System Level Synthesis-based Robust Model Predictive

Control through Convex Inner Approximation

Shaoru Chen, Nikolai Matni, Manfred Morari, Victor M. Preciado ∗

Abstract

We propose a robust model predictive control (MPC) method for discrete-time linear time-invariant systems with norm-bounded additive disturbances and model uncertainty. In ourmethod, at each time step we solve a finite time robust optimal control problem (OCP) whichjointly searches over robust linear state feedback controllers and bounds the deviation of thesystem states from the nominal predicted trajectory. By leveraging the System Level Synthesis(SLS) framework, the proposed robust OCP is formulated as a convex quadratic program inthe space of closed-loop system responses. When an adaptive horizon strategy is used, weprove the recursive feasibility of the proposed MPC controller and input-to-state stability ofthe origin for the closed-loop system. We demonstrate through numerical examples that theproposed method considerably reduces conservatism when compared with existing SLS-basedand tube-based robust MPC methods, while also enjoying low computational complexity.

1 Introduction

In model predictive control (MPC), a finite time constrained optimal control problem (OCP) issolved at each time step and the first optimal control input is applied. When the system model isuncertain, robust model predictive control explicitly takes the system uncertainty into account bysolving a robust OCP at each time step to guarantee that the state and control input constraints arerobustly satisfied for the closed-loop system. Although in theory dynamic programming [1, Chapter15] can exactly solve the resulting robust OCPs, such methods suffer from prohibitive computationalcomplexity, making them impractical. This has motivated the development of alternative solutionsto the robust OCPs which aim to reach a reasonable compromise between conservatism as measuredby the size of the set of feasible initial states, and computational complexity.

For linear time-invariant (LTI) systems with only uncertain additive disturbances, variousclosed-loop methods have been proposed to solve the robust OCPs with feedback policies as decisionvariables [2–9]. When model uncertainty is present, robust MPC becomes more challenging sincethe deviation from the nominal predicted trajectory depends on the system states and the feedbackpolicy to be designed. For polytopic model uncertainty, robust MPC methods bsed on linear matrixinequalities (LMI) are presented in [2, 10, 11] where affine state feedback policies are considered.For system with both model uncertainty and additive disturbances, the tube-MPC method in [5]designs a feedback policy that contains all possible trajectories inside a tube. In [12,13], linear statefeedback policies are considered and the robust OCPs are reformulated in the space of closed-loopsystem responses through the System Level Synthesis (SLS) framework [14]. The authors in [15]

∗The authors are with the Department of Electrical and Systems Engineering, University of Pennsylvania. Email:{srchen, nmatni, morari, preciado}@seas.upenn.edu. Our codes are publicly available at https://github.com/

ShaoruChen/Lumped-Uncertainty-SLS-MPC.

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apply a disturbance feedback policy and incorporate the policy parameters into constraint tighten-ing of the robust OCP. In [16], the authors propose to lump both model uncertainty and additivedisturbance into a net-additive disturbance with larger size and calculate a disturbance feedbackpolicy for robust MPC design to address the enlarged disturbances.

Inspired by [16], in this work we describe the uncertain system dynamics as the sum of thenominal dynamics and the lumped uncertainty that captures the deviation from the nominal pre-dicted state at each time instant. When model uncertainty is present, the lumped uncertainty is afunction of the system state and control input, and therefore of the feedback policy to be designed.Unlike [16], which uses a conservative uniform over-approximation of the lumped uncertainty forrobust MPC design, we exactly characterize the dynamics of the lumped uncertainty under a linearstate feedback policy. This is made possible through SLS, which gives a transparent descriptionof the interaction between the lumped uncertainty and the system states and control inputs inclosed-loop. Based on the dynamics of the lumped uncertainty, our proposed robust MPC method,which we call lumped uncertainty SLS MPC, solves a convex inner approximation 1of the robustOCP which jointly optimizes over a linear state feedback policy and the norm upper bounds onsystem state deviation. Our contributions are as follows.

1. We propose a novel robust MPC method, lumped uncertainty SLS MPC, for uncertain LTIsystems with norm-bounded model uncertainty and additive disturbances. At each time step,lumped uncertainty SLS MPC searches over linear state feedback policies for prediction andsolves a convex inner approximation of the robust OCP in the space of system responsesusing SLS. The convex inner approximation is derived based on the dynamics of the lumpeduncertainty and is shown to achieve significant reduction in conservatism compared withexisting SLS-based and tube-based robust MPC methods.

2. We prove the recursive feasibility and input-to-state stability (ISS) of the lumped uncertaintySLS MPC when an adaptive horizon strategy is used.

3. We numerically compare our proposed method with representative baseline robust MPCapproaches such as tube-MPC [5], the SLS-based MPC method in [12], and the lumped-uncertainty method in [16]. Our proposed method is shown to outperform all baseline meth-ods by a significant margin in conservatism as measured by the sets of feasible initial states.In addition, we show that the computational complexity of our method is better than orcomparable to that of existing baselines.

The rest of the paper is organized as follows. The uncertain LTI system and the problemformulation are introduced in Section 2. With the SLS framework introduced in Section 3, wepresent the design of lumped uncertainty SLS MPC in Section 4 followed by the proof of recursivefeasibility (Section 5) and ISS (Section 6). A detailed comparison with baseline robust MPCmethods is given in Section 7 through numerical examples. Section 8 concludes the paper.

Notation Let R≥0 denote the set of non-negative real numbers. For a dynamical system, wedenote the system state at time t by x(t) and the t-step prediction of the state in an MPC loopby xt. For two vectors x and y, x ≤ y denotes element-wise comparison. For a symmetric matrixQ, Q � 0 denotes that Q is positive semidefinite. The notation xi:j is shorthand for the set{xi, xi+1, · · · , xj}. The norm ‖·‖∞→∞ is the `∞ induced norm. S = blkdiag(S1, · · · , SN ) denotesthat S is a block diagonal matrix whose diagonal blocks consist of S1, · · · , SN arranged in the order.

1A convex inner approximation of a robust OCP is a convex OCP that searches over a smaller set of robustcontrollers compared with the original one. It usually trades off conservatism for numerical efficiency.

2

We represent a linear, causal operator R defined over a horizon of T by the block-lower-triangularmatrix

R =

R0,0

R1,1 R1,0

.... . .

. . .

RT,T · · · RT,1 RT,0

(1)

where Ri,j ∈ Rp×q is a matrix of compatible dimension. The set of such matrices is denoted byLT,p×qTV and we will drop the superscript T, p×q when it is clear from the context. Let R(i, :) denotethe i-th block row of R, and R(:, j) denote the j-th block column of R, both indexing from 02.

Recall from [17] that a function γ(·) : R≥0 7→ R≥0 is a class-K function if it is continuous,strictly increasing and γ(0) = 0. The function γ(·) is a class-K∞ function if it is class-K andlimr→∞ γ(r) = ∞. A function β(·, ·) : R≥0 × R≥0 7→ R≥0 is a class-KL function if for each fixedt ≥ 0, β(·, t) is a class-K function, and for each fixed s ≥ 0, β(s, ·) is decreasing and β(s, t)→ 0 ast→∞.

2 Problem Formulation

Consider an uncertain discrete-time LTI system

x(k + 1) = (A+ ∆A)x(k) + (B + ∆B)u(k) + w(k),∀k ≥ 0 (2)

where x(k) ∈ Rnx is the system state, u(k) ∈ Rnu is the control input, and w(k) ∈ Rnx modelsthe additive disturbance at time k. The matrices (A, B) denote the nominal dynamics of thesystem, and ∆A,∆B are the parametric model uncertainty. In this work, we assume ‖∆A‖∞→∞ ≤εA, ‖∆B‖∞→∞ ≤ εB where ‖·‖∞→∞ is the `∞ → `∞ induced norm, and the additive disturbancew(k) satisfies ‖w(k)‖∞ ≤ σw for all k ≥ 0. For simplicity of notation, we denote these uncertaintysets by

PA = {∆A ∈ Rnx×nx |‖∆A‖∞→∞ ≤ εA}, PB = {∆B ∈ Rnx×nu |‖∆B‖∞→∞ ≤ εB},W = {w ∈ Rnx |‖w‖∞ ≤ σw}.

(3)

In robust MPC, a finite time constrained robust OCP is solved repeatedly with the currentstate x(k) as the initial condition, and the first control input in the solution is then applied to thesystem. Let T denote the horizon of the robust OCP in which we use xt, 1 ≤ t ≤ T to denotethe predicted states with the initial condition x0 = x(k). The predicted control inputs ut anddisturbances wt are defined similarly.

Problem 1 (Finite time robust optimal control) At each time step of the robust MPC, solvethe following finite time constrained robust OCP with horizon T :

minπ

JT (x(k), π)

s.t. xt+1 = (A+ ∆A)xt + (B + ∆B)ut + wt

ut = πt(x0:t)

xt ∈ X , ut ∈ U , xT ∈ XT , t = 0, 1, · · · , T − 1

∀∆A ∈ PA,∀∆B ∈ PB,∀wt ∈ W, t = 0, 1, · · · , T − 1

x0 = x(k)

(4)

2In this paper, we refer to a block matrix in a block-lower-triangular matrix R using its superscripts shown inEqn. (1). If we let R(i, j) denote the block matrix in the i-th row and j-th column, then we have R(i, j) = Ri,i−j

with (i, j) indexing from 0.

3

where the search is done over causal linear time-varying (LTV) state feedback control policies π =π0:T−1. The sets X ,U , and XT are polytopic state, input, and terminal constraints, defined as

X = {x ∈ Rn | Fxx ≤ bx}, U = {u ∈ Rm | Fuu ≤ bu},XT = {x ∈ Rn | FTx ≤ bT }.

