Set Robust Control Invariance for Linear Discrete Time Systems

Abstract

This paper introduces set robust control invariance, a concept that generalizes robust control invariance for systems described by difference equations to systems described by difference inclusions of special structure. The concept is useful for the analysis and synthesis of uncertain systems where a given control policy results in, for each initial state, a tube of trajectories rather than a sing trajectory; it also reveals the properties required for the terminal constraint set in receding horizon control of constrained linear systems with bounded disturbances and shows how improved terminal sets may be constructed. The family of set robust control invariant sets is characterized and the most important members of this family, the minimal and the maximal, are identified.

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Set Robust Control Invariance for Linear Discrete Time Systems
S. V. Rakovi´ c and D. Q. Mayne
Abstract—This paper introduces set robust control invariance,
a concept that generalizes robust control invariance for systems
described by difference equations to systems described by
difference inclusions of special structure. The concept is useful
for the analysis and synthesis of uncertain systems where a
given control policy results in, for each initial state, a tube of
trajectories rather than a single trajectory; it also reveals the
properties required for the terminal constraint set in receding
horizon control of constrained linear systems with bounded
disturbances and shows how improved terminal sets may be
constructed. The family of set robust control invariant sets is
characterized and the most important members of this family,
the minimal and the maximal, are identified.
I. INTRODUCTION
Set invariance theory, excellently surveyed in
vides, inter alia, useful tools for the synthesis of reference
governors[2] and predictive controllers
guaranteed invariance, stability and convergence properties.
A robust control (positively) invariant set, in particular the
minimal robust positively invariant set, is also a suitable
target set in robust time-optimal control [7]–[9] and plays an
integral part in the novel robust predictive control methods,
recently proposed in [10]–[12].
In this paper we introduce the concept of set robust
control invariance that generalizes robust control invariance
for systems described by difference equations to systems
described by difference inclusions of special structure (a
set X ⊂ Rnis robust control invariant for the system
x+= f(x,u,w) and constraint set (X,U,W) if, X ⊆ X
and for every x ∈ X, there exists an admissible control
u ∈ U such that f(x,u,w) ∈ X for all x ∈ X, all w ∈ W).
The motivation for this generalization lies in the fact that,
when uncertainty is present in the controlled system, we
are forced to consider a tube of trajectories (a sequence
of sets of states) instead of a single trajectory (a sequence
of states); each trajectory of the system, arising from a
particular realization of the uncertainty, lies in the tube.
Tubes have been extensively studied in [13]–[18], mainly
in the context of constrained continuous-time systems. In
practice, determination of ‘exact’ (or non-conservative) tubes
is difficult, so ‘outer bounding tubes’ that bound ‘exact’ tubes
are employed. If we wish to establish asymptotic/exponential
stability of an uncertain system under receding horizon
control using techniques, similar to those conventionally
employed for deterministic systems [19] when only a local
[1], pro-
[3]–[6] with
This research is supported by the Engineering and Physical Sciences
Research Council, UK.
S. V. Rakovi´ c and D. Q. Mayne are with Imperial College London,
London SW7 2BT, United Kingdom, e-mail: sasa.rakovic,
d.mayne@imperial.ac.uk
Control Lyapunov function is available for the terminal cost,
we need to generalize appropriately the concept of (robust)
control invariance. In particular, we wish to generalize the
requirement in receding horizon control that, if the last
element xN of a feasible trajectory {x0,x1,...,xN} lies in
a terminal constraint set T, there exists an admissible control
action u such that the successor element x+
lies in T. Since the elements of a tube are sets, it is necessary
to find a set Φ of sets that is set robust control invariant for
a system x+= f(x,u,w), w ∈ W in the following sense:
for any set X ∈ Φ, there exists a control law θ : X → U
such that X+= f(X,θ,W) ⊂ Y for some Y ∈ Φ, where
f(X,θ,W) ? {f(x,θ(x),w) | x ∈ X, w ∈ W}. Thus, if
the last element XN of a tube {X0,X1,...,XN} lies in
Φ, it is possible to find an admissible local control law π
such that the successor element X+
the new outer-bounding tube {X1,X2,...,XN,Y } has its
last element lying in Φ; this is an appropriate generalization
(with XN and θ(·) replacing xN and u respectively) of the
condition that the terminal constraint set in deterministic
receding horizon control is control (positively) invariant.
