Constrained Linear Systems with Hard Constraints ... - Semantic Scholar

Constrained Linear Systems with Hard Constraints and Disturbances : An Extended Command Governor with Large Domain of Attraction Elmer G. Gilbert a, Chong-Jin Ong b a b

Department of Aerospace Engineering, University of Michigan, Ann Arbor

Department of Mechanical Engineering, National University of Singapore and Singapore-MIT Alliance, 117576, Singapore

Abstract This paper considers a nonlinear feedback control policy that is an extension of those provided by command governors and reference governors. As in these control approaches it applies to discrete-time linear systems with hard constraints and set bounded disturbances. The control policy retains the main properties of traditional governors, such as straightforward direct implementation and finite-settling-time response to arbitrarily specified set points. Its principal advantage over traditional governors is a significantly larger domain of attraction, that may compete in size with those obtained by dynamic programming. A procedure for obtaining large reductions in the number of online computations is given that applies to both command governors and extended command governor. Connections to an interesting model predictive control strategy are made. Numerical examples illustrate advantageous features of the proposed approach.

Key words: Command governor, reference governor, hard constraints, disturbance input, domain of attraction, robust invariant set, constrained systems with disturbances

1

Introduction

Traditional command governors (CGs) or reference governors (RGs) offer a useful alternative to model predictive control (MPC) for feedback control of plants that are linear systems with hard constraints. They apply without great complication to plants with disturbances and have other important advantages. Unlike MPC, the governor approach involves two performance criteria and implements the overall feedback control by a linear control loop within an outer nonlinear control loop. The first performance criterion concerns linear system characteristics near equilibrium conditions where hard constraints are inactive and usual design goals, such as good static and dynamic response and disturbance rejection, are paramount. Incorporating the performance criterion in a suitable design procedure, leads to a linear feedback controller (static or dynamic) and its corresponding linear closed-loop system. The second criterion is typically expressed by a quadratic cost function whose purpose is to drive the linear closed-loop system to near equilibrium conditions quickly while enforcing the hard constraints. Minimizing the cost function determines the nonlinear feedback function of system state and reference command that implements the outer closed-loop system. Key features of CGs and RGs include: straightforward computation of the nonlinear feedback function, finite settling time to reference commands that are constant set points, explicitly-defined large domains of attraction. This paper introduces an extended command generator (ECG) that retains desirable features of RGs and CGs while significantly increasing the size of the domain of attraction. As in prior literature on governors, the ECG applies to discrete-time linear plants with hard constraints and with or without set-bounded disturbances. Email addresses: [email protected] (Elmer G. Gilbert), [email protected] (Chong-Jin Ong).

Preprint submitted to ...

3 January 2012

To be more specific, suppose the equilibrium-based linear feedback controller has been designed and the corresponding asymptotically stable closed loop system is described by: x(t + 1) = Ax(t) + Bu(t) + Ex w(t), y(t) = Cx(t) + Du(t) + Ey w(t) ∈ Y, t ∈ Z + ,

(1) (2)

where x(t) ∈ Rn , u(t) ∈ Rm , w(t) ∈ W ⊂ Rℓ , Y ⊂ Rp and Z + is the set of non-negative integers. The state x(t) includes the states of the plant and the linear controller (if the linear controller is dynamic). Disturbance sequences, {w(t)}, belong to the set: W := {w(t) ∈ W : t ∈ Z + }. Hard constraints are imposed by (2) where the expression for y(t) takes into account the hard constraints on the plant and any additional hard constraints within the linear controller. More specialized constraints such as x(t) ∈ X and u(t) ∈ U can be expressed by (2). The nonlinear controller, defined implicitly as the solution of a constrained quadratic optimization problem, takes the form u(t) = U (x(t), r(t)),

(3)

where {r(t)} ∈ R := {r(t) ∈ Rm : t ∈ Z + } is an arbitrary reference sequence for the input u(t). The discussion that follows compares the RG and CG with the ECG. Emphasis is on the differences in the domains of attraction. Justification of stated results is deferred to subsequent sections. Several definitions associated with system (1)-(3) are needed in the discussion. They apply to a suitable set of constant reference commands or set points: r(t) ≡ rs ∈ Ω ⊂ Rm . For each rs ∈ Ω the maximal set of constraint admissible initial conditions is D(rs ) := {x(0) : (1) − (3) are satisfied, r(t) = rs ∀t ∈ Z + and {w(t)} ∈ W}.