We assume that the sets X ,U , and XT are compact and contain the origin in their interior. Thedefinition of the uncertainty sets PA,PB, andW are given in (3). The objective function JT (x(k), π)is chosen to be the nominal cost function

JT (x(k), π) =∑T−1

t=0 (x>t Qxt + u>t Rut) + x>TQT xTs.t. xt+1 = Axt + But, ut = πt(x0:t)

x0 = x(k), ∀t = 0, 1, · · · , T − 1

(5)

with x0:T denoting the nominal trajectory. Here Q � 0, R � 0, and QT � 0 are the state, input,and terminal weight matrices, respectively.

In the robust OCP (4), we aim to find an LTV state feedback controller π0:T−1 which guaranteesthe robust satisfaction of all state and input constraints under the norm-bounded model uncertaintyand additive disturbances (3). After solving the robust OCP (4), the first control input u0 = π∗0(x0)is applied to drive the system to the next state x(k + 1), and the robust OCP (4) is then solvedagain with x0 = x(k + 1). We denote the MPC control law by ukMPC(x(k)) = π∗0(x(k)). Next,we aim to characterize the closed-loop behavior with ukMPC(·) in terms of recursive feasibility andclosed-loop stability.

Problem 2 (Closed-loop properties) Let ukMPC(·) denote the robust MPC controller arisingfrom solving the robust OCP (4) at each time step.

1. Show that ukMPC(·) is recursively feasible.

2. Show that ukMPC(·) renders the closed-loop system input-to-state (ISS) stable (the definitionof ISS will be given in Section 6).

In Section 3 and 4, we show how the SLS framework can be used to provide an efficient solutionto Problem 1. Then we approach Problem 2 by introducing an adaptive horizon strategy for ourproposed MPC method. We prove the recursive feasibility of ukMPC in Section 5 and the ISS of theclosed-loop system under ukMPC in Section 6. The relevant notations and assumptions for solvingeach problem are introduced in the corresponding sections.

3 Finite horizon System Level Synthesis

In this section, we introduce relevant concepts from SLS and show how to use finite horizon SLSto transform the robust OCP (4) over linear state feedback controllers into one over closed-loopsystem responses.

We first consider the LTI system

xt+1 = Axt + But + ηt (6)

over the horizon t = 0, 1, · · · , T where η0:T−1 is considered as an additive disturbance. By stackingall the relevant states, control inputs and disturbances as

x =[x>0 x>1 · · · x>T

]>, u =

[u>0 u>1 · · · u>T

]>, η =

[x>0 η>0 · · · η>T−1

]>, (7)

4

the system dynamics over the horizon T can be written as

x = ZAx + ZBu + η (8)

where A = blkdiag(A, · · · , A, 0) ∈ LT,nx×nx

TV , B = blkdiag(B, · · · , B, 0) ∈ LT,nx×nu

TV , and Z is theblock-downshift operator with identity matrices of size nx × nx in the first block sub-diagonal andzeros everywhere else. Note that the initial state x0 is embedded as the first component of thedisturbance process η. Next, we parameterize an LTV state feedback controller by an operatorK ∈ LT,nu×nx

TV with u = Kx. In other words, at time t, we have ut =∑t

i=0Kt,t−ixi. The closed-

loop dynamics of the linear system with the controller u = Kx is now given by

x = Z(A + BK)x + η (9)

from which we can derive the transfer function from η to (x,u) as[xu

]=

[(I − Z(A + BK))−1

K(I − Z(A + BK))−1

]η. (10)

Such maps from η to (x,u) are called system responses. Since Z is a block-downshift operator, thematrix inversion in (10) is well-defined. Additionally, we observe that the system responses are alsoblock-lower-triangular structure, which allows us to let Φx ∈ LT,nx×nx

TV and Φu ∈ LT,nu×nx

TV denotethe system responses from η to x and to u, so that[

xu

]=

[Φx

Φu

]η. (11)

In this work, we are particularly interested in the case when η can be represented as a filteredsignal, i.e., η = Σw with an invertible matrix Σ ∈ LT,nx×nx

TV and w = [x>0 w>0 · · · w>T−1]>. The mapΣ can be interpreted as a causal LTV filter acting on the disturbance signal w. A simple exampleis to choose Σ = blkdiag(I, σ0I, · · · , σT−1I) with σt > 0 and ‖wt‖∞ ≤ 1 for t = 0, · · · , T −1. Then,η = Σw models a disturbance signal with time-varying norm bounds σt. When the filtered signalη = Σw is considered, the closed-loop dynamics under the controller u = Kx is given by

x = Z(A + BK)x + Σw (12)

and the system responses mapping w to (x,u) under the controller u = Kx are denoted as x =Φxw, u = Φuw, respectively. In SLS, the task of controller synthesis is shifted to the designof the closed-loop system responses {Φx, Φu}. This is made possible by the following theoremwhich establishes the relationship between the system responses {Φx, Φu} and the controller Kthat achieves the desired response (11).

Theorem 1 Let Σ ∈ LT,nx×nx

TV be invertible and K ∈ LT,nu×nx

TV be a state feedback controller. Then,for the closed-loop dynamics (12) over the horizon t = 0, · · · , T , we have

1. The affine subspace defined by

[I − ZA −ZB

] [Φx

Φu

]= Σ, Φx ∈ LT,nx×nx

TV , Φu ∈ LT,nu×nx

TV (13)

parameterizes all possible system responses x = Φxw,u = Φuw for system (12).

5

2. For any block-lower-triangular matrices {Φx, Φu} satisfying (13), the controller K = ΦuΦ−1x

achieves the desired response.

Proof 1. For a given controller K, similar to (10), the system responses from w to (x,u) are givenby Φx = (I−Z(A+BK))−1Σ, Φu = K(I−Z(A+BK))−1Σ and satisfy the affine constraint (13).

2. First note that the block diagonal of Φx and Σ are equal according to (13). Then, Σ beinginvertible indicates that Φ−1x exists. For any {Φx, Φu} satisfying (13), we synthesize the statefeedback controller K = ΦuΦ

−1x which satisfies I−ZA−ZBK = ΣΦ−1x by multiplying equation (13)

with Φ−1x from right. Then we have (I − Z(A + BK))−1Σ = ΦxΣ−1Σ = Φx and K(I − Z(A +BK))−1Σ = ΦuΦ

−1x Φx = Φu.

Remark 1 Let Σ = I in Theorem 1. Then[I − ZA −ZB

] [Φx

Φu

]= I, Φx ∈ LT,nx×nx

TV ,Φu ∈ LT,nu×nx

TV (14)

parameterizes all achievable system responses mapping η to (x,u), i.e., x = Φxη,u = Φuη. Thisrecovers the result in [14, Theorem 2.1] which does not consider η as a filtered signal. By pluggingin η = Σw, we have x = Φxη = ΦxΣw = Φxw and u = Φuη = ΦuΣw = Φuw. Therefore, weobtain the mapping between the system responses with respect to η and w as

Φx = ΦxΣ, Φu = ΦuΣ. (15)

In addition, both system responses {Φx,Φu} and {Φx, Φu} are achievable by the same state feedbackcontroller K = ΦuΦ

−1x = (ΦuΣ)(ΦxΣ)−1 = ΦuΦ

−1x .

Theorem 1 states any LTV state feedback controller K can be parameterized by a pair of systemresponses {Φx, Φu} satisfying the affine constraint (13). Therefore, the search for the controllerK can be transformed into the search for the closed-loop system responses {Φx, Φu} with anadditional affine constraint (13). The maps x = Φxw,u = Φuw transparently describe the effectsof the disturbance w on x and u under a state feedback controller; this can be used to translateany constraint on (x,u) into an equivalent one on {Φx, Φu}. See [14] for examples of solvingcontrol problems through SLS. Note that the constraint (13) is jointly affine in (Φx, Φu,Σ). Byrepresenting η = Σw, we can use the filter Σ as a design parameter in addition to {Φx, Φu} inrobust controller synthesis.

In the next section, we take the model uncertainty (∆A,∆B) from (2) into account and proposean SLS-based convex inner approximation of the robust OCP (4) which jointly optimizes over LTVstate feedback controllers and upper bounds on the deviation of the system states from the nominalpredicted trajectories through the parameterization of the system responses {Φx, Φu} and the filterΣ.

4 Robust MPC through convex inner approximation

Inspired by [16], we derive a convex inner approximation of the robust OCP (4) in the space ofsystem responses by describing the uncertain system dynamics as

xt+1 = Axt + But + ∆Axt + ∆But + wt

= Axt + But + ηt,(16)

6

where ηt := ∆Axt + ∆But + wt lumps the effects of the model uncertainty {∆A,∆B} and additivedisturbance wt. The perturbation ηt can be interpreted as the deviation of the system from thepredicted nominal trajectory. We call the signal η0:T−1 the lumped uncertainty to distinguish itfrom the additive disturbance signal w0:T−1. For robust control, we want to bound the lumpeduncertainty η0:T−1 in the presence of the feedback controller to be designed. Since each ηt is afunction of the states x0:t, the disturbances w0:t and the underlying controller K, the bounds on ηtmust be time-varying in order to be tight. In our proposed method, at each time step we solve arobust OCP that jointly optimizes over the state feedback controller K and the time-varying upperbounds on η0:T−1. The derivation of our proposed robust OCP is presented step-by-step below.

4.1 Characterization of the lumped uncertainty

Recall that A = blkdiag(A, · · · , A, 0), B = blkdiag(B, · · · , B, 0). We concatenate the lumpeduncertainties ηt over the horizon T as shown in (7). By Theorem 1, the affine subspace[

I − ZA −ZB] [Φx

Φu

]= I, Φx ∈ LT,nx×nx

TV ,Φu ∈ LT,nu×nx

TV (17)

parameterizes all achievable closed-loop system responses that map the lumped uncertainty η to xand u as shown in Remark 1. By stacking the model uncertainty parameters over the horizon Tas ∆A = blkdiag(∆A, · · · ,∆A, 0) ∈ LT,nx×nx

TV , ∆B = blkdiag(∆B, · · · ,∆B, 0) ∈ LT,nx×nu

TV , we havethat η = Z∆Ax + Z∆uu + w from the definition of lumped uncertainty in (16). Combined withthe achievable system responses {Φx,Φu} given in (17), we describe the dynamics of η as

η = Z[∆A ∆B

] [xu

]+ w = Z

[∆A ∆B

] [Φx

Φu

]η + w. (18)

We can view equations (11) and (18) as a linear dynamical system with ηt as the state, wt as theexogenous input and xt, ut as the system output. Note that Z[∆A ∆B][Φ>x Φ>u ]> is strictly causal,and therefore the dynamics of η in (18) is well-defined. In the next subsection, we propose aninner-approximation of the robust OCP (4) based on the dynamics of the lumped uncertainty η.