Practicality enforces the use of ‘outer bounding tubes’ having
a simple structure; for example each element of Φ has the
form z ⊕ R.
This paper is organized as follows. Section 2 is concerned
with preliminaries. Section 3 provides a characterization of a
family of set robust control invariant sets. Section 4 discusses
dynamical behavior of trajectories of a sequence of sets
of states starting from a set of states that is element of a
set robust control invariant set. Section 5 provides a set of
constructive simplifications. Section 6 gives an illustrative
example. Finally, Section 7 presents conclusions.
NOTATIONAND BASIC DEFINITIONS:
{0,1,2,...} and N+ ? {1,2,...}. Given two sets U and
V, such that U ⊂ Rnand V ⊂ Rn, the Minkowski set
addition is defined by U ⊕ V ? {u + v | u ∈ U, v ∈ V},
the Minkowski (Pontryagin or geometric) set difference is:
U ? V ? {x | x ⊕ V ⊆ U}; we employ x ⊕ X where
X is a set to denote {x} ⊕ X. Given the sequence of sets
{Ui⊂ Rn}b
hyperball in Rnis denoted by Bn
(r > 0) where given a vector x ∈ Rn, |x|pdenotes the vector
p–norm.
N= f(xN,u) also
N⊂ Y ∈ Φ in which case
Let
N
?
i=a, we define?b
i=aUi? Ua⊕···⊕Ub. A closed
p(r) ? {x ∈ Rn| |x|p≤ r}
II. PRELIMINARIES
We consider the following discrete-time linear time-
invariant (DLTI) system:
x+= Ax + Bu + w
(1)
Proceedings of the
44th IEEE Conference on Decision and Control, and
the European Control Conference 2005
Seville, Spain, December 12-15, 2005
MoB09.5
0-7803-9568-9/05/$20.00 ©2005 IEEE
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where x ∈ Rnis the current state, u ∈ Rmis the current
control input, x+is the successor state and w is the additive
and bounded disturbance. The system is subject to hard
constraints:
(x,u,w) ∈ X × U × W
(2)
The sets U and W are compact, the set X is closed; each
set is convex and contains the origin as an interior point. We
also define the corresponding nominal system:
z+= Az + Bv,
(3)
where z ∈ Rnis the current state, v ∈ Rmthe current control
action and z+the successor state of the nominal system. We
make the standing assumption that:
Assumption 1: The couple (A,B) is controllable.
We first recall a few standard definitions in set invariance
theory [1].
Definition 1: A set Ω ⊂ Rnis control invariant (CI) for
system x+= f(x,u) and constraint set (X,U) if Ω ⊂ X and,
for all x ∈ Ω, there exists a u ∈ U such that f(x,u) ∈ Ω.
A set Ω ⊂ Rnis robust control invariant (RCI) for system
x+= f(x,u,w) and constraint set (X,U,W) if Ω ⊆ X and,
for all x ∈ Ω, there exists a u ∈ U such that f(x,u,w) ∈ Ω
for all w ∈ W.
Definition 2: A set Ω ⊂ Rnis positively invariant (PI)
for system x+= f(x) and constraint set X if Ω ⊂ X if, for
all x ∈ Ω, f(x) ∈ Ω.
A set Ω ⊂ Rnis robust positively invariant (RPI) for
system x+f(x,w) and constraint set (X,W) if Ω ⊆ X and
if, for all x ∈ Ω, f(x,w) ∈ Ω for all w ∈ W.
