Robust Control Correction Synthesis: A Set Theoretic Approach

Sasa V. Rakovic, Michael P. Kearney and Peter Ross McAree

Abstract— This paper analyzes the robust control correctionsynthesis problem for constrained discrete–time control sys-tems. The robust control correction synthesis problem is firstintroduced and discussed in a general non–linear compact set-ting under two interpretations on the uncertainty leading to inf–sup and sup–inf control correction syntheses. The solution ofthe corresponding robust control correction synthesis problemis obtained by utilizing set theoretic methods. In addition,motivated by numerical considerations, this paper also discussesthe problem formulated in the linear convex compact settingand comments on plausible computational procedures.

I. INTRODUCTION

The control synthesis under constraints and uncertainty is,in general, a complex problem and its solution is usuallyobtained by utilizing either set theoretic approaches or gametheoretic methods in the spirit of the dynamic program-ming [1]–[8]. Computational complexity present in exact dy-namic programming based synthesis methods has motivatedthe utilization of approximate design procedures such as,for example, simplified robust control synthesis, robust andtube receding horizon control synthesis and robust referencegovernors based techniques [9]–[11]. One of the crucial rolesin the control synthesis under constraints and uncertainty isplayed by the information available to the controller for theimplementation of adequate control actions. The nature ofthe underlying information pattern results in two differentclasses of problems, namely the inf–sup, or simply min–max, control synthesis [1], [2], [4], [6], [7] and the sup–inf, or simply max–min, control synthesis [4], [8]. The maindifference between the two frequently encountered classesof control synthesis problems for discrete–time systemssubject to constraints and uncertainty lies in the utilizationof feedback control laws employing available informationadequately that, in general, belong to different functionalspaces and result in different values of the underlying costfunction.

The deterministic control correction synthesis problemconsidered recently in [12] was concerned with the cor-rection, in an optimal fashion, of an existing controllerso as to ensure that the overall, corrected control ruleproduces control actions satisfying control constraints and

Sasa V. Rakovic is, currently, a Scientific Associate at the Institutefor Automation Engineering of Otto–von–Guericke–UniversitatMagdeburg, Germany and a Visiting Academic Fellow at theDepartment of Engineering Sciences of the University of Oxford,UK. Michael P. Kearney and Peter Ross McAree are with theDivision of Mechanical Engineering, University of Queenslandand with the Cooperative Research Centre for Mining, KenmoreEast, Australia. Michael Kearney is supported by an AustralianPostgraduate Award. e-mail:svr@sasavrakovic.com,m.kearney|p.mcaree@uq.edu.au

guarantees that the evolution of the controlled dynamicsremains indefinitely long within a suitable subset of the stateconstraint set. The control correction synthesis problem wasmotivated by a variety of problems encountered in engi-neering practice including control correction in automotiveapplications (for example, correction of driver’s commandsoperating a vehicle) and human–operated processes (forinstance, correction of human control commands imposed onrobots and manipulated machines) as well as correction andenhancement of existing and/or malfunctioning controllersin industrial processes. In this note, we formalize the robustcontrol correction synthesis problem and offer its solutionby using set theoretic methods. We analyze and providesolutions to both the inf–sup and the sup–inf control correc-tion synthesis problems in the non–linear compact setting.In addition, motivated by computational considerations, wediscuss corresponding problems in the linear convex compactsetting.

Paper Overview: Section II provides necessary prelim-inaries and introduces robust control correction synthesisproblems. Sections III and IV analyze, respectively, min–maxand max–min robust control correction synthesis problems.Section V treats corresponding syntheses problems in thelinear convex compact setting. Sections VI and VII presentan illustrative example and concluding remarks.

Basic Nomenclature and Definitions: The sets of non–negative, positive integers and non–negative real numbersare denoted, respectively, by N, N+ and R+, i.e. N :={0, 1, 2, . . .}, N+ := {1, 2, . . .} and R+ := {x ∈ R :x ≥ 0}. For q1, q2 ∈ N such that q1 < q2 we denoteN[q1:q2] := {q1, q1 + 1, . . . , q2 − 1, q2}. For q ∈ N+, Nq

stands for N[0:q]. For two sets X ⊂ Rn and Y ⊂ Rn, theMinkowski set addition is defined by X ⊕ Y := {x + y :x ∈ X, y ∈ Y } and the Minkowski set subtraction isX � Y := {z ∈ Rn : z ⊕ Y ⊆ X}. A set X ⊂ Rn

is a C set if it is compact, convex, and contains the origin.A set X ⊂ Rn is a proper C set if it is a C set and theorigin is in its non–empty interior. We denote by |x|L thenorm of the vector x induced by a symmetric, proper C

set L. A polyhedron is a set described by the intersectionof finitely many half–spaces. A polytope is a closed andbounded polyhedron. Given a set X and a real matrix M ofcompatible dimensions (possibly a scalar) we denote by MX

the image of X under M , so that MX := {Mx : x ∈ X},and by M−1X the inverse image of X under M so thatM−1X := {x : Mx ∈ X}. If f (·) is a set-valued functionfrom, say, X into U , namely, its values are subsets of U ,then its graph is the set {(x, y) : x ∈ X, y ∈ f(x)}.