Remark 2 From (11) and (18) we obtain the map w 7→ (x,u):[xu

]=

[Φx

Φu

](I − Z

[∆A ∆B

] [Φx

Φu

])−1w. (19)

Eqn. (19) describes the system responses from w to (x,u) under the model uncertainty (∆A,∆B).Such a relationship has been exploited in [12, 13] to solve robust OCPs through SLS. However, thecomplex structure of the system responses in (19) makes it difficult to assess the effects of the systemuncertainty and thus renders the resultant robust MPC method conservative. We provide a detailedcomparison between our method and [12] in Section 7.

4.2 Inner approximation of the robust OCP

In this subsection, we propose a convex inner approximation of the robust OCP (4) by jointlyoptimizing over the closed-loop system responses and the norm bounds on the lumped uncertaintyη. Our proposed method is motivated by the following observations:

1. By over-approximating the lumped uncertainties η0:T−1 by `∞ norm balls, we can easilytighten the polytopic state and control input constraints of the robust OCP (4) throughHolder’s inequality as in the additive disturbance case [3, 13,16].

7

2. To reduce the conservatism of the `∞ ball over-approximation, it is desirable to upper bound‖ηt‖∞ for each time step t separately based on the interdependence between the lumpeduncertainty η and (x,u).

Next, we first over-approximate the lumped uncertainty η using `∞ norm balls, and then derivenorm upper bounds of these `∞ balls by exploiting the dynamics of η. Finally, we obtain a convexinner approximation of the robust OCP (4) by tightening the state and input constraints based onthe `∞ ball over-approximation of η.

4.2.1 Over-approximating lumped uncertainty through `∞ balls

We first assume each lumped uncertainty ηt can be over-approximated by an `∞ norm ball withradius σt, i.e., ‖ηt‖∞ ≤ σt for 0 ≤ t ≤ T − 1. Equivalently, we can represent ηt as ηt = σtwt with‖wt‖∞ ≤ 1. Let w = [x>0 w>0 · · · w>T−1]> and Σ = blkdiag(I, σ0I, · · · , σT−1I). Then the lumpeduncertainty η can be considered as the filtered signal η = Σw. By Theorem 1, we have

[I − ZA −ZB

] [Φx

Φu

]= Σ, Φx ∈ LT,nx×nx

TV , Φu ∈ LT,nu×nx

TV (20)

parameterizes all achievable closed-loop system responses {Φx, Φu} that map w to (x,u), i.e.,x = Φxw,u = Φuw. As a result, the dynamics (18) of the lumped uncertainty η can be equivalentlywritten as

η = Z[∆A ∆B

] [Φx

Φu

]w + w. (21)

Compared with (18), the representation (21) reveals the dependence of the lumped uncertaintyη on the normalized disturbance signal w where the scale of each ηt is absorbed in {Φx, Φu} byconstraint (20) and the construction of Σ. Since constraint (20) is affine in {Φx, Φu,Σ}, we canjointly optimize over the closed-loop system responses {Φx, Φu} and the norm bounds {σt}T−1t=0

encoded in Σ while maintaining the convexity of the overall formulation.

4.2.2 Upper-bounding the magnitude of lumped uncertainty

Recall that we let {σ0, σ1, · · · , σT−1} be a set of upper bounds on the `∞ norm of the lumpeduncertainties ηt, i.e., ‖ηt‖∞ ≤ σt for t = 0, · · · , T − 1. From Eqn. (21), we have η0 = ∆AΦ0,0

x x0 +∆BΦ0,0

u x0 + w0, and for 1 ≤ t ≤ T − 1,

ηt = ∆A(Φt,tx x0 +

t∑i=1

Φt,t−ix wi−1) + ∆B(Φt,t

u x0 +t∑i=1

Φt,t−iwi−1) + wt. (22)

By the triangle inequality and submultiplicativity of ‖·‖∞, we can upper bound ‖ηt‖∞ by

‖ηt‖∞ ≤ ‖∆A‖∞→∞(‖Φt,tx x0‖∞ +

t∑i=1

‖Φt,t−ix ‖∞→∞‖wi−1‖∞)+

‖∆B‖∞→∞(‖Φt,tu x0‖∞ +

t∑i=1

‖Φt,t−iu ‖∞→∞‖wi−1‖∞) + ‖wt‖∞

≤ εA(‖Φt,tx x0‖∞ +

t∑i=1

‖Φt,t−ix ‖∞→∞) + εB(‖Φt,t

u x0‖∞ +t∑i=1

‖Φt,t−iu ‖∞→∞) + σw

(23)

8

using ‖wt‖∞ ≤ 1 for 0 ≤ t ≤ T − 1. The upper bound (23) on the norm of the lumped uncertaintyat time t depends on the system uncertainty parameters εA, εB, σw, and the closed-loop systemresponses up to time t− 1. For t = 0, the upper bound on ‖η0‖∞ can be derived similarly.

Lemma 1 For the closed-loop system responses {Φx, Φu} satisfying constraint (20) with a setof positive real numbers {σt}T−1t=0 , and the uncertainty parameters described in (3), a sufficientcondition for {σt}T−1t=0 to be upper bounds on the norm of the lumped uncertainty ηt is given by

εA‖Φ0,0x x0‖∞ + εB‖Φ0,0

u x0‖∞ + σw ≤ σ0

εA(‖Φt,tx x0‖∞ +

t∑i=1

‖Φt,t−ix ‖∞→∞) + εB(‖Φt,t

u x0‖∞ +t∑i=1

‖Φt,t−iu ‖∞→∞) + σw ≤ σt

(24)

for t = 1, · · · , T − 1. In other words, constraints (24) guarantee that ‖ηt‖∞ ≤ σt for 0 ≤ t ≤ T − 1.

Proof Since η0 = ∆AΦ0,0x x0 +∆BΦ0,0

u x0 +w0, by the triangle inequality and the submultiplicativityof the `∞ norm, the first inequality in (24) guarantees ‖η0‖∞ ≤ σ0. Then, by induction and theupper bounds derived in (23), ‖ηt‖∞ ≤ σt holds for t = 1, · · · , T − 1.

4.2.3 Constraint tightening of the robust OCP

With the system responses {Φx, Φu} characterized by constraint (20), the map from the normalizeddisturbances w to (x,u) is given by x = Φxw and u = Φuw, which allows us to tighten thestate and control input constraints of the robust OCP (4). We illustrate the constraint tighteningprocedure using the state constraint xt ∈ X – an analogous argument can be applied to thecontrol and terminal constraint sets. Recall that X = {x|Fxx ≤ bx} and Fx ∈ RnX×nx , i.e., thereare nX linear constraints in defining the set X . We use f>x ≤ b to denote an arbitrary linearconstraint in the definition of X and introduce the notation (f, b) ∈ facet(X ) with facet(X ) ={(Fx(i, :), b(i))|i = 1, · · · , nX } being the set of all linear constraint parameters of X . Since xt =Φt,tx x0 +

∑ti=1 Φt,t−i

x wi−1, the tightening of the constraint xt ∈ X is given by

f>Φt,tx x0 +

t∑i=1

‖f>Φt,t−ix ‖1 ≤ b, ∀(f, b) ∈ facet(X ), t = 0, · · · , T − 1. (25)

This is due to f>Φt,t−ix wi−1 ≤ ‖f>Φt,t−i

x ‖1‖wi−1‖∞ ≤ ‖f>Φt,t−ix ‖1 where the first inequality follows

from Holder’s inequality and the second is from the fact ‖wt‖∞ ≤ 1 for 0 ≤ t ≤ T − 1. Similarly,the tightened constraints for the terminal state and control inputs are given by

f>ΦT,Tx x0 +

T∑i=1

‖f>ΦT,T−iu ‖1 ≤ b, ∀(f, b) ∈ facet(XT ), (26)

f>Φt,tu x0 +

t∑i=1

‖f>Φt,t−iu ‖1 ≤ b, ∀(f, b) ∈ facet(U), t = 0, · · · , T − 1. (27)

9

Theorem 2 Consider the following convex quadratic program

minimizeΦx,Φu,{σt}T−1

t=0

∥∥∥ [Q1/2

R1/2

][Φx(:, 0)

Φu(:, 0)

]x0

∥∥∥22

subject to[I − ZA −ZB

] [Φx

Φu

]= Σ, Φx ∈ LT,nx×nx

TV , Φu ∈ LT,nu×nx

TV ,

Σ = blkdiag(I, σ0I, · · · , σT−1I),

disturbance bound constraints (24),

tightened constraints (25), (26), and (27),

x0 = x(k),

(28)

where Q = blkdiag(Q, · · · , Q,QT ), and R = blkdiag(R, · · · , R, 0) encode the weights of the stage andterminal costs in Eqn. (5), respectively. For any feasible solution {Φx, Φu, {σt}T−1t=0 } of problem (28),

the synthesized controller K = ΦuΦ−1x guarantees the robust satisfaction of all state and control

input constraints of the robust OCP (4).

Proof The proof follows from Theorem 1 and the derivation of the constraints (24) to (27) pre-sented in this subsection. First, from constraint (24), we have {σt}T−1t=0 are all lower-bounded by thenorm of the additive disturbance σw > 0. Therefore, any solution Σ of problem (28) is invertibleand Theorem 1 can be applied. By Theorem 1 and Lemma 1, the affine constraint (20) and the up-per bound constraint (24) guarantee that ‖ηt‖∞ ≤ σt under the synthesized controller K = ΦuΦ

−1x .