Robust control invariance is illustrated in Figure 1.
x ∈ Ω
Ω ⊆ X
{f(x, u1, w) | w ∈ W} ⊆ Ω, u1∈ U
{f(x, u2, w) | w ∈ W} ?⊆ Ω, u2∈ U
Fig. 1.Illustration of Robust Control Invariance
If a given set Ω is RCI for system (1) and constraint set
(X,U,W), then there exists a control law ν : Ω → U such
that the set Ω is a RPI for the system x+= Ax+Bν(x)+w
and constraint set (Xν,W) where
Xν? X ∩ {x | ν(x) ∈ U}.
(4)
A suitable control law ν(·) is a selection from set valued
map U(x) defined for each x ∈ Ω by:
U(x) ? {u ∈ U | Ax + Bu + w ∈ Ω, ∀w ∈ W}
(5)
In other words, the control law ν(·) is any control law
satisfying:
ν(x) ∈ U(x), x ∈ Ω
(6)
Similar statement may be made for control and positively
invariant sets.
Given an initial state x ∈ Ω, the set of possible state
trajectories (each trajectory corresponding to a particular
realization of the disturbance process) lies in the ‘exact
tube’ {X0,X1,...} defined by the following set recursion
for i ∈ N+:
Xi(x) ? {Ay + Bν(y) + w | y ∈ Xi−1(x), w ∈ W} (7)
with X0(x) ? {x}. The set sequence {Xi(x)} is the exact
‘tube’ containing all the possible state trajectory realizations
due to the uncertainty and it contains the actual state trajec-
tory corresponding to a particular uncertainty realization. If
x ∈ Ω, the sets Xi(x), i ∈ N satisfy Xi(x) ⊆ Ω,∀i ∈ N
because x ∈ Ω and Ω is RPI for the system x+= Ax +
Bν(x)+w and constraint set (Xν,W). The shapes of the sets
Xi(x), i ∈ N change with time i and they are, in general,
complex geometrical objects (depending on the properties
of the couple (A,B), control law ν(·) and the geometry of
constraint set (X,U,W)). In the sequel we demonstrate that
it is possible to generate a tube (sequence of state sets) with
elements of fixed, simple shape that bounds the exact tube
and possesses robust control invariance properties.
First, we introduce the concepts of set robust positive
invariance and set robust control invariance:
Definition 3: A set of sets Φ is set robust positively
invariant (SRPI) for system x+= f(x,w) and constraint
set (X,W) if, for any set X ∈ Φ, (i) X ⊆ X and, (ii)
X+= f(X,W) ? {f(x,w) | x ∈ X, w ∈ W} ⊆ Y for
some set Y ∈ Φ.
Definition 4: A set of sets Φ is set robust control invariant
(SRCI) for system x+= f(x,u,w) and constraint set
(X,U,W) if, for any set X ∈ Φ, (i) X ⊆ X and, (ii)
there exists a policy θX :
f(X,θX,W) ? {f(x,θX(x),w) | x ∈ X, w ∈ W} ⊆ Y
for some set Y ∈ Φ.
In the sequel we consider the simple case when each
element X of Φ has the form z⊕R, i.e. Φ ? {z⊕R | z ∈ Z}
and refer to R as the shape of X. Note that any arbitrary
X ∈ Φ satisfies that X ⊆ Z ⊕ R by definitions of Φ and
Minkowski set addition. An appropriate illustration of set
robust control invariance is given in Figure 2.
It follows from the definition of Φ and the fact that R is
fixed that each X ∈ Φ is parameterized by z; accordingly
we use Xzto denote z ⊕ R.
X → U such that X+=
III. A FAMILY OF SET ROBUST CONTROL INVARIANT
SETS
As stated above, we restrict attention to a set of sets
defined as follows:
Φ ? {z ⊕ R | z ∈ Z}
(8)
where R ⊂ Rnand Z ⊂ Rn. We are interested in char-
acterizing all those Φ that are set robust control invariant.
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Z ⊕ R
Z
f(X, θX, W) ⊆ Y ∈ Φ
X ∈ Φ
Fig. 2. Illustration of Set Robust Control Invariance
Thus the sets X and Y in definitions 3 and 4 have the form
X = z1⊕ R and Y = z2⊕ R with z1,z2∈ Z.