2009 IEEE International Conference on Control and AutomationChristchurch, New Zealand, December 9-11, 2009

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II. PRELIMINARIES & PROBLEM FORMULATION

We consider the discrete–time, time invariant, system:

x+ = f(x, u, w), (2.1)

where x ∈ Rn is the current state, u ∈ Rm is the currentcontrol, w ∈ Rp is the disturbance input, x+ ∈ Rn is thesuccessor state and f (·, ·, ·) : Rn × Rm × Rp → Rn isthe state transition mapping. The system variables, state x,control u and disturbance w, are subject to constraints:

x ∈ X ⊂ Rn, u ∈ U ⊂ Rm and w ∈ W ⊂ Rp. (2.2)

The standing technical assumption in this note is:Assumption 1: The function f (·, ·, ·) is continuous. The

system variable constraints sets are compact subsets in thecorresponding spaces, i.e. X is a compact subset of Rn, U isa compact subset of Rm and W is a compact subset of Rp.

The min–max control correction synthesis problem arisesunder the following interpretation:

Interpretation 1: At any time instance k ∈ N, the statexk and the operator’s command uk ∈ U are known whenthe corrected control action uk is determined, while futureoperator’s commands uk+i, i ∈ N+ are unknown and cantake arbitrary values uk+i ∈ U, i ∈ N+. The currentdisturbance wk and future disturbances wk+i, i ∈ N+ are notknown and can take any arbitrary values wk+i ∈ W, i ∈ N.

The min–max control correction synthesis can be seenas a game between the corrector, the operator and theadversary with the following verbal description. At any timek ∈ N, first, the operator chooses control action uk ∈ U andintends to apply it to the system x+ = f(x, u, w). Second,the corrector modifies the operator’s control action uk tothe actual control action uk ∈ U by utilizing correctionrules of the form uk (·, ·) : X × U → U. Third, theadversary chooses a disturbance wk ∈ W. Hence, if at timeinstance k the state is xk, the operator’s control is uk and thedisturbance is wk the state at time k +1 is given by xk+1 =f(xk, uk(xk, uk), wk). The corrector’s prime objectives are(i) to ensure that all state sequences {xk}

∞

k=0 generated byxk+1 = f(xk, uk(xk, uk), wk), uk ∈ U, wk ∈ W remainwithin the state constraint set X and (ii) to modify optimallythe operator’s control actions.

The max–min control correction synthesis problem arisesunder the following interpretation:

Interpretation 2: At any time instance k ∈ N, the statexk, the operator’s command uk ∈ U and the disturbancewk ∈ W are known when the corrected control action uk isdetermined, while future operator’s commands uk+i, i ∈ N+

and disturbances wk+i, i ∈ N+ are unknown and can takearbitrary values uk+i ∈ U, i ∈ N+ and wk+i ∈ W, i ∈ N.

A brief verbal description of the max–min control cor-rection synthesis is as follows: At any time k ∈ N, first, theoperator chooses control action uk ∈ U and intends to applyit to the system x+ = f(x, u, w). Second, the adversarychooses a disturbance wk ∈ W. Third, the corrector modifiesthe operator’s control action uk to the actual control actionuk ∈ U by utilizing correction rules of the form uk (·, ·, ·) :

X×U×W → U. Hence, if at time instance k the state is xk,the operator’s control is uk and the disturbance is wk the stateat time k +1 is given by xk+1 = f(xk, uk(xk, uk, wk), wk).As in the min–max setting, the corrector aims to ensurethat all state sequences {xk}

∞

k=0 generated by xk+1 =f(xk, uk(xk, uk, wk), wk), uk ∈ U, wk ∈ W remain withinthe state constraint set X and to modify optimally operator’scontrol actions.

In the min–max setting, main roles in the controllabilityunder constraints and uncertainty are played by the min–maxpreimage mapping BI–S (·), given for any set X ⊆ Rn, by:

BI–S(X) := {x : ∃u ∈ U s.t. ∀w ∈ W

f(x, u, w) ∈ X} ∩ X, (2.3)

and by the min–max set–valued control map U I–S (·):

U I–S(x) := {u ∈ U : ∀w ∈ W, f(x, u, w) ∈ X}, (2.4)

defined for all x ∈ BI–S(X). Evidently, given a set X , theset of all states x ∈ X such that there exists a control u ∈ U

(here u = u(x)) ensuring that all possible successor statesf(x, u, w), w ∈ W lie in the set X is precisely given by theset BI–S(X). Similarly, given a state x ∈ BI–S(X), the set ofall controls u ∈ U ensuring that all possible successor statesf(x, u(x), w), w ∈ W lie in the set X is given by U I–S(x).

In the controllability under constraints and uncertainty inthe max–min setting, the main focus is on the max–minpreimage mapping BS–I (·), given for any set X ⊆ Rn, by:

BS–I(X) := {x : ∀w ∈ W, ∃u ∈ U

s.t. f(x, u, w) ∈ X} ∩ X, (2.5)

and on the max–min set–valued control map US–I (·, ·):

US–I(x, w) := {u ∈ U : f(x, u, w) ∈ X}, (2.6)

defined for all (x, w) ∈ BS–I(X) × W. Note that givena set X , the set of all states x ∈ X such that for anydisturbance w ∈ W there exists a control u ∈ U (hereu = u(x, w)) ensuring that all possible successor statesf(x, u(x, w), w), w ∈ W lie in the set X is precisely givenby the set BS–I(X). Similarly, given any state–disturbancepair (x, w) ∈ BS–I(X) × W, the set of all controls u ∈ U

ensuring that any possible successor state f(x, u(x,w), w)is in the set X is given by US–I(x, w).