Then, constraints (25) to (27) guarantee robust constraint satisfaction of the robust OCP (4) bytightening the state and control input constraints based on the `∞ norm ball over-approximation ofthe lumped uncertainty η. By letting ηt = 0 for 0 ≤ t ≤ T − 1, we have Σ = blkdiag(I, 0, · · · , 0)and Φx(:, 0) = Φx(:, 0), Φu(:, 0) = Φu(:, 0) by Remark 1. Therefore, Φx(:, 0)x0 = Φx(:, 0)x0 andΦu(:, 0)x0 = Φu(:, 0)x0 denote the nominal states and control inputs, respectively, and the objectivefunction in (28) coincides with the nominal cost function given in (5).

Problem (28) is jointly convex over the closed-loop system responses {Φx, Φu} and the normupper bounds on the lumped uncertainty {σt}T−1t=0 , allowing them to be simultaneously optimized.We call problem (28) a convex inner approximation of the robust OCP (4) since the constraintsin (28) define an inner approximation to the feasible set of the robust OCP (4). Therefore, prob-lem (28) is in general more conservative than the robust OCP (4). However, we note that in thespecial case of horizon T = 1, problem (28) is non-conservative in that problem (28) solves therobust OCP (4) exactly.

Lemma 2 For horizon T = 1, the optimization problem (28) is a non-conservative inner approxi-mation of the robust OCP (4) over LTV state feedback controllers.

Proof The proof relies on showing that any feasible solution of the robust OCP (4) over statefeedback controllers constructs a feasible solution of the inner approximation (28) when T = 1.This is because when T = 1, the upper bound σ0 on ‖η0‖∞ given by (23) is tight. For details of theproof, see Appendix A.1.

For horizons T > 1, the convex inner approximation (28) is conservative because in general thereare not enough degrees of freedom to choose ∆A,∆B and w0:t−1 adversarially to achieve the upperbounds on ‖wt‖∞ in (23). We note that for a feasible solution of problem (28), under the synthesized

10

robust controller K = ΦuΦ−1x , all possible trajectories of the uncertain system are contained in a

tube consisting of `∞ balls centered around the nominal trajectory with varying radii {σt}T−1t=0 . The`∞ ball is a natural choice in bounding the lumped uncertainty under the uncertainty parameterassumption (3) and it allows efficient constraint tightening in problem (28). In comparison, thetube-MPC method [5] does not consider LTV state feedback controllers; instead, it parameterizesfeedback controllers implicitly by a tube with fixed shape, and uses vertex enumeration over boththe uncertainty parameter and the tube to tighten the constraints.

Finally, our proposed MPC controller is given by

ukMPC(x(k)) = Φ0,0∗u x(k) (29)

with Φ0,0∗u being the Φ0,0

u -component of the solution of (28). Since the robust optimal controlproblem (28) is obtained by tightening the constraints using the lumped uncertainty, we call ourproposed method lumped uncertainty SLS MPC.

Remark 3 When εA = εB = 0, the formulation (28) recovers the SLS-based robust MPC methodshown in [8,13], as the upper bound constraint (24) is trivially satisfied. In this special case whereonly additive disturbances are present, the lumped uncertainty SLS MPC is equivalent to the affinestate feedback3 or disturbance feedback MPC approaches [3], as shown in [8].

5 Recursive feasibility

Our robust MPC policy (29) is obtained by solving the convex inner approximation (28) of therobust OCP (4) at each time k. Although our goal is to find a robust state feedback controllerK through (28), the optimization is over the system responses with tightened constraints. As aresult, even if a controller K is a feasible solution to the robust OCP (4), there is no guarantee thatthe system responses {Φx, Φu} it generates is feasible for the convex inner approximation (28).This prevents us from applying standard controller shifting argument [1, Chapter 12] to prove therecursive feasibility of our proposed robust MPC method when the horizon T is constant.

To overcome this difficulty, we allow the horizon of the robust MPC problem (4) to vary withtime. We denote Tk as the MPC horizon at time k. To guarantee the recursive feasibility of lumpeduncertainty SLS MPC, we use an adaptive horizon strategy described as

Tk = arg minT∈{1,··· ,Tmax}

J∗T (x(k)) (30)

where Tmax is a fixed horizon upper bound and J∗T (x(k)) is the optimal objective of (28) with horizonT . We let J∗T (x(k)) = ∞ if the convex tightening (28) is infeasible. The following assumptionsabout the terminal set are made to ensure recursive feasibility.

Assumption 1 A linear state feedback controller κ(x) = Kx is known such that the A + ∆A +(B + ∆B)K is stable for all ∆A ∈ PA,∆B ∈ PB.

Assumption 2 The terminal set XT is a robust forward invariant set under the local controllerκ(x) = Kx, i.e., for all x ∈ XT , we have x ∈ X ,Kx ∈ U , and ((A+ ∆A) + (B+ ∆B)K)x+w ∈ XTfor all ∆A ∈ PA,∆B ∈ PB, w ∈ W.

3An affine state-feedback controller is used in [3] while SLS only parameterizes a linear state feedback controller.However, we can always augment system (2) with state x = [x; 1] to make them equivalent.

11

These two assumptions are standard in the analysis of MPC and computational tools [1, Chapter10] are available to compute the local feedback controller κ(x) = Kx and find the required terminalset XT . We first show that, when lumped uncertainty SLS MPC is feasible at x(0) with horizonT0 ≥ 2, solving the convex inner approximation (28) in a decreasing horizon manner drives thesystem into the terminal set in T0 steps.

Theorem 3 Let the robust OCP inner approximation (28) be feasible at time k with horizon Tk ∈{2, 3, · · · , Tmax} and x0 = x(k). Let x(k+1) be the state at time k+1 with the MPC controller (29).Then the inner approximation (28) is feasible at time k + 1 with horizon Tk+1 = Tk − 1 andx0 = x(k + 1).

Proof The proof relies on an explicit construction of a feasible solution Φx, Φu, {σt}Tk+1−1t=0 for

problem (28) with the initial condition x(k + 1) and horizon Tk+1. See Appendix A.2 for details.

By Theorem 3, problem (28) is always feasible with Tk = T0 − k ≥ 0 at x(k). As a result, attime k = T0 − 1, the lumped uncertainty SLS MPC with horizon Tk = 1 is feasible and guaranteesthe next state x(T0) will lie in the terminal set XT . Now with the construction of the terminal set,we can guarantee that ukMPC(·) is recursively feasible inside the terminal set with horizon Tk = 1and renders the terminal set robustly forward invariant.

Lemma 3 Under Assumption 1 and 2, for any state x(k) ∈ XT , the convex inner approxima-tion (28) is feasible with horizon Tk = 1 and x(k) as the initial condition. In addition, the terminalset XT is a robust forward invariant set under the lumped uncertainty SLS MPC controller ukMPC(·)with horizon Tk = 1.

Proof The recursive feasibility of ukMPC(·) inside the terminal set holds because the inner approx-imation (28) with horizon one is not conservative (see Lemma 2) and a terminal set XT and localcontroller κ(x) exist which satisfy Assumptions 1 and 2. With horizon Tk = 1, lumped uncertaintySLS MPC guarantees the next state x(k+1) is always in XT , proving the robust forward invarianceof the terminal set XT .

Theorem 4 (Recursive feasibility) A lumped uncertainty SLS MPC policy which solves the ro-bust OCP inner approximation (28) and adopts the adaptive horizon strategy (30) is recursivelyfeasible. In other words, if problem (28) is feasible at time k = 0 with horizon T0, it is feasible alltime steps k ≥ 1 with horizon Tk selected by (30).

Proof We divide the discussion into two cases. At time k, if the inner approximation (28) isfeasible with horizon Tk ≥ 2, then by Theorem 3 it is feasible at time k + 1 with horizon Tk+1 =Tk − 1. We refer the reader to the proof of Theorem 3 for the construction of a feasible solutionof problem (28) at time k + 1. If at time k, problem (28) is feasible with horizon Tk = 1, thenby the proof of Lemma 3 we know that the local controller κ(x) = Kx from Assumption 2 yields afeasible solution to (28) at time k + 1 with horizon Tk+1 = 1. Summarizing the above two cases,we conclude that lumped uncertainty SLS MPC is recursively feasible.

5.1 Horizon design choices

Adaptive horizon strategies have been applied to guarantee recursive feasibility in various MPCdesign methods such as tube MPC [5], learning MPC [18], and tightening-based MPC [15]. Inaddition to the strategy (30) which chooses the horizon that minimizes the cost function alongthe prediction horizon, other horizon selection procedures are available, such as the decreasing

12

horizon [5], minimum time [3, 19], and single policy [3, 5] strategies. They induce different compu-tational complexities and performances for the synthesized MPC controller, and a comprehensivecomparison between these adaptive horizon strategies is beyond the scope of this paper. No matterwhich of the aforementioned adaptive horizon strategies is used, the proposed lumped uncertaintySLS MPC is guaranteed to be recursively feasible thanks to Theorem 3 and Lemma 3.

In this work, we apply the adaptive horizon strategy described in (30) to guarantee the ISSof the closed-loop system shown in the next section. In practice, we can keep running lumpeduncertainty SLS MPC with a fixed horizon T until infeasibility of problem (28) occurs when wedecrease the horizon to T − 1 and repeat the process. This will make sure that at each time stepproblem (28) looks ahead long enough for planning. In the next section, we prove the ISS of lumpeduncertainty SLS MPC with horizons selected by the strategy (30).

6 Input-to-state stability

In this section, we prove the input-to-state stability of the closed-loop system under the proposedMPC controller (29) which adopts the adaptive horizon strategy (30). Recall that the closed-loopdynamics can be written as

x(k + 1) = Ax(k) + BukMPC(x(k)) + η(k) with η(k) = ∆Ax(k) + ∆BukMPC(x(k)) + w(k). (31)

We first introduce the following definition of ISS.

