can necessitates online updates of the reference trajectory.
be used to constrain the reference r and the terminal The most challenging problems are those, where the op-
state x, compare [18, Ass. 2]. In [31] the complexity of erating conditions change at a similar time scale to the
9system dynamics, while the target signal and hence the The result essentially follows from the continuously dif-
optimal system operation is determined based on long ferentiable parametrization and the triangular inequal-
term considerations that involve a significantly larger ity. In the special case of constant matrices Pf , condi-
time scale, i.e., the period length T is very large. An tion (26) is satisfied with Lp = 0. In addition to the
example of such a multi time-scale problem would be continuity property (26), we use the fact that the termi-
energy systems, compare e.g. [17], where real time deci- nal cost is contractive with some factor ρ ∈ (0, 1), com-
sions are made every 5 min, while the planning horizon pare (24).
is 7 days yielding T ≥ 2 · 103 . For such problems, it is
vital that the reference r is updated frequently, while at In the following, we summarize the basic approach to
the same time it may be computationally too expensive partially decouple the optimization problem (22). Sup-
to solve the joint planning and regulation problem (8) pose at time ti , we have trajectories x·|ti , u·|ti , r·|ti ,
in each time step t. αtsi , that satisfy the constraints in (22). For the next
Continuity terminal cost: In the following, we demon- M ∈ N time-steps t = ti , . . . , ti + M − 1, the track-
strate how the optimization problem (8) can be decom- ing MPC considers the shifted reference r·|t ∗
, i.e., rk|t =
i
posed into two partially decoupled optimization prob- ∗
rmod(k+t−t and the following updated terminal set
lems that may be solved at different time scales by using i ,T )|ti
the following continuity property of the terminal cost. size
∗
Proposition 15 Suppose that the conditions in Lemma 5 αttr =ρ2(t−ti ) max{αmin , Vf (xN |ti , rN |ti )}, (27)
are satisfied with Zr = Z, Pf = X −1 , Kf = Y Pf ,
with the contraction rate ρ according to (24). The closed-
p p
X X loop input is computed based on the following reference
X(r) = X0 + θj (r)Xj , Y (r) = Y0 + θj (r)Yj , tracking MPC (similar to (1))
j=1 j=1
min JN (x·|t , u·|t , r·|t ) (28a)
with θj continuously differentiable. Then there exists a u·|t
constant Lp , such that for any r, r̃ ∈ Z the terminal cost s.t. xk+1|t = f (xk|t , uk|t ), (xk|t , uk|t ) ∈ Z, (28b)
Vf satisfies the following continuity condition
Vf (xN |t , rN |t ) ≤ αttr , x0|t = xt , (28c)
k = 0, . . . , N − 1.
q
Vf (x, r̃) (26)
q
≤ Vf (x, r)(1 + Lp kr − r̃k) + kxr − x̃r kPf (r̃) . Note that the contractive terminal constraint (28c) with
αttr according to (27) is similar to a contractive MPC [8].
In parallel, the following reference optimization problem
is solved at time ti in order to obtain an updated refer-
PROOF. The fact that θj is continuously differentiable
ence at ti+1 = ti + M
directly implies that X is continuously differentiable
w.r.t. r. Thus, also the matrix Pf = X −1 is continuously e
differentiable in r for any r ∈ Z, using the fact X is posi- min JT (r·|ti+1 , y·|t i+1
)
r·|ti+1 ,αst
i+1
tive definite with uniform lower and upper bounds. This q
M ∗
property in combination with compact constraints and s.t. ρ αttri (1
+ Lp krN +M |ti − rN |ti+1 k)
uniform bounds on Pf ensures that there exists a local q
s
Lipschitz constant Lp ≥ 0, such that + Vf (xr∗
N +M |ti , rN |ti+1 ) ≤ αti+1 , (29a)
Pf (r̃) − Pf (r) ≤ Lp Pf (r)kr − r̃k, xrj+1|ti+1 = f (xrj|ti+1 , urj|ti+1 ), (29b)
rl+T |ti+1 = rl|ti+1 , (29c)
which implies Li rj|ti+1 + LP K,i (rj|ti+1 )αtsi+1 ≤ li , (29d)
√ √
q αmin ≤ αtsi+1 ≤ α1 , (29e)
kxkPf (r̃) ≤ kxk2Pf (r) + Lp kxk2Pf (r) kr − r̃k
q i = 1, . . . , nz , j = 0, . . . , T − 1,
=kxkPf (r) 1 + Lp kr − r̃k ≤ kxkPf (r) (1 + Lp kr − r̃k), l = 0, . . . , max{0, N + M − T }.