Necessary set invariance notions are recalled next:Definition 1: A set Ω ⊆ X is a min–max robust control

invariant set for the system x+ = f(x, u, w) and constraintset (X, U, W) if and only if Ω ⊆ BI–S(Ω).

Definition 2: A set Ω ⊆ X is a max–min robust controlinvariant set for the system x+ = f(x, u, w) and constraintset (X, U, W) if and only if Ω ⊆ BS–I(Ω).

For typographical convenience we simply use terms min–max and max–min control invariant sets rather than the wholecorresponding expressions as no confusion should arise.

Definition 3: A min–max (max–min) control invariant setΩI–S (ΩS–I) is said to be the maximal min–max (max–min)control invariant set if and only if it contains all min–max(max–min) control invariant sets.

501

In this paper we consider min–max and max–min additivecontrol correction rules of the form:

uI–Sk (xk, uk) = uk + πI–S

k (xk, uk) (2.7a)

uS–Ik (xk, uk, wk) = uk + πS–I

k (xk, uk, wk) (2.7b)

where, for any k ∈ N, functions πI–Sk (·, ·) and πS–I

k (·, ·, ·)are referred to as min–max and max–min control modi-fication rules, respectively. Note that min–max and max–min control modification rules πI–S

k (·, ·) and πS–Ik (·, ·, ·), are

such that for any k ∈ N and all (x, u) ∈ X × U, itholds that πI–S

k (xk, uk) ∈ U ⊕ (−U) and, similarly, forany k ∈ N and all (x, u, w) ∈ X × U × W it holds thatπS–I

k (xk, uk, wk) ∈ U ⊕ (−U). We need some additionalnecessary notation. Given an integer k ∈ N+, let uk :={u0, u1, . . . , uk−1} and wk := {w0, w1, . . . , wk−1} denote,respectively, admissible sequences of operator’s inputs anddisturbances over the horizon of length of k, i.e. ui ∈U and wi ∈ W for all i ∈ Nk−1. Let also ΠI–S

k :={πI–S

0 (·, ·) , πI–S1 (·, ·) , . . . , πI–S

k−1 (·, ·)} denote the min–maxcontrol modification policy over the horizon of length of k,i.e. πI–S

i (x, u) : X × U → U ⊕ (−U), for all i ∈ Nk−1,are the min–max control modification rules. Similarly, letΠS–I

k := {πS–I0 (·, ·, ·) , πS–I

1 (·, ·, ·) , . . . , πS–Ik−1 (·, ·, ·)} denote

the max–min control modification policy over the horizonof length of k, i.e. πS–I

i (x, u, w) : X×U×W → U⊕ (−U),for all i ∈ Nk−1, are the max–min control modification rules.Sets of all admissible sequences of operator’s inputs anddisturbances over the horizon of length of k are, clearly, setsU

k and Wk, respectively. Sets of all admissible min–max

and max–min control modification policies over the horizonof length of k are denoted, respectively, by Π

I–Sk and Π

S–Ik .

Finally, let φ(i, x, uk,ΠI–Sk ,wk) and φ(i, x, uk,ΠS–I

k ,wk)denote, respectively, solutions to (2.1) at time i ∈ Nk whenthe initial state (at time i = 0) is x, the sequence of operator’sinputs is uk ∈ U

k, the disturbance sequence is wk ∈ Wk

and min–max and max–min control modification policies areΠI–S

k ∈ ΠI–Sk and ΠS–I

k ∈ ΠS–Ik , respectively. We adapt the

same notational convention for the case when k =∞.We examine in this note the following problems:Problem 1: A: k–horizon min–max control correction

synthesis problem: Given an integer k ∈ N+, find the setof states x ∈ X and the corresponding min–max controlmodification policies ΠI–S

k ∈ ΠI–Sk which ensures that:

∀uk ∈ Uk, ∀wk ∈ W

k, ∀i ∈ Nk, φ(i, x, uk,ΠI–Sk ,wk) ∈ X

and ui + πI–Si (φ(i, x, uk,ΠI–S

k ,wk), ui) ∈ U. (2.8)

B: k–horizon max–min control correction synthesis problem:Given an integer k ∈ N+, find the set of states x ∈ X

and the corresponding max–min control modification policiesΠS–I

k ∈ ΠS–Ik which ensures that:

∀uk ∈ Uk, ∀wk ∈ W

k, ∀i ∈ Nk, φ(i, x, uk,ΠS–Ik ,wk) ∈ X

and ui + πS–Ii (φ(i, x, uk,ΠS–I

k ,wk), ui, wi) ∈ U. (2.9)Problem 2: A: ∞–horizon min–max control correction

synthesis problem: Solve Problem 1–A for k = ∞.B: ∞–horizon max–min control correction synthesis prob-lem: Solve Problem 1–B for k = ∞.