Definition 1 (Input-to-state stability, adapted from [20]) Given a compact set R ⊆ Rnx

including the origin in its interior, the system (31) is said to be input-to-state stable in R if R is arobust forward invariant set for system (31) and there exists a class-KL function β and a class-Kfunction γ such that for any initial state x(0) ∈ R and bounded lumped uncertainties {η(k)}∞k=0

with ‖η(k)‖∞ ≤ C <∞, k ≥ 0, the system state x(k) exists for all k ≥ 0 and satisfies

‖x(k)‖∞ ≤ β(‖x(0)‖∞, k) + γ(

sup0≤t≤k

‖η(t)‖∞). (32)

The set R is called the region of attraction (ROA) of system (31).

Unlike the standard notion of ISS [21,22], the ISS given in the above definition is with respectto the lumped uncertainties {η(k)}∞k=0 where η(k) includes both the exogenous disturbance inputw(k) and the uncertainty terms involving states x(k). However, the implication of ISS in Defini-tion 1 remains the same: (i) the origin is asymptotically stable for the closed-loop system when nouncertainty is present, i.e., when ∆A = 0,∆B = 0, σw = 0; (ii) all the closed-loop trajectories arebounded when the lumped uncertainties are bounded; (iii) the closed-loop trajectory approachesthe origin if η(k)→ 0 as k →∞. Next, we define the region of attraction (ROA) of the closed-loopsystem (31) using the method proposed in [16].

Definition 2 (N-step robust controllable set) For the MPC controller and the closed-loop dy-namics (31), we define the N -step robust controllable set CN to a given set S recursively as

C0 = S, Ck+1 = Pre(Ck) ∩ X , k = 1, · · · , N − 1

where Pre(S) is the preset of S defined as the set of states from which system (31) evolve into Sin one step for all possible realizations of model uncertainty and additive disturbances.

13

Definition 3 (ROA of the adaptive horizon robust MPC) The region of attraction R forthe proposed robust MPC method is defined as the union of the N -step robust controllable set to theterminal set XT for N ∈ {0, · · · , Tmax}

By the recursive feasibility of the proposed robust MPC method with the adaptive horizonstrategy (30), we know that the ROA R ⊆ X is robust forward invariant for the uncertain closed-loop dynamics (31). Therefore, the state of the closed-loop system (31) is always bounded, and ISSadditionally provides a qualitative description of the asymptotic behavior of the closed-loop systemwhen the uncertainty parameters approach zero.

In standard MPC with a quadratic cost function and a fixed horizon, the value function ofthe finite time optimal control problem is continuous since it can be formulated as a parametricquadratic program [23]. Then, the continuity of the value function can be used to prove the ISS ofthe closed-loop system, as shown in [3]. However, in our proposed robust MPC method, the valuefunction, which we denote as J∗MPC(x(k)), is not given by the solution to a parametric quadraticprogram; instead, we obtain it from

J∗MPC(x(k)) = minT∈{1,··· ,Tmax}

J∗T (x(k)), (33)

and therefore cannot guarantee the continuity of J∗MPC(x(k)). We can overcome this challenge usingthe results from [24], which states that the existence of a dissipative-form ISS-Lyapunov functionverifies ISS of a discrete-time system under no regularity assumptions on the system dynamics orthe ISS-Lyapunov function.

Definition 4 (ISS-Lyapunov function [24]) A function V (x) : Rnx 7→ R≥0 is called an ISS-Lyapunov function for system (31) with a ROA R if there exist class-K∞ functions α1(·), α2(·), α3(·),a class-K function σ(·) such that

α1(‖x‖∞) ≤ V (x) ≤ α2(‖x‖∞), ∀x ∈ R (34a)

V (x(k + 1))− V (x(k)) ≤ −α3(x(k)) + σ(‖η(k)‖∞), ∀x(k) ∈ R. (34b)

As proved in [24, Theorem 2.3], the closed-loop system (31) is ISS if and only if there exists anISS-Lyapunov function. We prove the ISS of system (31) in Theorem 5 by showing that the valuefunction J∗MPC(x(k)) is an ISS-Lyapunov function.

Assumption 3 Let `(x, u) = x>Qx+u>Ru denote the stage cost in the nominal cost function (5).The cost weights Q and R satisfy Q � 0, R � 0.

Assumption 4 The terminal cost weight QT in (5) satisfies QT � 0 and

x>(−QT + (Q+K>RK) + (A+ BK)>QT (A+ BK))x ≤ 0 (35)

for all x ∈ XT where K is the local stabilizing controller defined in Assumption 2.

With the given locally stabilizing controller κ(x) = Kx satisfying Assumption 1, the terminalcost weight QT satisfying Assumption 4 can be obtained by solving a discrete-time Lyapunovequation [1, Chapter 7] for the linear system x(k + 1) = (A+ BK)x(k).

Theorem 5 Let Assumptions 1, 2, 3, 4 hold and x(0) ∈ R where R denotes the ROA fromDefinition 3. The value function J∗MPC(x(k)) given in (33) is an ISS-Lyapunov function for theclosed-loop system (31) and therefore proves the ISS of system (31).

Proof See Appendix A.3.

14

7 Numerical examples

7.1 Comparison to existing methods

In this section, we compare lumped uncertainty SLS MPC with other robust MPC methods inthe literature that can handle both model uncertainty and additive disturbances. Specifically, thefollowing methods are considered for comparison:

1. lumped-SLS-MPC : Our proposed method which describes the effects of uncertainty throughlumped uncertainties and synthesizes robust feedback controllers using SLS.

2. unif-df-MPC : the method from [16] which uses a disturbance feedback approach based on auniform over-approximation of the lumped uncertainty.

3. grid-SLS-MPC : the SLS-based robust MPC method from [12] which bounds the effects ofuncertainty by grid-searching over a set of hyperparameters.

4. tube-MPC : the method proposed in [5] which bounds the trajectories of system under uncer-tainty through a tube.

lumped-SLS-MPC and unif-df-MPC Similar to lumped-SLS-MPC, unif-df-MPC also lumpsmodel uncertainty and additive disturbances into a net-additive disturbance which we denote asthe lumped uncertainty in this paper. However, unif-df-MPC ignores the dependence of the lumpeduncertainties on the system states and control inputs; instead, it over-approximates all the lumpeduncertainties by a single norm ball, i.e., by choosing a large-enough upper bound σ such that‖ηt‖∞ ≤ σ holds for all xt ∈ X , ut ∈ U . With this over-approximation, the disturbance feedbackapproach [3] is applied in [16] to synthesize a robust controller.

The method unif-df-MPC achieves good balance between simplicity and performance: indeed,its performance is comparable to tube-MPC in many examples shown in Section 7.2. The drawbackis that the uniform norm upper bound σ may be conservative. For example, when x0 = 0, the actuallumped uncertainty ηt are expected to be small since the system trajectory is near the origin. Butwith a large state constraint, e.g., X = {x|‖x‖∞ ≤ 10}, the uniform norm upper bound σ can beorders of magnitude larger than the true values of ‖ηt‖∞ since all states in X are considered to derivethe upper bound. In contrast, lumped-SLS-MPC does not suffer from this over-conservatism as itconsiders the dependence of ηt on (xt, ut, wt) along the prediction horizon. Further, unif-df-MPC isa special case of lumped-SLS-MPC that sets σt = σ, 0 ≤ t ≤ T − 1. Therefore, lumped-SLS-MPCis guaranteed to be less conservative than unif-df-MPC.

lumped-SLS-MPC and grid-SLS-MPC Both methods use SLS to derive a convex inner ap-proximation of the robust OCP but with different formulations. lumped-SLS-MPC relies on thedynamics of the lumped uncertainty (18) to derive an inner approximation of the robust OCP (4)while grid-SLS-MPC directly exploits the mapping (19). In grid-SLS-MPC, a tuple of three hyper-parameters have to be grid-searched in order to make the robust OCP inner approximation convex.These hyperparameters can be interpreted as uniform upper bounds on the effects of the uncer-tainty along the prediction horizon. Although not directly comparable, lumped-SLS-MPC does notuse uniform bounds for constraint tightening and the upper bounds σt are treated as optimizationvariables instead of hyperparameters in the proposed inner approximation (28). These featuresare desirable to derive more computationally efficient and less conservative robust optimal controlformulations: this is confirmed by the numerical examples in Section 7.2. In [13], an SLS-based

15

inner approximation similar to grid-SLS-MPC is proposed but is more conservative as shown in [12].Therefore, we do not compare our method with [13].

lumped-SLS-MPC and tube-MPC The tube-MPC method [5] for handling both model un-certainty and additive disturbances requires first fixing the shape of a polytopic set, and thenjointly synthesizing a tube and a feedback controller that constrains the system trajectories insidethe tube by solving a convex program. The design of lumped-SLS-MPC can be interpreted in asimilar fashion: we synthesize an LTV state feedback controller such that all trajectories are con-strained in a tube characterized by a sequence of `∞ balls with varying sizes centered at the nominalpredicted trajectory. However, the underlying feedback controllers are parameterized differently intube-MPC and lumped-SLS-MPC which lead to different robust OCP formulations. We comparethe conservatism and computational complexities of these two methods through numerical exam-ples shown in the next subsection. Note that tube-MPC can handle polytopic model uncertaintywhich we leave for future research.

7.2 Experiments

We now compare the conservatism and computational complexity of the four robust MPC methodsdescribed above numerically. Our codes are publicly available at https://github.com/ShaoruChen/Lumped-Uncertainty-SLS-MPC.

Implementation details For grid-SLS-MPC, we follow the implementation details describedin [12]. There are three hyperparameters in grid-SLS-MPC. In all the experiments, we use bisec-tion [12, Algorithm 1] in the interval [0.01 100] to determine the lower and upper bounds on thesehyperparameters and then apply a 5× 5× 5 grid search between these bounds to look for a feasibletuple of hyperparameters. For tube-MPC, we choose the tube as the disturbance invariant set withan LQR controller. See [12] for more details. All the experiments were implemented in MATLABR2019b with YALMIP [25] and MOSEK [26] on an Intel i7-6700K CPU.