for any r, r̃ ∈ Z. Thus, condition (26) follows from Note, that the constraints on the reference r can be fur-
ther relaxed using the formula in Remark 14. Since we
e
start to solve (29) at time ti , the target signal y·|t is
q
Vf (x, r̃) = kx − x̃r kPf (r̃) i+1
not yet available and instead the currently available tar-
≤kx − xr kPf (r̃) + kxr − x̃r kPf (r̃) e
get signal y·|t i
needs to be shifted by M time steps (as-
q
suming it is T -periodic). The overall algorithm is sum-
≤ Vf (x, r)(1 + Lp kr − r̃k) + kxr − x̃r kPf (r̃) .
marized in Alg. 1.
10Algorithm 1 Partially decoupled reference updates Theorem 3 and the contractivity (24). Correspondingly,
Execute at each time step ti = i · M , i ∈ N at time ti+1 , the candidate solution satisfies
∗
Obtain r·|t i
from reference planner (29).
∗ 2M tr
Get xN |ti from Tracking MPC (28). Vf (xN |ti+1 , rN +M |ti ) ≤ ρ αti . (30)
Compute αttri using (27).
Tracking MPC The constraint (29a) ensures
for t = ti , . . . , ti + M − 1 do
Update r·|t , αttr .
q
∗
Vf (xN |ti+1 , rN |ti+1 )
Solve tracking MPC (28).
Apply control input ut = u∗0|t . (26)(30) q
∗ ∗
≤ ρM αttri (1 + Lp krN +M |ti − rN |ti+1 k)
end for
(29a)
Reference planner
q
∗ s∗
e + Vf (xr∗ N +M |ti , rN |ti+1 ) ≤ αti+1 ,
Obtain target signal y·|t i+1
.
Solve trajectory planning problem (29).
which in combination with the update (27) and (29e)
The optimization problem (28) represents a standard implies αttri+1 ≤ αts∗i+1 . At time ti+1 a feasible solution
tracking MPC (Sec. 2) that is executed in each time to (29) is given by the previous reference r shifted by
∗
step t with a fixed (periodic) reference trajectory r and M steps, i.e., rj|ti+1 = rmod(j+M,T )|ti , j = 0, . . . , T − 1,
a shrinking terminal set. On the other hand, the op- with the candidate terminal set size
timization problem (29) can be solved in the interval
[ti , ti+M ], thus allowing to solve larger planning prob-
q √ √
αtsi+1 = max{ρM αttri + 0.5(1 − ρM ) αmin , αmin }
lems (T >> 1) by updating the reference r less fre-
quently (M ≥ 1). Condition (29a) constrains how the
updated reference r may deviate from the previous so- and condition (29a) strictly satisfied using the fact hat
lution, which partially couples the planning (29) and Vf (xr∗
N +M |ti , rN |ti+1 ) = 0 by definition.
regulation problem (28). Compared to a joint optimiza- Part II. Convergence: Consider the auxiliary candidate
tion, as in (8), the practical convergence under chang- reference r̂ based on ŷ r from (17) with some βti ∈ [0, 1].