III. MIN–MAX CONTROL CORRECTION SYNTHESIS

In this section we provide solutions to Problems 1–Aand 2–A arising under Interpretation 1 (assumed to holdthroughout this section). Our first result is an adequatemodification of standard results [3], [6], [12]:

Proposition 1: Let X be an arbitrary, non–empty subset ofX. There exists an x ∈ X and a min–max control correctionrule uI–S(x, u) such that for all u ∈ U:

uI–S(x, u) ∈ U and ∀w ∈ W, f(x, uI–S(x, u), w) ∈ X

if and only if BI–S(X) = ∅ where BI–S(X) is given by (2.3).If, in addition, BI–S(X) = ∅ then there exists a min–maxadditive control correction rule uI–S(x, u) = u + vI–S(x, u)such that for all (x, u) ∈ BI–S(X)× U it holds that:

u + vI–S(x, u) ∈ U and

∀w ∈ W, f(x, u + vI–S(x, u), w) ∈ X. (3.1)

Furthermore, any min–max additive control correction ruleensuring (3.1) satisfies for all (x, u) ∈ BI–S(X)× U:

uI–S(x, u) = u + vI–S(x, u), vI–S(x, u) ∈ V I–S(x, u),

where the min–max set–valued control modification mapV I–S (·, ·) is given, for all (x, u) ∈ BI–S(X)× U, by:

V I–S(x, u) := {v ∈U⊕ (−U) : u + v ∈ U and

∀w ∈ W, f(x, u + v, w) ∈ X}. (3.2)

A. k-horizon min–max control correction problem

We utilize the semi–group property of the min–max preim-age mapping BI–S (·) and consider the set iteration, for anyk ∈ N+:

X I–Sj+1 := BI–S(X I–S

j ), j ∈ Nk−1 and X I–S0 := X. (3.3)

With the sequence of min–max controllable sets {X I–Sj }k

j=0,we associate a sequence of min–max set–valued controlmodification maps given, for any k ∈ N+, all j ∈ N[1:k]

and all (x, u) ∈ X I–Sj × U, by:

V I–Sj (x, u) := {v ∈ U⊕ (−U) : u + v ∈ U and

∀w ∈ W, f(x, u + v, w) ∈ X I–Sj−1}. (3.4)

Standard results [6], [12] yield:Proposition 2: Pick a k ∈ N+ and consider sequences

of min–max controllable sets {X I–Sj }k

j=0 and min–max set–valued control modification maps {V I–S

j (·, ·)}kj=1 specified

in (3.3)–(3.4). Then the k–horizon min–max control cor-rection synthesis problem is solvable, i.e. there exists astate x ∈ X and a min–max control modification policyΠI–S

k ∈ ΠI–Sk such that (2.8) holds if and only if X I–S

k =∅. Furthermore, the min–max control modification policyΠI–S

k = {πI–S0 (·, ·) , πI–S

1 (·, ·) , . . . , πI–Sk−1 (·, ·)} is such that:

∀(y, u) ∈ X I–Sk−i × U, ∀i ∈ Nk−1, πI–S

i (y, u) ∈ V I–Sk−i(y, u).

As in the deterministic case [12], the k–horizon min–maxcontrol correction synthesis problem is solvable for all initialstates for which the standard k–horizon min–max controlsynthesis problem is solvable.

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Proposition 3: Suppose Assumption 1 holds, pick a k ∈N+ and consider sequences of min–max controllable sets{X I–S

j }kj=0 and min–max set–valued control modification

maps {V I–Sj (·, ·)}k

j=1 specified in (3.3)–(3.4). Then min–maxcontrollable sets X I–S

j , j ∈ Nk and graphs of min–max set–valued control modification maps V I–S

j (·, ·) , j ∈ N[1:k] arecompact when non–empty.

B. ∞-horizon min–max control correction problem

As is customary [3], [6] we examine the limiting behaviorof the sequence of min–max controllable sets {X I–S

j }∞j=0.Proposition 4: Suppose Assumption 1 holds and consider

the sequence of min–max controllable sets {X I–Sj }∞j=0 gen-

erated by (3.3) as k goes to ∞. Then (i) the sequence{X I–S

j }∞j=0 converges to the set X I–S :=⋂∞

j=0 XI–Sj , (ii) If for

some k∗ ∈ N it holds that X I–Sk∗ = X I–S

k∗+1 then the limit X I–S

of the sequence {X I–Sj }∞j=0 satisfies X I–S = X I–S

k∗ , (iii) theset X I–S is compact when non–empty and it is the maximalmin–max control invariant set.Assumption 1 and Proposition 4 yield our next result.

Proposition 5: Suppose Assumption 1 holds. Then the∞–horizon min–max control correction synthesis problemis solvable, i.e. there exist a state x ∈ X and a min–maxcontrol modification policy ΠI–S

∞∈ Π

I–S∞

such that for allu∞ ∈ U

∞, all w∞ ∈ W∞ and all i ∈ N it holds that

φ(i, x, u∞,ΠI–S∞

,w∞) ∈ X and

ui + πI–Si (φ(i, x, u∞,ΠI–S

∞,w∞), ui) ∈ U

if and only if the limit X I–S of the sequence of min–maxcontrollable sets {X I–S

j }∞j=0 is a non–empty set. Furthermore,if X I–S = ∅, the min–max control modification policy ΠI–S

∞

is any time invariant, min–max control modification policyΠI–S = {πI–S (·, ·) , πI–S (·, ·) , . . .} such that:

∀(x, u) ∈ X I–S × U, πI–S(x, u) ∈ V I–S∞

(x, u),

where, for all (x, u) ∈ X I–S × U the min–max set–valuedcontrol modification map V I–S

∞(·, ·) is given by:

V I–S∞

(x, u) := {v ∈ U⊕ (−U) : u + v ∈ U and

∀w ∈ W, f(x, u + v, w) ∈ X I–S}, (3.5)

and it has compact graph.Corollary 1: Suppose Assumption 1 holds and assume

that Ω ⊆ X is compact and a min–max control invariantset. Then there exists a time invariant min–max controlmodification rule πI–S (·, ·) such that for all (x, u) ∈ Ω×U:

u+πI–S(x, u) ∈ U and ∀w ∈ W, f(x, u+πI–S(x, u), w) ∈ Ω.