7.2.1 Conservatism evaluation

We use the two dimensional example from [15] to evaluate the performances of each robust MPCmethod. The nominal dynamics and the state and control input constraints are given as

A =

[1 0.15

0.1 1

], B =

[0.11.1

], X =

{x ∈ R2|

[−8−8

]≤ x ≤

[88

]}, U = {u ∈ R| − 4 ≤ u ≤ 4}.

(36)We note that A is unstable. The model uncertainties {∆A,∆B} are bounded by ‖∆A‖∞ ≤ εA = 0.1,‖∆B‖∞ ≤ εB = 0.1. The additive disturbances satisfy ‖w(k)‖∞ ≤ 0.1, i.e., σw = 0.1. We solve therobust OCP (4) with the terminal constraint XT chosen as the maximal robust control invariantset (the shaded polytope in Figure 1) found by the iterative algorithm [27, Algorithm 2]. The MPChorizon is set to T = 5 and the cost weights are chosen as Q = 10I,R = 1, QT = 10I.

Comparison with XT Since the terminal set XT is chosen as the maximal robust control invariantset, the robust OCP (4) is infeasible for any initial state x0 /∈ XT and is feasible for all x0 ∈ XT witha robust piecewise affine controller as the solution. However, to find this robust piecewise affinecontroller we need to solve dynamic programming and multi-parametric quadratic programmingproblems with robustified constraints, which has an exponential complexity in the worst case [28]

16

Figure 1: Conservatism evaluation of robust MPC methods. The shaded blue region denotes themaximum robust invariant set XT . The convex hull of the feasible initial conditions for each methodis plotted.

and is considered intractable in practice. We can evaluate the conservatism of each robust MPCmethod by estimating their feasible regions (the set of x0 for which the robust MPC method isfeasible) and comparing them with the maximum robust control invariant set XT .

To do so, we first sample initial conditions from a uniform grid of 288 states within XT . Thenwe take the convex hull of all the feasible initial states of each method to estimate their feasibleregions as shown in Figure 1. We observe the following: (i) the feasible region of lumped-SLS-MPC almost matches the maximal robust control invariant set. In particular, lumped-SLS-MPC isfeasible for all sampled initial states. This indicates that lumped-SLS-MPC is not conservative forthe considered example, and furthermore only requires solving a computationally efficient convexquadratic program, in contrast to the dynamic programming approach; (ii) the feasible region ofunif-df-MPC is a subset of that of lumped-SLS-MPC: as described in Section 7.1, this is expected;(iii) grid-SLS-MPC gives the most conservative result in this example; (iv) the feasible regions ofunif-df-MPC and tube-MPC are similar in area but cover different states.

Comparison using varying uncertainty parameters We now stress test all four robust MPCmethods for increasing uncertainty sizes. In the first test, we fix εB = 0.1, σw = 0.1 and vary εAfrom 0.05 to 0.25 with step size 0.01. In the second test, we fix εA = εB = 0.1 and vary σw from0.05 to 0.8 with step size 0.05. For each of these tests, we sample 225 uniformly-spaced states fromthe state constraint X and evaluate the feasibility of the robust MPC methods on these states.Since the maximal robust control invariant set XT found in the previous subsection is no longervalid for the changed uncertainty parameters, we do not impose terminal constraints in these tests.

We evaluate the conservatism of each robust MPC method by its approximate feasible regioncoverage, which we define as the ratio of the feasible states to the sampled 225 states. The resultsare shown in Fig. 2, and we observe that lumped-SLS-MPC outperforms all the other methods bya significant margin. Note that as the sampled states are from the state constraint X instead of themaximal robust invariant set for each tuple of uncertainty parameters (εA, εB, σw), it is expectedthat the coverage of each robust MPC method decreases as the uncertainty parameter increases.

17

0.05 0.1 0.15 0.2 0.25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) Coverage of feasible regions for varying εA withεB = 0.1, σw = 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(b) Coverage of feasible regions for varying σw withεA = 0.1, εB = 0.1.

Figure 2: The coverage of the feasible regions (ratio of feasible sampled initial states) of the robustMPC methods with varying εA (Left) or σw (Right). Our proposed method lumped-SLS-MPCconsistently outperforms other candidate methods by a non-trivial margin.

In Fig. 2a, lumped-SLS-MPC is shown to be the most resistant to the increasing level of modeluncertainty since the slope of its coverage curve is the flattest. unif-df-MPC and tube-MPC achievessimilar coverages with small εA but both of their coverages drop quickly as εA increases. grid-SLS-MPC is still the most conservative method for all cases. At εA = 0.1, we recover the setup in theprevious experiment, and confirm the observations (iii) and (iv) therein. In Fig. 2b, lumped-SLS-MPC achieves the largest coverage for all values of σw tested. Compared with Fig. 2a, the gapbetween unif-df-MPC and tube-MPC in coverage becomes even larger when we increase the normof the additive disturbances.

Randomly generated systems We randomly generate 50 two dimensional systems with nom-inal dynamics A ∈ R2×2, B ∈ R2×1. Each entry of A and B is uniformly sampled from the interval[−2, 2] and [−1, 1], respectively. The state and control input constraints are the same as in (36)with no terminal constraint imposed. The MPC horizon is set as T = 5 and the uncertainty pa-rameters are chosen as εA = εB = 0.1, σw = 0.1. For each of the randomly generated system, wedo a 10× 10 uniform grid search over X and run the robust MPC methods on each sampled state.The coverage of each robust MPC method is computed as described above and is plotted in Fig. 3for each randomly generated system. We arrange the results in ascending order according to thecoverage of lumped-SLS-MPC. Among the 50 randomly generated systems, 25 of them are open-loop unstable while the rest are open-loop stable. From Fig. 3, we observe that lumped-SLS-MPChas the largest coverage except at one example (No. 38) where the coverage of tube-MPC exceedsthat of lumped-SLS-MPC by 2%.

7.2.2 Computational complexity

All four robust MPC methods solve a convex quadratic program at each MPC iteration, but theyhave different computational complexities. The number of variables in lumped-SLS-MPC, unif-df-MPC and grid-SLS-MPC are quadratic in the system dimension nx and the MPC horizon T whilebeing linear in nx and T for tube-MPC. However, when considering `∞ → `∞ norm-bounded model

18

0 5 10 15 20 25 30 35 40 45 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3: Coverage of each robust MPC method for 50 randomly generated systems. Largercoverage indicates less conservatism. lumped-SLS-MPC outperforms all the other methods exceptat example No. 38 where tube-MPC exceeds lumped-SLS-MPC by 2% of coverage.

uncertainty, the number of constraints in tube-MPC is linear in the horizon and exponential in thestate and input dimensions since vertex enumeration is applied in tube-MPC to constrain the systemtrajectories inside the tubes. In contrast, the other three methods only tighten the constraintsthrough norm-based inequalities: in these methods the number of constraints grows quadraticallywith the system dimension and the horizon. A detailed comparison is shown in [12, Table 1].

In Fig. 4a, we plot the solver time of each robust MPC method for the problem setup con-sidered above with varying horizons. The initial condition is chosen as x0 = [1 0]>. The solvertime of tube-MPC is significantly larger than that of lumped-SLS-MPC or unif-df-MPC mainlydue to its large number of constraints in the quadratic programming formulation. grid-SLS-MPCbecomes infeasible for horizon larger than 5 and its solver time includes the bisection and gridsearch steps [12]. Thus we see lumped-SLS-MPC and unif-df-MPC enjoy similar computationalcomplexity.

Next, we randomly generate a hundred 2-dimensional systems to compare the computationaltime. We generate (A, B) randomly as shown in Section 7.2.1. The robust OCP constraints arefrom (36) with no terminal constraints used, and the cost function is given by Q = I,R = 1, QT = I.For each sampled system (A, B), we fix the initial condition as x0 = [2 − 1]> and the MPChorizon as T = 10. The uncertainty parameters are chosen as εA = εB = σw = 0.05. Then weapply all four robust MPC methods on the 100 randomly generated system (A, B) (82 of themare open-loop unstable) and compare their solver time in Fig. 4b. We mention that among the100 randomly generated systems, lumped-SLS-MPC is feasible for 55 examples while unif-df-MPCtube-MPC, grid-SLS-MPC are feasible on 48, 51, 37 examples, respectively. In Fig. 4b, the solvertime is reported only for feasible solutions of each robust MPC methods. We observe that lumped-SLS-MPC and unif-df-MPC are comparable, and both of them have much lower computationalcomplexity than tube-MPC and grid-SLS-MPC.

8 Conclusion

In this paper, we propose lumped uncertainty SLS MPC, a novel SLS-based robust MPC methodfor uncertain LTI systems subject to norm-bounded model uncertainty and additive disturbance.

19

0 5 10 15

10-1

100

101

(a) Solver time of each robust MPC method for thenumerical example considered in (36) with varyinghorizons.

lumped-SLS unif-df tube grid-SLS

10-1

100

101

(b) Box plot of the solver time of each robust MPCmethod on a set of 100 randomly generated systems.

Figure 4: Left: With increasing horizion, grid-SLS-MPC becomes infeasible for horizon larger than5. The solver time of tube-MPC is significantly larger than that of lumped-SLS-MPC or unif-df-MPC, while the latter two methods achieve comparable computational time. Right: Statistics ofthe solver time of each robust MPC method on 100 randomly generated systems with fixed horizonT = 10. We observe that lumped-SLS-MPC and unif-df-MPC are much more computationallycheaper to run than the rest two methods.

lumped uncertainty SLS MPC solves a convex inner approximation of the robust optimal controlproblem which guarantees the robust satisfaction of all state and control input constraints. It alsoenjoys recursive feasibility and input-to-state stability when combined with an adaptive horizonstrategy. By jointly optimizing over a linear time-varying state feedback controller and the normbounds of the state deviation in the prediction, our proposed method achieves significant improve-ment in conservatism compared with other baseline robust MPC methods, such as tube MPC, whilebeing numerically efficient to solve. In future work, we will extend lumped uncertainty SLS MPCto accommodate other forms of model uncertainty and investigate output feedback MPC.