Assumption 2:
(i) The set R is RCI for system (1) and
constraint set (αX,βU,W) where (α,β) ∈ [0,1) ×
[0,1).
(ii) The control law ν : R → βU is such that R is RPI
for system x+= Ax + Bν(x) + w and constraint set
(Xν,W), where Xν= αX ∩ {x | ν(x) ∈ βU}.
Note that the control law ν(·) in assumption 2 (ii) exists by
assumption 2 (i). Let the sets Uν, Z and V be defined by:
Uν? {ν(x) | x ∈ R},
(9)
Z ? X ? R, V ? U ? Uν
(10)
Assumption 2 implies that the set Z and V are non-empty
sets and contain the origin in their interiors. We also assume:
Assumption 3:
(i) The set Z is a CI set for the nominal
system (3) and constraint set (Z,V).
(ii) The control law ϕ : Z → V is such that Z is PI for
system z+= Az+Bϕ(z) and constraint set Zϕ, where
Zϕ? Z ∩ {z | ϕ(z) ∈ V}.
Existence of the control law ϕ(·) in assumption 3 (ii) is
guaranteed by assumption 3 (i).
We can now establish the following result:
Theorem 1: Suppose that Assumptions 2 and 3 are sat-
isfied. Then Φ ? {z ⊕ R | z ∈ Z} is set robust control
invariant for system x+= Ax + Bu + w and constraint set
(X,U,W).
Proof: Let X ∈ Φ, then X = z⊕R for some z ∈ Z so
that X ⊆ Z ⊕ R. By assumption 3, Z ⊆ Z so that, by (10),
X ⊆ Z ⊕ R ⊆ Z ⊕ R ⊆ X. For every x ∈ X we have
x = z + y, where y = yz(x) ? x − z ∈ R. Let the control
law θz: z⊕R → U be defined by θz(x) ? ϕ(z)+ν(y), y =
yz(x) and let u = θz(x). Then the successor state to x is
x+? Ax+Bθz(x)+w = A(z+y)+B(ϕ(z)+ν(y))+w =
z++Ay +Bν(y)+w where z+? Az +Bϕ(z). It follows
from assumption 3 that z+∈ Z and, by assumption 2, that
Ay + Bν(y) + w ∈ R, ∀(y,w) ∈ R × W. We conclude
that x+= Ax + Bθz(x) + w ∈ Y, ∀(y,w) ∈ R × W where
Y ? z+⊕R ∈ Φ. That θz(x) ∈ U for all x ∈ X follows from
assumptions 2 and 3, since ϕ(z) ∈ V = U ? Uν, ∀z ∈ Z
and ν(y) ∈ Uν, ∀y ∈ R so that θz(x) ∈ (U ? Uν) ⊕ Uν⊆
U, ∀x ∈ X and every X ∈ Φ.
The (local) control law θz: Xz→ U where Xz= z⊕R ∈
Φ is defined in the proof of Theorem 1:
θz(x) = ϕ(z) + ν(yz(x)), yz(x) ? x − z
(11)
Theorem 1 exploits linearity and state decomposition as
illustrated in Figure 3.
z
yz(x)
x
z+
y+
z(x)
x+
X = z ⊕ R, z ∈ Z
X+= z+⊕ R, z ∈ Z
Fig. 3.Exploiting Linearity – Theorem 1
Generally, there exists an infinite number of set robust
control invariant sets Φ, since, given a set R satisfying
assumption 2 (i), there exists an infinite number of CI sets
Z satisfying assumption 3 (i). Our attention is therefore
restricted to important cases such as the minimal and the
maximal set robust control invariant set for a given set R
satisfying assumption 2 (i). Before proceeding we need to
define the maximal control invariant set.
Definition 5: A set Ωmax ⊂ Rnis the maximal control
invariant set for the system x+= f(x,u) and constraint
set (X,U) if Ωmax is control invariant set for the system
x+= f(x,u) and constraint set (X,U) and Ωmaxcontains
all control invariant sets for the system x+= f(x,u) and
constraint set (X,U).