ing operation conditions may be slower, as the reference Suppose βti is chosen, such that kr̂ − r·|ti+1 k ≤ , with
is updated less frequently and the constraint (29a) lim- some constant . There exists a constant 1 > 0, such
its the rate of change, compare the numerical example that for ≤ 1 this auxiliary reference satisfies the con-
in Sec. 4. However, the partially decoupled updates in straint (29a), with
Alg. 1 can significantly reduce the computational de-
mand, especially in case of longer planning horizons T . q q
ρMαttri (1 + Lp krN |ti+1 − r̂N k) + Vf (xrN |ti+1 , r̂N )
Proposition 16 Suppose the conditions in Prop. 12 q √ √
and 15 hold and Alg. 1 is initialized with r·|t ∗
, αts∗0 sat- ≤ρM αttri + ( α1 ρM Lp + cu )1
0
isfying (29b)–(29e), αttr0 ≤ αts∗0 and xt0 such that (28)
q √
=ρM αttri + (1 − ρM )0.5 αmin ≤ αtsi+1 .
is feasible. Then Alg. 1 is recursively feasible for the re-
sulting closed-loop system. Assume further 4 that there
exists a constant c > 0, such that for every constraint We show satisfaction of (29d) for the auxillary reference
i = 1, . . . , nz , we have either inf r∈Z̃r LP K,i (r) ≥ c or r̂ with a case distinction. √
supr∈Z̃r LP K,i (r) = 0. For a T -periodic target signal Case 1: Suppose that αtsi+1 = αmin . In this case con-
∗
y e , the resulting reference r·|t i
converges to the optimal dition (29d) is equivalent to r̂ ∈ Z̃r , which is guaranteed
T∗ by the convexity condition (Ass. 6) as in Prop. 12.
reachable trajectory r in finite time and the state xt
converges exponentially fast to xTt ∗ . s M
q √
Case 2: αti+1 = ρ αti + 0.5(1 − ρM ) αmin . Given
tr
Pf , Kf and Pf−1 continuous, Z compact and continuity
PROOF. Part I. Recursive feasibility: First, for t = of the quadratic norm, there exists a function δ ∈ K∞ ,
ti + k, k = 1, . . . , M − 1 the reference r·|t satisfies the such that for any r, r̃ ∈ Z:
tightened constraints (29d) with αts∗i . Thus, feasibility
of (28) at time ti implies recursive feasibility of (28) at LP K,i (r) − LP K,i (r̃) ≤ δ(kr − r̃k), i = 1, . . . , nz .
t with the updated terminal set size αttr ≤ αts∗i accord-
ing to (27), the standard MPC candidate solution from For constraints i with LP K,i (r̂(j)) = 0 feasibility
of (29d) is independent of αs and thus follows from
4
This condition excludes the special case where Li,x = convexity (Ass. 6). For the other constraints i satisfac-
−Li,u Kf (r) for some (but not all) r ∈ Z̃r . tion of condition (29d) at ti+1 follows from feasibility
11of (29d) at ti together with the definition of the candi- to uniform stability in Thm. 8. It is possible to adjust
∗ tr s∗
date reference rj|ti+1 = rmod(j+M,T )|ti , αti ≤ αti and Alg. 1, such that the reference planner does not require
LP K,i (r̂j ) > 0: explicit information from the system, by replacing the
update αttr in (27) and just using the fact that the termi-
LP K,i (r̂j )αtsi+1 + Li r̂j nal set is ρ-contractive. Although this may simplify the
q √ computation, the closed-loop convergence of the tracking
≤LP K,i (r̂j ) ρM αttri + 0.5(1 − ρM ) αmin MPC is typically significantly faster, which is why the
q update (27) can speed up the convergence rate of the ref-
+ Li (r̂j − rj|ti+1 ) + li − LP K,i (rj|ti+1 ) αttri erence planner. Furthermore, it is possible to implement
√ Alg. 1 in an asynchronous fashion with M changing on-
≤li + α1 δ(kr·|ti+1 − r̂k) + kLi kkr·|ti+1 − r̂k
line, if the constraint (29a) is adjusted to hold for any
q √ M ∈ [Mmin , Mmax ] ⊂ N. This way, the reference plan-
− (1 − ρM ) LP K,i (r̂j ) ( αttri − 0.5 αmin ) ≤ li ,
| {z } | {z } ner needs to solve (29) until ti + Mmax , but the reference
≥c √
≥0.5 αmin can also be updated earlier starting at ti + Mmin .