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A Appendix

A.1 Proof of Lemma 2

With horizon T = 1, the state feedback controller is parameterized by u0 = Kx0 with K ∈ Rnu×nx .The robust state constraint in the robust OCP (4) can be written as

f>(Ax0 + BKx0 + ∆Ax0 + ∆BKx0 + w0) ≤ b, ∀∆A ∈ PA,∆B ∈ PB, w0 ∈ W (37)

for an arbitrary linear constraint (f, b) ∈ facet(X ). A tight upper bound on the left-hand side(LHS) of the above inequality is given by

LHS ≤ f>(A+ BK)x0 + ‖f>‖1‖∆Ax0‖∞ + ‖f>‖1‖∆BKx0‖∞ + ‖f>‖1‖w0‖∞≤ f>(A+ BK)x0 + ‖f>‖1‖∆A‖∞→∞‖x0‖∞ + ‖f>‖1‖∆B‖∞→∞‖Kx0‖∞ + ‖f>‖1‖w0‖∞≤ f>(A+ BK)x0 + εA‖f>‖1‖x0‖∞ + εB‖f>‖1‖Kx0‖∞ + ‖f>‖1σw

(38)by applying Holder’s inequality, the submultiplicativity of the `∞ norm, and the definition of theuncertainty set PA,PB,W. It is easy to show that the upper bound given in (38) is achievable andthus tight. Therefore, the robust state constraint (37) is equivalent to

f>(A+ BK)x0 + ‖f>‖1(εA‖x0‖∞ + εB‖Kx0‖∞ + σw) ≤ b. (39)

We can similarly rewrite the robust control input and terminal constraints in this way.Now consider the robust OCP inner approximation (28). For a given linear state constraint

(f, b) ∈ facet(X ), the tightened constraint and the uncertainty norm upper bound constraint aregiven by

f>(A+ BΦ0,0u )x0 + ‖f>Φ0,0

x ‖1σ0 ≤ b,

εA‖Φ0,0x x0‖∞ + εB‖Φ0,0

u x0‖∞ + σw ≤ σ0.(40)

22

By the affine constraint (20), we have Φ0,0x = I and Φ0,0

u is a free variable. As a result, thefeasibility of (39) indicates feasibility of (40) by trivially matching Φ0,0

u = K,σ0 = εA‖Φ0,0x x0‖∞ +

εB‖Φ0,0u x0‖∞+σw. Since this relationship holds for all state, control input and terminal constraints,

we have every feasible solution of the robust OCP (4) constructs a feasible one for (28). Since (28)is an inner approximation of the robust OCP (4) by construction, we prove the equivalence betweenthe convex inner approximation (28) and the robust OCP (4).

A.2 Proof of Theorem 3

We denote problem (28) at time k as SLSOCP(k) with the initial state x(k) and horizon Tk ∈{2, 3, · · · , Tmax}. Correspondingly, at time k + 1, problem (28) is denoted as SLSOCP(k + 1) withinitial state x(k + 1) and horizon Tk+1. We want to show that when SLSOCP(k) is feasible withhorizon Tk, then SLSOCP(k+1) is also feasible with horizon Tk+1 = Tk−1. In this proof, notationswith subscripts k and k + 1, such as Φx,k and Φx,k+1, refer to those used in problems SLSOCP(k)and SLSOCP(k + 1), respectively.

Sketch of the proof Assume SLSOCP(k) is feasible with a feasible solution (Φx,k, Φu,k, {σi,k}Tki=0).

Then, the synthesized linear state feedback controller Kk = Φu,kΦ−1x,k is a robust one for the robust

OCP (4) at time k and we denote this control policy by

Uk = {u∗k|k, u∗k+1|k(·), · · · , u

∗k+Tk−1|k(·)} (41)

We use xt|k and ut|k(·) to denote the predicted state and control policy at time t in SLSOCP(k).Note that Uk is just a different representation of the controller Kk with the correspondenceu∗t|k(xk:t|k) =

∑t−ki=0 K

t−k,t−k−ik xk+i|k. Then, at time k + 1, a candidate robust feedback policy

for the robust OCP (4) is given by

Uk+1 = {u∗k+1|k(·), · · · , u∗k+Tk+1|k(·)} (42)

with horizon Tk+1 = Tk − 1 which truncates (41). However, since problem (28) is only an innerapproximation of the robust OCP (4), whether the truncated policy Uk+1 will generate a feasiblesolution to the inner approximation (28) in the space of system responses remains to be shown.

In this proof, we resolve this issue in three steps. The first step is to recover the robust linearstate feedback controller Kk+1 from the truncated policy sequence Uk+1 and derive the correspond-ing system responses {Φx,k+1, Φu,k+1}. In the second step, we illustrate the relationship between

the constructed {Φx,k+1, Φu,k+1} and the existing feasible solution {Φx,k, Φu,k} to SLSOCP(k).By exploiting the connection between these two sets of system responses, in the third step, we

verify that {Φx,k+1, Φu,k+1} together with the generated norm upper bound sequence {σi,k+1}Tk+1

i=0

is indeed a feasible solution to SLSOCP(k + 1), hence proving the feasibility of SLSOCP(k + 1).

Step 1: candidate solution construction Recall that we use xt|k, ut|k(·) to denote the pre-dicted state and control policy at time t with k ≤ t ≤ k+Tk in SLSOCP(k) where the current stateis xk|k = x(k). The terms xt|k+1, ut|k+1(·) for k+ 1 ≤ t ≤ k+ 1 + Tk+1 are defined similarly. When

SLSOCP(k) is feasible, its solution {Φx,k, Φu,k, {σi,k}Tki=0} generates a robust state feedback con-

troller Kk = Φu,kΦ−1x,k ∈ L

TkTV which gives ut|k(xk:t|k) =

∑t−ki=0 K

t−k,t−k−ik xk+i|k for k ≤ t ≤ k + Tk.

We let Σk be the block diagonal matrix consisting of the norm upper bounds {σi,k}Tki=0 as shownin (28), and {Φx,k,Φu,k} be the corresponding system responses directly acting on the lumped

23

uncertainties (see (11)). As shown in Remark 1, we know that Φx,k = Φx,kΣ−1k , Φu,k = Φu,kΣ

−1k .

We choose Tk+1 = Tk − 1 as the horizon of SLSOCP(k + 1).At time k + 1, both x(k) and x(k + 1) are known. The truncated policy (42) generates a state

feedback controller Kk+1 ∈ LTk+1

TV which satisfies

ut|k+1(xk+1:t|k+1) = ut|k(xk:t|k), k + 1 ≤ t ≤ k + Tk − 1 (43)

with xk|k = x(k), xk+1|k = xk+1|k+1 = x(k + 1). Eqn. (43) says that ut|k+1(·) returns the samecontrol input as ut|k(·) given the same predicted trajectory. With the parameterization of Kk andKk+1, Eqn. (43) is equivalent to

K0,0k+1xk+1|k+1 = K1,1

k xk|k +K1,0k xk+1|k+1

K1,1k+1xk+1|k+1 +K1,0

k+1xk+2|k+1 = K2,2k xk|k +K2,1

k xk+1|k+1 +K2,0k xk+2|k+1

· · ·

KTk+1−1,Tk+1−1k+1 xk+1|k+1 + · · ·+K

Tk+1−1,0k+1 xk+Tk+1|k+1 =

KTk−1,Tk−1k xk|k +KTk−1,Tk−2

k xk+1|k+1 +KTk−1,Tk−3k xk+2|k+1 + · · ·+KTk−1,0

k xk+Tk−1|k+1

(44)

for all possible xt|k+1, k + 2 ≤ t ≤ k + Tk − 1. This immediately leads to

Ki,jk+1 = Ki+1,j

k , for all i = 0, 1, · · · , Tk+1, j = 0, 1, · · · , i− 1 (45a)

Ki,ik+1xk+1|k+1 = Ki+1,i+1

k xk|k +Ki+1,ik xk+1|k+1, for all i = 0, 1, · · · , Tk+1 (45b)

Under the condition xk+1|k+1 = x(k + 1) 6= 04, the system of linear equations (45) always has afeasible solution Kk+1 which is our candidate robust linear state feedback controller realizing thetruncated policy (42). By (10), a candidate system response solution {Φx,k+1,Φu,k+1} acting onthe lumped uncertainties can be explicitly constructed from Kk+1 by

Φx,k+1 = (I − Z(A + BKk+1))−1, Φu,k+1 = Kk+1(I − Z(A + BKk+1))

−1 (46)

where the size of block diagonal matrices A, B are adapted according to the horizon Tk+1. We choose

the candidate solutions of the uncertainty norm upper bounds {σi,k+1}Tk+1

i=0 for SLSOCP(k + 1)as σi,k+1 = σi+1,k for 0 ≤ i ≤ Tk+1, and Σk+1 be the corresponding block diagonal matrix.

Then, we can construct a candidate solution {Φx,k+1, Φu,k+1} by letting Φx,k+1 = Φx,k+1Σk+1,

Φu,k+1 = Φu,k+1Σk+1 for SLSOCP(k + 1). Before we show the feasibility of the constructed

candidate solution {Φx,k+1, Φu,k+1, {σi,k+1}Tk+1

i=0 }, the relationship between {Φx,k+1,Φu,k+1} and{Φx,k,Φu,k} has to be established.

Step 2: properties of the candidate solution At time k+1, we denote the lumped uncertaintyat time k by ηk|k = x(k + 1) − x(k). Since we use the truncated control policy at time k + 1 asshown in (43), we have that

xt|k+1 = xt|k, ut|k+1 = ut|k, t = k + 1, · · · , k + Tk+1 (47)

under the controller Kk+1 in the case that the same predicted uncertainty parameters are given fromtime k to time k+Tk+1, i.e., we consider the case when ∆A|k+1 = ∆A|k,∆B|k+1 = ∆B|k, wt|k+1 = wt|k

4We can provide non-zero nominal control inputs Kk+1(:, 0)x(k + 1) on the left-hand side of (45b) only whenx(k + 1) is non-zero. In this case the state feedback controller u = Kx is equivalent to an affine feedback controllerused in [3]. We can remove the non-zero state assumption by augmenting system (2) with state x = [x; 1].