Furthermore, any such time invariant min–max control mod-ification rule πI–S (·, ·) satisfies:

∀(x, u) ∈ Ω× U, πI–S(x, u) ∈ V I–S(x, u),

where, for all (x, u) ∈ Ω × U, the corresponding min–maxset–valued control modification map V I–S (·, ·) is given by:

V I–S(x, u) := {v ∈U⊕ (−U) : u + v ∈ U and

∀w ∈ W, f(x, u + v, w) ∈ Ω}, (3.6)

and it has compact graph.The selection of min–max control modification and additivecontrol correction rules is made based on an optimizationproblem P(x, u) given, for all (x, u) ∈ Ω× U, by:

Π0(x, u) := arg infv

supw∈W

{V (x, u, v, w) : v ∈ V I–S(x, u)},

V 0(x, u) := infv

supw∈W

{V (x, u, v, w) : v ∈ V I–S(x, u)}, (3.7)

where the set–valued map V I–S (·, ·) is given in (3.6) and thecost function V (·, ·, ·, ·) : Ω×U× (U⊕ (−U))×W → R+

reflects the desired selection criterion. Employing results onparametric mathematical programming [13] we have:

Proposition 6: Suppose Assumption 1 holds and assumethat Ω ⊆ X is compact and a min–max control invariant set.Consider the optimization problem P(x, u) given by (3.7)and assume that the cost function V (·, ·, ·, ·) : Ω × U ×(U⊕(−U))×W → R+ is a lower–semi continuous function.Then (i) the optimal cost function V 0 (·, ·) : Ω×U → R+

is a lower–semi continuous function, (ii) for all (x, u) ∈Ω × U it holds that Π0(x, u) = ∅ and, consequently, thereexists a single–valued function π0 (·, ·) such that ∀(x, u) ∈Ω×U, π0(x, u) ∈ Π0(x, u) (iii) the single–valued min–maxadditive control correction rule:

μI–S(x, u) := u + π0(x, u)

is such that for all (x, u) ∈ Ω× U:

μI–S(x, u) ∈ U and ∀w ∈ W, f(x, μI–S(x, u), w) ∈ Ω.

IV. MAX–MIN CONTROL CORRECTION SYNTHESIS

In this section we provide solutions to Problems 1–Band 2–B arising under Interpretation 2 (assumed to holdthroughout this section). An analogue of Proposition 1 holdstrue:

Proposition 7: Let X be an arbitrary, non–empty subset ofX. There exists an x ∈ X and a max–min control correctionrule uS–I(x, u, w) such that for all (u, w) ∈ U×W:

uS–I(x, u, w) ∈ U and f(x, uS–I(x, u, w), w) ∈ X

if and only if BS–I(X) = ∅, where BS–I(X) is given by (2.5).If, in addition, BS–I(X) = ∅, there exists a max–min additivecontrol correction rule uS–I(x, u, w) = u+vS–I(x, u, w) suchthat for all (x, u, w) ∈ BS–I(X)× U×W, it holds that:

u + vS–I(x, u, w) ∈ U and f(x, u + vS–I(x, u, w), w) ∈ X.

(4.1)Furthermore, any max–min additive control correction ruleensuring (4.1) satisfies, for all (x, u, w) ∈ BS–I(X)×U×W:

uS–I(x, u, w) = u+vS–I(x, u, w), vS–I(x, u, w) ∈ VS–I(x, u, w),

where, for all (x, u, w) ∈ BS–I(X)×U×W, the max–min set–valued control modification map VS–I (·, ·, ·) takes the form:

VS–I(x, u, w) := {v ∈U⊕ (−U) : u + v ∈ U and

f(x, u + v, w) ∈ X}. (4.2)

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A. k-horizon max–min control correction problem

Similarly as in Section III, we utilize the semi–groupproperty of the max–min preimage mapping BS–I (·) andconsider the set iteration, for any k ∈ N+,:

X S–Ij+1 := BS–I(X S–I

j ), j ∈ Nk−1 and X S–I0 := X. (4.3)

With the sequence of max–min controllable sets {X S–Ij }k

j=0,we associate a sequence of max–min set–valued controlmodification maps given, for any k ∈ N+, all j ∈ N[1:k],and all (x, u, w) ∈ X S–I

j × U×W, by:

VS–Ij (x, u, w) := {v ∈ U⊕ (−U) : u + v ∈ U and

f(x, u + v, w) ∈ X S–Ij−1} (4.4)

A relevant analogue of Proposition 2 is:Proposition 8: Pick a k ∈ N+ and consider sequences

of max–min controllable sets {X S–Ij }k

j=0 and max–min set–valued control modification maps {VS–I

j (·, ·, ·)}kj=1 speci-

fied in (4.3)–(4.4). Then the k–horizon max–min controlcorrection synthesis problem is solvable, i.e. there exists astate x ∈ X and a max–min control modification policyΠS–I

k ∈ ΠS–Ik such that (2.9) holds if and only if X S–I

k =∅. Furthermore, the max–min control modification policyΠS–I

k = {πS–I0 (·, ·, ·) , πS–I

1 (·, ·, ·) , . . . , πS–Ik−1 (·, ·, ·)} is such

that for all i ∈ Nk−1:

∀(y, u, w) ∈ X S–Ik−i × U×W, πS–I

i (y, u, w) ∈ VS–Ik−i(y, u, w).