We now introduce:
Definition 6: Given a RCI set R satisfying assump-
tion 2 (i), a set Φmax = {z ⊕ R | z ∈ Z} is a maximal
set robust control invariant (MSRCI) set for system x+=
Ax+Bu+w and constraint set (X,U,W) if Z is the maximal
control invariant set satisfying assumption 3 (i).
In order to define the minimal set robust control invariant
set for a given set R satisfying assumption 2 (i), we define:
ρ(Z) ? sup
z∈Z|z|p
(12)
Any equilibrium point for the difference equation x+=
f(x,u), (¯ x = f(¯ x, ¯ u)) is a minimal, control invariant set the
system x+= f(x,u) and constraint set (X,U) providing
that (¯ x, ¯ u) ∈ (X,U); however one can always extract the
equilibrium point such that |¯ x|pis minimal.
Definition 7: Given a RCI set R satisfying assump-
tion 2 (i), a set Φmin = {z ⊕ R | z ∈ Z} is a minimal
set robust control invariant (mSRCI) set for system x+=
Ax + Bu + w and constraint set (X,U,W) if the set Z
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is a control invariant set satisfying assumption 3 (i) and
Z minimizes ρ(Z) over all control invariant sets satisfying
assumption 3 (i) (i.e. Z is contained in the minimal p–norm
ball).
The following observation is a direct consequence of
Definitions 6 and 7.
Proposition 1: Let a set R satisfying assumption 2 (i) be
given. Then: (i) the minimal set robust control invariant set
is Φmin = 0 ⊕ R and, (ii) the maximal set robust control
invariant set is Φmax= {z ⊕R | z ∈ Zmax}, where Zmaxis
the maximal control invariant set satisfying assumption 3 (i).
Proof of this observation follows directly from Defini-
tions 6 and 7, the facts that {0} is a control invariant set
satisfying assumption 3 (i), ρ({0}) = 0, and Zmax, the
maximal control invariant set satisfying assumption 3 (i),
exists and is unique [14], [20].
If W = {0}, then R = {0} satisfies assumption 2 (i), so
that {0} is the minimal and Zmax the maximal set robust
control invariant set for the system x+= Ax + Bu + w
and constraint set (X,U,{0}). The set Zmax, in this case,
corresponds to the maximal control invariant set for the
system x+= Ax + Bu and constraint set (X,U).
IV. DYNAMICAL BEHAVIOR OF X ∈ Φ ? {z ⊕ R | z ∈ Z}
Weconsidernowa
{Xz0,Xz1,Xz2,...}, Xzi? zi⊕ R, where the sequence
{z0,z1,z2...} satisfies:
tube(sequenceofsets)
zi+1= Azi+ Bϕ(zi), z0∈ Z.
(13)
It follows from assumption 3 and the fact z0∈ Z that that
zi∈ Z for all i ∈ N implying that Xzi∈ Φ for all i ∈ N.
Next we consider a sequence of sets {Yi} with initial set
Y0 ⊆ Xz0∈ Φ generated using the control policy πz0?
{θz0,θz1,θz2,...} where, for each z ∈ Z, θzis specified as
in Theorem 1 (see (11)). The sets {Yi} satisfy the following
difference equation for i ∈ N+:
Yi? {Ax + Bθzi(x) + w | x ∈ Yi−1, w ∈ W}
(14)
with initial condition Y0⊆ Xz0. The tube {Yi} is a forward
reachable tube [14], [18]; any trajectory of the system x+=
Ax + Bu + w with initial state x ∈ Y0, control policy πz0
and admissible disturbance sequence lies in this tube.
Proposition 2: Suppose assumptions 2 and 3 hold. The
sequences {Yi} and {Xzi} satisfy
Yi⊆ Xzi, ∀i ∈ N.
Suppose, for some i ∈ N, Yi ⊆ Xzi. By
Theorem 1, Yi+1 ⊆ Xzi+1. Since Y0 ⊆ Xz0, the desired
result follows by induction.