where the last inequality holds for ≤ 2 with some
2 > 0. The reference satisfies 4 Numerical examples
kr·|ti+1 − r̂k ≤ Lg kŷ r − y·|t
r
k
i S The following examples show the general applicability
(18) T∗ r of the proposed method and illustrate advantages com-
= (1 − βt )Lg ky·|t i+1
− y·|t k ,
i+1 S pared to existing approaches. We first benchmark the
performance of the proposed approach at the example
with some Lipschitz constant Lg , compare Ass. 6. Thus, of setpoint tracking of a continuous stirred tank reac-
r T∗
choosing βt = max{1 − /(Lg ky·|t i+1
− y·|t i+1
kS ), 0} < 1 tor (CSTR). Then, we demonstrate the applicability of
with = min{1 , 2 }, the candidate reference r̂ is feasi- nonlinear periodic reference tracking at the example of
r T∗
ble. In case ky·|ti+1
− y·|t kS ≥ /Lg , this implies a ball and plate system. The offline and online compu-
tation is done using SeDuMi-1.3 [32] and CasADi [1],
∗
JT (r̂, ti+1 ) − JT (r·|t , ti ) respectively.
i
(19) √
≤ − (1 − βt )(/Lg )2 ≤ −3 /(L3g ymax ), 4.1 Setpoint tracking - CSTR
r
with ymax as defined on page 6. In case ky·|t i+1
− The following example demonstrates the performance
T∗
y·|t kS ≤ /Lg , the candidate reference converges benefits of the proposed method, especially the param-
to the optimal reference trajectory in one step, i.e., eterized terminal ingredients (Lemma 5) and the online
βt = 0, JT (r̂, ti+1 ) = VT . This ensures conver- optimized terminal set (Prop. 12), at the example of set
gence of JT√to VT and thus r to rT ∗ in at most point tracking (T = 1).
Tmax = M (( ymax Lg /)3 + 1) time steps. Exponential System model: We consider a CSTR model 5
convergence of x to xT ∗ follows from exponential stabil-
M
ity (Thm. 3) and finite time convergence of the reference (1 − x1 ) − kx1 e− x2
!
1
ẋ1 θf
r. ẋ = = M
ẋ2 1
θf (xf − x2 ) + kx1 e− x2 − αf u(x2 − xc )
The result in Proposition 16 essential builds on three
with the concentration x1 , the temperature x2 and the
properties. First, the constraints on the reference plan-
coolant flow rate u, taken from [23]. The discrete-time
ner (29a) and the tracking MPC (28) with the contract-
model is defined with an Euler discretization and the
ing terminal set size (27) are such that the proposed al-
sampling time h = 0.1s. We consider the setpoint track-
gorithm is recursively feasible. Second, in each time step
ing problem using the nonlinear MPC schemes (8), (22)
ti , it is possible to incrementally reduce the tracking cost
in Section 3 with T = 1 and the output y = x2 .
JT , which in combination with convexity (Ass. 6) and
Offline computations and terminal set: The constraint
compact constraints implies that the reference planner
set is given by Z = [0, 1]2 ×[0, 2], the stage cost is Q = I2 ,
converges to the optimal reference rT ∗ in finite time. Fi-
R = 0.01, the output weighting is S = 103 and the fea-
nally, once the reference planner converged in finite time,
sible reference manifold is
the tracking MPC (28) ensures exponential convergence
to the reference r = rT ∗ (c.f. Thm. 3) and thus expo-
nential convergence of xt to xT ∗ . Zr = {(x, u)| f (x, u) = x, x2 ∈ [0.43, 0.86]}.
Remark 17 We point out that we only showed con- 5
The parameters are θf = 20, k = 300, M = 5, xf =
vergence for this partially coupled approach, as opposed 0.3947, xc = 0.3816, αf = 0.117.
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