Similarly to [12] and Section III, Proposition 8 asserts thatthe k–horizon max–min control correction synthesis problemis solvable for all initial states for which the standard k–horizon max–min control synthesis problem is solvable. Atopologically relevant analogue of Proposition 3 holds true.

Proposition 9: Suppose Assumption 1 holds, pick a k ∈N+ and consider sequences of max–min controllable sets{X S–I

j }kj=0 and max–min set–valued control modification

maps {VS–Ij (·, ·, ·)}k

j=1 specified in (4.3)–(4.4). Then max–min controllable sets X S–I

j , j ∈ Nk and graphs of max–minset–valued control modification maps VS–I

j (·, ·, ·) , j ∈ N[1:k]

are compact when non–empty.

B. ∞-horizon max–min control correction problem

Relevant analogues of Propositions 4 and 5 address-ing the limiting behavior of sequences {X S–I

j }∞j=0 and{VS–I

j (·, ·, ·)}∞j=0 also hold true.Proposition 10: Suppose Assumption 1 holds and con-

sider the sequence of max–min controllable sets {X S–Ij }∞j=0

generated by (4.3) as k goes to ∞. Then (i) the sequence{X S–I

j }∞j=0 converges to the set X S–I :=⋂∞

j=0 XS–Ij , (ii) If for

some k∗ ∈ N it holds that X S–Ik∗ = X S–I

k∗+1 then the limit X S–I

of the sequence {X S–Ij }∞j=0 satisfies X S–I = X S–I

k∗ , (iii) theset X S–I is compact when non–empty and it is the maximalmax–min control invariant set.

Proposition 11: Suppose Assumption 1 holds. Then the∞–horizon max–min control correction synthesis problemis solvable, i.e. there exist a state x ∈ X and a max–min

control modification policy ΠS–I∞∈ Π

S–I∞

such that for allu∞ ∈ U

∞, all w∞ ∈ W∞ and all i ∈ N it holds that

φ(i, x, u∞,ΠS–I∞

,w∞) ∈ X and

ui + πS–Ii (φ(i, x, u∞,ΠS–I

∞,w∞), ui, wi) ∈ U

if and only if the limit X S–I of the sequence of max–mincontrollable sets {X S–I

j }∞j=0 is a non–empty set. Furthermore,if X S–I = ∅, the max–min control modification policy ΠS–I

∞

is any time invariant, max–min control modification policyΠS–I = {πS–I (·, ·, ·) , πS–I (·, ·, ·) , . . .} such that:

∀(x, u, w) ∈ X S–I × U×W, πS–I(x, u, w) ∈ VS–I∞

(x, u, w),

where, for all (x, u, w) ∈ X S–I × U×W the max–min set–valued control modification map VS–I

∞(·, ·, ·) is given by:

VS–I∞

(x, u, w) := {v ∈U⊕ (−U) : u + v ∈ U and

f(x, u + v, w) ∈ X S–I}, (4.5)

and it has compact graph.Corollary 2: Suppose Assumption 1 holds and assume

that Ω ⊆ X is compact and a max–min control invariantset. Then there exists a time invariant max–min controlmodification rule πS–I (·, ·, ·) such that for all (x, u, w) ∈Ω× U×W:

u + πS–I(x, u, w) ∈ U and f(x, u + πS–I(x, u, w), w) ∈ Ω.

Furthermore, any such time invariant max–min control mod-ification rule πS–I (·, ·, ·) satisfies:

∀(x, u, w) ∈ Ω× U×W, πS–I(x, u, w) ∈ VS–I(x, u, w),

where, for all (x, u, w) ∈ Ω × U × W, the correspondingmax–min set–valued control modification map VS–I (·, ·, ·) isgiven by:

VS–I(x, u, w) := {v ∈U⊕ (−U) : u + v ∈ U and

f(x, u + v, w) ∈ Ω}, (4.6)

and it has compact graph.The selection of max–min control modification and additivecontrol correction rules is made, for all (x, u, w) ∈ Ω×U×W, by utilizing optimization problems:

Π0(x, u, w) := arg infv{V (x, u, v, w) : v ∈ VS–I(x, u, w)},

V ∗(x, u, w) := infv{V (x, u, v, w) : v ∈ VS–I(x, u, w)},

V 0(x, u) := supw

{V ∗(x, u, w) : w ∈ W}, (4.7)

where the set–valued map VS–I (·, ·, ·) is given in (4.6) and thecost function V (·, ·, ·, ·) : Ω×U× (U⊕ (−U))×W → R+

reflects the desired selection criterion. Utilizing results onparametric mathematical programming [13] we have:

Proposition 12: Suppose Assumption 1 holds and assumethat Ω ⊆ X is compact and a max–min control invariantset. Consider optimization problems specified in (4.7) andassume that the cost function V (·, ·, ·, ·) : Ω × U × (U ⊕(−U)) × W → R+ is a lower–semi continuous function.Then (i) functions V ∗ (·, ·, ·) : Ω × U × W → R+ andV 0 (·, ·) : Ω×U → R+ are lower–semi continuous, (ii) for

504

all (x, u, w) ∈ Ω×U×W it holds that Π0(x, u, w) = ∅ and,consequently, there exists a single–valued function π0 (·, ·, ·)such that ∀(x, u, w) ∈ Ω×U×W, π0(x, u, w) ∈ Π0(x, u, w)(iii) the single–valued max–min additive control correctionrule:

μS–I(x, u, w) := u + π0(x, u, w)

is such that for all (x, u, w) ∈ Ω× U×W:

μS–I(x, u, w) ∈ U and f(x, μS–I(x, u), w) ∈ Ω.

V. LINEAR CONVEX COMPACT CASE

The underlying state transition mapping is, throughout thissection, f(x, u, w) = Ax + Bu + Dw so that:

x+ = Ax + Bu + Dw, (5.1)

with (A,B,D) ∈ Rn×n×Rn×m×Rn×p. Assumption 1 is,throughout this section, replaced by:

Assumption 2: The state, control and disturbance con-straint sets X ⊂ Rn, U ⊂ Rm and W ⊂ Rp are convex,compact and have non-empty interior.

A. Explicit Characterizations and Refined Properties

In this case, the min–max preimage mapping is given by:

BI–S(X) = (A−1((X �DW)⊕ (−BU))) ∩ X, (5.2)

while the corresponding min–max set–valued control mod-ification and control maps are given, respectively, for all(x, u) ∈ BI–S(X)× U, and all x ∈ BI–S(X) by:

V I–S(x, u) = {v ∈ U⊕ (−U) : u + v ∈ U and

Ax + B(u + v) ∈ X �DW},

U I–S(x) = {u ∈ U : Ax + Bu ∈ X �DW}. (5.3)

Likewise, the max–min preimage mapping is given by:

BS–I(X) = (A−1((X ⊕ (−BU))�DW)) ∩ X (5.4)

while the max–min set–valued control modification andcontrol maps are given, respectively, for all (x, u, w) ∈BS–I(X)× U×W, and all (x,w) ∈ BS–I(X)×W by:

VS–I(x, u, w) = {v ∈ U⊕ (−U) : u + v ∈ U and

Ax + B(u + v) + Dw ∈ X},

US–I(x,w) = {u ∈ U : Ax + B(u + v) + Dw ∈ X}. (5.5)

As set operations involved in (5.2)– (5.5) preserve convex-ity [14], [15], we summarize briefly consequent refinementsof assertions established in Sections III and IV.

Proposition 13: Suppose Assumption 2 holds and assumealso that the set X is a convex and compact subset of X.Then the set BI–S(X) (BS–I(X)) and the graph of the corre-sponding min–max (max–min) set–valued control modifica-tion map V I–S (·, ·) (VS–I (·, ·, ·)) given, respectively, by (5.2)and (5.3) ( (5.4) and (5.5)) are convex and compact (whennon–empty). Furthermore, if the set BI–S(X) (BS–I(X)) isnon–empty, then the min–max (max–min) set–valued controlmodification map V I–S (·, ·) (VS–I (·, ·, ·)) is compact– andconvex– valued and, is, in fact, continuous with respect tothe Hausdorff distance.

Proposition 14: Suppose Assumption 2 holds and con-sider the sequence of min–max (max–min) controllable sets{X I–S

j }∞j=0 ( {X S–Ij }∞j=0) generated by (3.3) ( (4.3)) with

BI–S (·) (BS–I (·)) given by (5.2) ( (5.4)). Then, (i) each X I–Sj

(X S–Ij ) is convex and compact (when non–empty) and (ii) the

limit X I–S (X S–I) of the sequence {X I–Sj }∞j=0 ( {X S–I

j }∞j=0)is a convex and compact set (when non–empty).

B. Minimum Effort Control Correction and Computations

The synthesis of the minimum effort min–max controlcorrection rule utilizes the optimization problem P

I–SME(x, u):

V 0(x, u) = minv{|v|L : v ∈ V I–S(x, u)}

Π0(x, u) = arg minv{|v|L : v ∈ V I–S(x, u)}, (5.6)

where for all (x, u) ∈ ΩI–S×U, with ΩI–S being a convex andcompact min–max control invariant set, the min–max set–valued control modification map V I–S (·, ·) is given by (3.2)with the set X replaced by the set ΩI–S.

Likewise (with some abuse of notation), the synthesis ofthe minimum effort max–min control correction rule utilizesthe optimization problem P

S–IME(x, u, w):

V 0(x, u, w) = minv{|v|L : v ∈ VS–I(x, u, w)}

Π0(x, u, w) = arg minv{|v|L : v ∈ VS–I(x, u, w)}, (5.7)

where for all (x, u, w) ∈ ΩS–I × U × W, with ΩS–I beinga convex and compact max–min control invariant set, themax–min set–valued control modification map VS–I (·, ·, ·) isgiven by (4.2) with the set X replaced by the set ΩS–I.