We now make the assumption:
Assumption 4: The origin is exponentially stable for the
system z+= Az + Bϕ(z) with a region of attraction Z,
where Z and ϕ(·) satisfy assumption 3.
Definition 8: A set R is robustly exponentially stable for
x+= f(x,w), w ∈ W, with a region of attraction S if there
exists a c > 0 and a γ ∈ (0,1) such that any solution {xi}
of x+= f(x,w) with initial state x(0) ∈ S, and admissible
(15)
Proof:
disturbance sequence {wi} (wi∈ W for all i ∈ N) satisfies
d(x(i),R) ≤ cγid(x(0),R) for all i ∈ N.
Theorem 2: Suppose assumptions 2, 3 and 4 are satisfied
and that R is compact. Then R is robustly exponentially
stable for the system x+= Ax + Bθzi(x) + w where {zi}
satisfies (13) with any initial state z0such that x0∈ z0⊕R.
The region of attraction is Z ⊕ R.
Proof:
If x ∈ Xz ? z ⊕ R, d(x,R) ≤ |z|. Since
xi ∈ Xzifor all i ∈ N and zi → 0 exponentially, it
follows that d(xi,R) → 0 exponentially. Hence R is robustly
exponentially stable for the system x+= Ax+Bθzi(x)+w
with a region of attraction Z ⊕ R.
An illustration of Theorem 2 is given in Figure 4.
Z ⊕ R
Z
Xz1s.t. Y1⊆ Xz1
Y0⊆ Xz0
Fig. 4.Illustration of Theorem 2
V. CONSTRUCTIVE SIMPLIFICATIONS
Here we provide possible choices for the sets R and Z
and the corresponding control laws.
A. A set R and feedback control law ν(·) satisfying assump-
tion 2
It is desirable to confine the spread of trajectories due to
the bounded disturbance. This can be achieved if R is chosen
to be minimal in some sense. The set R can be constructed
using recent results established in [21]–[23].
1) Case I: A possible choice for ν(·) is ν(x) = Kx. The
corresponding controlled nominal system is:
x+= AKx + w, AK? (A + BK),
(16)
where K is chosen so that all the eigenvalues of AK are
strictly inside the unit disk. In the absence of constraints,
the minimal RPI (mRPI) set F∞ (for the system (16)
and constraint set (Rn,W)) exists, is unique, compact and
contains the origin [24, Sect. IV]; it is given by
F∞=
∞
?
i=0
Ai
KW.
(17)
Thus F∞ is mRPI for system (16) and constraint set
(XK,W), XK ? X ∩ {x | Kx ∈ U} if F∞ ⊆ XK. It is
impossible, in general, to obtain an explicit characterization
of the mRPI set F∞except when AKis nilpotent or As
αI for (α,s) ∈ [0,1) × N [9], [24]. In [21] a method for
K=
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computation of an ε (ε > 0) outer RPI approximation of F∞
is given:
Theorem 3: [21] If 0 ∈ interior(W), then for all ε > 0,
there exists ζ ∈ [0,1) and a corresponding integer s such
that the following set inclusions
As
KW ⊆ ζW and ζ(1 − ζ)−1Fs⊆ Bn
are true. Furthermore, if (18) is satisfied, then the set F(ζ,s)
defined by:
p(ε)
(18)
F(ζ,s)? (1 − ζ)−1Fs
s−1
?
j=0
(19)
Fs?
Aj
KW, i ∈ N+
(20)
is an RPI set for the system (16) and constraint set (Rn,W)
such that F∞⊆ F(ζ,s)⊆ F∞⊕ Bn
This result can be extended to case when the origin is in the
relative interior of W [23].
Thus, the first suitable candidate for a set R satisfying
assumption 2 is the set F(ζ,s)defined in (19) providing that
F(ζ,s)⊆ XK.