24

for k+ 1 ≤ t ≤ k+Tk+1, where ∆A|k,∆B|k denote the predicted model uncertainty parameters andwt|k denote the predicted disturbance at time t in the k-step of MPC. Now we consider differentrealizations of the uncertainty parameters to explore the connection between {Φx,k+1,Φu,k+1} and{Φx,k,Φu,k}.

First, we let all predicted model uncertainty parameters ∆A|k+1,∆B|k+1 and wt|k+1, k+1 ≤ t ≤k + Tk+1 be zero and so are the lumped uncertainties. By the mapping of system responses (11)and the matching constraint (47), we have

Φi,ix,k+1xk+1|k+1 = Φi+1,i+1

x,k xk|k + Φi+1,ix,k wk|k

Φi,iu,k+1xk+1|k+1 = Φi+1,i+1

u,k xk|k + Φi+1,iu,k wk|k

(48)

for i = 0, 1, · · · , Tk+1. Next we let ∆A|k+1 = 0,∆B|k+1 = 0, wt|k+1 = 0 for t 6= k + 1, but allowwt|k+1 to be non-zero at t = k + 1. Then the lumped uncertainties are all zero except wk+1|k+1,and wk+1|k+1 = wk+1|k+1. Note that we apply the same predicted uncertainty parameters inSLSOCP(k) such that wt|k = wt|k+1 for k+ 1 ≤ t ≤ k+Tk+1. By the matching constraint (47) andthe system responses map (11), we have

Φi,ix,k+1xk+1|k+1 + Φi,i−1

x,k+1ηk+1|k+1 = Φi+1,i+1x,k xk|k + Φi+1,i

x,k ηk|k + Φi+1,i−1x,k ηk+1|k

Φi,iu,k+1xk+1|k+1 + Φi,i−1

u,k+1ηk+1|k+1 = Φi+1,i+1u,k xk|k + Φi+1,i

u,k ηk|k + Φi+1,i−1u,k ηk+1|k

(49)

for i = 1, · · · , Tk+1. Subtracting Eqn. (48) from both sides of (49), we conclude that Φi,i−1x,k+1 =

Φi+1,i−1x,k ,Φi,i−1

u,k+1 = Φi+1,i−1u,k for i = 1, · · · , Tk+1 since the predicted lumped uncertainty ηk+1|k+1 =

ηk+1|k can be arbitrary. Similarly, by letting wt|k+1 be non-zero at time t = k+2 and zero otherwise,

we can prove Φi,i−2x,k+1 = Φi+1,i−2

x,k ,Φi,i−2u,k+1 = Φi+1,i−2

u,k for i = 2, · · · , Tk+1. Repeat this process andwe have

Φi,jx,k+1 = Φi+1,j

x,k , Φi,ju,k+1 = Φi+1,j

u,k , for i = 1, 2, · · · , Tk+1, j = 0, · · · , i− 1. (50)

Eqn. (48) and (50) describe the relationship between {Φx,k+1,Φu,k+1} and {Φx,k,Φu,k}. Next,

we will show that our proposed candidate solution {Φx,k+1, Φu,k+1, {σi,k+1}Tk+1

i=0 } is feasible forSLSOCP(k + 1).

Step 3: feasibility of the candidate solution Recall that Φx,k+1 = Φx,k+1Σk+1, Φu,k+1 =Φu,k+1Σk+1. Since {Φx,k+1,Φu,k+1} in (46) satisfies the affine constraint (17) by construction, we

have {Φx,k+1, Φu,k+1, {σi,k+1}Tk+1

i=0 } satisfy the affine constraint in (28).For the tightened constraints (25), (26), (27), we take the state constraint tightening (25) as an

example for analysis. For an arbitrary linear constraint (f, b) ∈ facet(X ), we have

f>Φt,tx,k+1xk+1|k+1 +

t∑i=1

‖f>Φt,t−ix,k+1‖1

=f>Φt,tx,k+1xk+1|k+1 +

t∑i=1

‖f>Φt,t−ix,k+1‖1σi−1,k+1 (by Φx,k+1 = Φx,k+1Σk+1)

=f>(Φt+1,t+1x,k xk|k + Φt+1,t

x,k ηk|k) +

t∑i=1

‖f>Φt+1,t−ix,k ‖1σi,k (by Eqn.(48) and (50))

≤f>Φt+1,t+1x,k xk|k + ‖f>Φt+1,t

x,k ‖1‖ηk|k‖∞ +

t∑i=1

‖f>Φt+1,t−ix,k ‖1σi,k (by Holder’s inequality)

25

≤f>Φt+1,t+1x,k xk|k +

t+1∑i=1

‖f>Φt+1,t+1−ix,k ‖1σi−1,k (by the upper bound ‖wk|k‖∞ ≤ σ0,k)

=f>Φt+1,t+1x,k xk|k +

t+1∑i=1

‖f>Φt+1,t+1−ix,k ‖1 (by Φx,k = Φx,kΣk)

≤b (by the feasibility of SLSOCP(k))

for t = 0, 1, · · · , Tk+1. Therefore, our constructed candidate solution satisfy all the tightened stateconstraints in (28) and we can prove the feasibility of the constructed solution for the tightenedterminal state (26) and control input constraint (27) similarly. It is also straight forward to apply the

above techniques to prove the the feasibility of {Φx,k+1, Φu,k+1, {σi,k+1}Tk+1

i=0 } for the upper bound

constraint (24). Hence, we prove that our constructed solution {Φx,k+1, Φu,k+1, {σi,k+1}Tk+1

i=0 } isfeasible for SLSOCP(k + 1).

A.3 Proof of Theorem 5

By Assumption 3, the stage cost `(x, u) = x>Qx + u>Ru is positive definite and there exists aclass-K∞ function α1(·) such that α1(‖x‖∞) ≤ `(x, 0) ≤ J∗MPC(x) for all x ∈ R where R is theregion of attraction given in Definition 3. Using an argument similar to [3, Proposition 17], we canshow that each J∗N (x) for N ∈ {1, · · · , Tmax} is a strictly convex, piecewise quadratic function inx. Since the value function J∗MPC(x) is the point-wise minimum of J∗N (x) for N = 1, · · · , Tmax,there exists a class-K∞ function α2(·) such that J∗MPC(x) ≤ α2(‖x‖∞) for all x ∈ R. Recall thatTk is the horizon of the MPC at time k which achieves the minimum nominal cost according tothe adaptive horizon strategy (30). By the recursive feasibility of our proposed MPC method, theROA R is a robust forward invariant set for the closed-loop system. To show J∗MPC(x(k)) satisfiescondition (34b), we divide our discussion into two cases:Case 1 Tk = 1: When Tk = 1, by Assumption 4 we have

J∗MPC(x(k)) = `(x∗k|k, u∗k|k) + x∗,>k+1|kQT x

∗k+1|k

≥ `(x∗k|k, u∗k|k) + (x∗k+1|k)

>(Q+K>RK)x∗k+1|k + (x∗k+1|k)>(A+ BK)>QT (A+ BK)x∗k+1|k

= `(x∗k|k, u∗k|k) + q(x∗k+1|k)

where q(x) is defined as a positive definite quadratic function q(x) = x>(Q + K>RK + (A +BK)>QT (A+ BK))x and we denote its Lipschitz constant with respect to ‖·‖∞ over the compactset R as Lq. The terms x∗k+1|k and u∗k+1|k denote the nominal predicted state and control input

at time k + 1 under the optimal controller synthesized at time k by solving (28). Since Tk = 1,we have x(k + 1) ∈ XT and the local controller u = Kx is a feasible solution for the robust OCPinner approximation (28) with horizon T = 1 and x0 = x(k + 1). Therefore, at time k + 1, theoptimal cost function of problem (28) satisfies J∗T (x(k + 1)) ≤ q(x(k + 1)) = q(x∗k+1|k + η(k)) ≤q(x∗k+1|k) +Lq‖η(k)‖∞ with T = 1. By construction, we have that J∗MPC(x(k+ 1)) ≤ J∗T (x(k+ 1))for T = 1 and therefore

J∗MPC(x(k)) ≥ `(x∗k|k, u∗k|k) + q(x∗k+1|k)

≥ `(x∗k|k, u∗k|k) + J∗MPC(x(k + 1))− Lq‖η(k)‖∞

≥ α3(‖x(k)‖∞) + J∗MPC(x(k + 1))− Lq‖η(k)‖∞

for some class-K∞ function α3(·) which satisfies condition (34b).

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Case 2 Tk ≥ 2: First, we note that

J∗MPC(x(k)) =

Tk−1∑i=0

`(x∗k+i|k, u∗k+i|k)) + (x∗k+Tk|k)

>QT x∗k+Tk|k

= `(x∗k|k, u∗k|k) + q(x∗k+1|k)

(51)

where q(·) in this case is defined as the sum of the quadratic terms in the tail under the truncatedcontroller (42), and the Lipschitz constant of q(·) with respect to ‖·‖∞ over the region R is denotedby Lq. By Theorem 3 on the recursive feasibility of the proposed MPC method, if Tk ≥ 2 at time k,then at time k+1 the robust OCP (28) is feasible with horizon Tk−1 with the shifted controller (42)as a feasible solution. Therefore, we have J∗MPC(x(k + 1) ≤ J∗Tk−1(x(k + 1)) ≤ q(x(k + 1)) whichleads to

J∗MPC(x(k + 1)) ≤ q(x(k + 1))

= q(x∗k+1|k + η(k))

≤ q(x∗k+1|k) + Lq‖η(k)‖∞= J∗MPC(x(k))− `(x∗k|k, u

∗k|k) + Lq‖η(k)‖∞

≤ −α3(‖x(k)‖∞) + Lq‖η(k)‖∞

(52)

for some class-K∞ function α3(·). Summarizing both cases, we conclude that J∗MPC(·) is an ISS-Lyapunov function which shows that the closed-loop system (31) is ISS.

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