Relevant computational remarks are provided jointly next.Remark 1: When Assumption 2 holds, the set ΩI–S (ΩS–I)

is min–max (max–min) control invariant polytope, the systemis linear and sets X, U and W are, in addition, polytopes, theoptimization problem P

I–SME(x, u) (PS–I

ME(x, u, w)) a parametricconvex mathematical program [13]. In particular, when theset L defining the cost |v|L is a symmetric, proper C poly-tope/ellipsoid in Rm, the problem P

I–SME(x, u) (PS–I

ME(x, u, w))is a parametric linear/quadratic programming problem and itssolution can be computed employing the standard computa-tional geometry tools as offered in [16], [17]. In this case, theoptimal cost function V 0 (·, ·) (V 0 (·, ·, ·)) is a continuous,convex piecewise affine/quadratic function defined over thepolytopic partition of the set Ω×U (Ω×U×W). Furthermore,there exists a continuous piecewise affine function π0 (·, ·)(π0 (·, ·, ·)) defined over the same polytopic partition of theset ΩI–S×U (ΩS–I×U×W) such that for all (x, u) ∈ ΩI–S×U

((x, u, w) ∈ ΩS–I×U×W ) it holds that π0(x, u) ∈ Π0(x, u)(π0(x, u, w) ∈ Π0(x, u, w)). The min–max and max–minadditive control correction rules μI–S (·, ·) and μS–I (·, ·, ·),defined by:

∀(x, u) ∈ ΩI–S × U, μI–S(x, u) = u + π0(x, u),

∀(x, u, w) ∈ ΩS–I × U×W, μS–I(x, u, w) = u + π0(x, u, w),

inherit the continuity and piecewise affine property (withrespect to the corresponding polytopic partition of setsΩI–S × U and ΩS–I × U × W) from functions π0 (·, ·) and

505

π0 (·, ·, ·). Furthermore, the min–max and max–min additivecontrol correction rules μI–S (·, ·) and μS–I (·, ·, ·) satisfy, forall (x, u) ∈ ΩI–S × U,

μI–S(x, u) ∈ U and Ax + BμI–S(x, u) ∈ ΩI–S �DW,

and for all (x, u, w) ∈ ΩS–I × U×W,

μS–I(x, u, w) ∈ U and Ax + BμS–I(x, u, w) + Dw ∈ ΩS–I.

In addition, for all x ∈ ΩI–S and all u ∈ U I–S(x) we have:

π0(x, u) = 0 and μI–S(x, u) = u, and, similarly,

for all (x,w) ∈ ΩS–I ×W and all u ∈ US–I(x,w) we have:

π0(x, u, w) = 0 and μS–I(x, u, w) = u.

Finally, an alternative implementation of the min–max (max–min) additive correction rule μI–S (·, ·) (μS–I (·, ·, ·)) is toutilize the solution of the optimization problem P

I–SME(x, u)

(PS–IME(x, u, w)) on–line at the pair (x, u) ∈ ΩI–S × U (at the

triplet (x, u, w) ∈ ΩS–I×U×W) encountered in the process.

VI. ILLUSTRATIVE EXAMPLES

Our numerical example is the sampled double integratorsystem (h = 0.2),

x+ =

[1 h

0 1

]x +

[h2

2h

]u + w,

with variable constraint sets X = [−2, 2] × [−1, 1], U =[−1, 1] and W = [−0.1, 0.1] × [−0.1, 0.1]. The min–maxcontrol correction rule is determined on–line at each timeinstance by solving the optimization problem P

I–SME(x, u)

specified in (5.6) with ΩI–S equal to the maximal min–maxcontrol invariant set, computed off–line utilizing results ofPropositions 4 and 14. The cost function utilized, in allcases of our illustrative example, is v2 and computations areperformed according to Remark 1. Likewise, the max–mincontrol correction rule is determined on–line at each timeinstance by solving the optimization problem P

S–IME(x, u, w)

specified in (5.7) with ΩS–I equal to the maximal max–min control invariant set, computed off–line utilizing re-sults of Propositions 10 and 14. The optimization problemsP

I–SME(x, u) and P

S–IME(x, u, w) are standard strictly convex

quadratic programming problems. Note that, in this partic-ular setting, the maximal min–max and max–min controlinvariant sets coincide (i.e. ΩS–I = ΩI–S). Figure 1 showsthe simulation results for min–max and max–min cases for aconstant operator’s command u = 1 (where both simulationsutilize the same realization of an admissible disturbance se-quence). Evidently, both the min–max and max–min controlcorrectors successfully keep the state trajectory within thecorresponding min–max and max–min control invariant sets.A closer inspection of Figure 1 reveals that the informationaladvantage of the max–min control corrector permits forthe safe operation closer to the boundary of the set ΩI–S

(sub–figure on the right) compared to the min–max controlcorrector that forces trajectory to stay a certain distance fromthe boundary of the set ΩI–S (sub–figure on the left).

Fig. 1. State trajectories induced by the min–max and max–min controlcorrections rules for a constant operator’s command u = 1 from the origin.

VII. CONCLUSIONS

We discussed the robust control correction synthesis prob-lems in two, information related, settings, leading to the min–max and max–min, control correction synthesis problems.We derived solutions to considered problems by utilizing settheoretic methods and employing set invariance concepts.

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