2) Case II: A significantly improved result appears
in [22], [23] where it is shown that a linear programming
problem can be posed the solution of which yields a piece-
wise affine control law ν0(·), parameters α0,β0, and a RPI
set R for system x+= Ax+Bν0(x)+w and constraint set
(Xν0,W) where Xν0 ? α0X ∩ {x | ν0(x) ∈ β0U}. If α0
and β0lie in (0,1), assumption 2 is satisfied.
p(ε).
B. A set Z and feedback control law ϕ(·) satisfying assump-
tion 3
1) Case I: Given a set R and control law ν(·) satisfying
assumprion 2, a set Z and control law ϕ(·) satisfying
assumption 3 can be chosen as follows. The control law
ϕ(·) can be chosen to be any exponentially stabilizing linear
state feedback control law for system z+= Az + Bv; thus
ϕ(z) = Kz and a suitable choice for Z is any positively
invariant set or system z+= (A + BK)z and the (tighter)
constraint set specfified in assumption 3. For any set X ∈ Φ
(X = Xz = z ⊕ R for some z ∈ Z), the corresponding
control law is θz(·) is then defined by:
θz(x) = Kz + ν(yz(x)), x ∈ Xz= z ⊕ R
(21)
with yz(x) ? x − z.
2) Case II: Further simplification is obtained if the control
law ν(·) is linear (ν(x) ? K1x) and the set R is chosen
to be robust positively invariant set for the system x+=
(A + BK1)x + w and constraint set (Xν,W). In this case
the control law ϕ(·) can be chosen to be any exponentially
stabilizing linear state feedback control law (ϕ(x) ? K2x)
and the set Z is chosen to be positively invariant set for the
system z+= (A+BK2)z and constraint set Zϕ? Z∩{z |
ϕ(z) ∈ U?Uν} where ν(·) is the control law defined in Case
I in Subsection V-A. For any set X ∈ Φ (X = Xz= z ⊕ R
for some z ∈ Z), the corresponding control law is θz(·) is:
θz(x) = K2z + K1yz(x), x ∈ Xz= z ⊕ R
(22)
with yz(x) ? x − z.
3) Case III: The simplest but most conservative case is
when the control law ϕ(·) is chosen to be identical to the
control law ν(·) which is chosen to be any exponentially
stabilizing linear state feedback control law (ν(x) = Kx).
In this case the set Z can be chosen to be any positively
invariant set Z for the system z+= (A+BK)z and tighter
constraint set specified in assumption 3.
For any set X ∈ Φ (X = Xz= z ⊕ R for some z ∈ Z),
the corresponding control law is θz(·) is then defined by:
θz(x) = Kz + Kyz(x) = Kx, x ∈ Xz= z ⊕ R
(23)
with yz(x) ? x − z.
4) Case IV: Further improvement is obtained if the feed-
back control law ϕ(·) is obtained by exploiting the robust
model predictive control schemes discussed in more detail
in [11], [12] and the set Z is defined as in [11], [12].
C. Ellipsoidal Sets
The theory outlined above does not require the sets R
and Z to have a particular shape but merely to possess
certain properties specified in assumptions 2 and 3; these
properties are illustrated in Figure 5 for ellipsoidal sets R
and Z. Sets R and Z with ellipsoidal shape can be computed
by solving Linear Matrix Inequalities, exploiting standard
results in [13], [17], [25]–[27].
−10−8−6−4−20246810
−20
−15
−10
−5
0
5
10
15
20
Z ⊕ R
Z
X0∈ Φmax
X∞∈ Φmin= R
x1
x2
Fig. 5. Sample Set Trajectory for a set X0∈ Φ – Ellipsoidal Sets
VI. ILLUSTRATIVE EXAMPLE
To illustrate and enable simple visualization of our results
we consider a simple, second order, linear discrete time
system defined by:
x+=
?
1
0
0.2
1
?
x +
?
0
1
?
u + w,
(24)
which is subject to the control constraints:
u ∈ U ? { u ∈ R | − 2 ≤ u ≤ 2}
(25)
and state constraints:
x ∈ X ? { x = (x1,x2)?∈ R2| − 10 ≤ x1≤ 1,
− 10 ≤ x2≤ 10}
(26)
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