(4)

Using x(0) = x and t = 0 in (1)-(4), (4) shows D(rs ) is robustly invariant (RI) for system (1)-(3). That is: x ∈ D(rs ) implies Ax + BU (x, rs ) + Ex w ∈ D(rs ) for all w ∈ W . The set D(rs ) is the maximal domain of attraction (MDA) if there exists an attractor set, F∞ (rs ), such that F∞ (rs ) ⊂ D(rs ) and x(0) ∈ D(rs ) implies x(t) → F∞ (rs ) as t → ∞ for all {w(t)} ∈ W. Generally, F∞ (rs ) is significantly smaller than D(rs ). System (1)-(3) has finite settling time if r(t) ≡ rs ∈ Ω and x(0) ∈ D(rs ) imply the existence of tf ∈ Z + such that u(t) = rs for all t ≥ tf . Consider first the situation where there is no nonlinear feedback and U (x, rs ) ≡ rs ∈ Ω in (3). Then u(t) ≡ rs and (1)-(2) is a well-studied system (Gilbert & Ong, 2008; Kolmanovsky & Gilbert, 1998; Blanchini & Miani, 2008; Kerrigan, 2000). Under reasonable assumptions, including the choice of Ω, it has a MDA, D(rs ) = O∞ (rs ) := {x(0) : (1), (2) are satisfied and u(t) = rs ∀t ∈ Z + and {w(t)} ∈ W} ⊂ Rn ,

(5)

F∞ (rs ) : = {Γrs + z : z ∈ F∞ } = {Γrs } + F∞ , Γ := (Im − A)−1 B,

(6)

where

is the attractor. Here, F∞ is defined by Minkowski set summation (Kolmanovsky & Gilbert, 1998), F∞ := lim Ft , Ft := t→∞

t−1 X

(Ai Ex W ), t ≥ 1, F0 = {0}

(7)

i=0

and Γrs is the equilibrium solution of (1) with w(t) ≡ 0. Command governors (Bemporad, Casavola & Mosca, 1997; Casavola, Mosca & Angeli, 2000; Casavola, Mosca & Papini, 2004), and reference governors (Gilbert, Kolmanovsky & Tan, 1995; Bemporad, 1998; Gilbert & Kolmanovsky, 1999; Gilbert & Kolmanovsky, 2002) have a larger MDA than O∞ (rs ). If Y is polyhedral, the standard CG generates U (x, r) by solving a quadratic programming problem with linear inequality constraints that depend in a simple way on x and r. For all rs ∈ Ω, the resulting system (1)-(3) has finite settling time and a MDA with attractor defined by (6). Unlike (4), the MDA is independent of rs and is given by D(rs ) = DCG := P∞ = ∪ρ∈Ω O∞ (ρ) for all rs ∈ Ω.

2

(8)

Clearly, O∞ (rs ) ⊂ DCG . In practice, DCG is typically much larger than O∞ (rs ). The RG approach includes a variety of implementations, both dynamic (Gilbert & Kolmanovsky, 1999), (Gilbert et al., 1995) and static (Ong & Gilbert, 2006a). Dynamic RG’s are arranged so they allow a full time step for the computation of U (x, r). Since they avoid the need for solving an online quadratic programming problem, RGs are simpler to implement than the CG. Appropriately initialized, RGs have the same MDA as CGs (D(rs ) = DRG = DCG ) and the same attractor. However, unlike the CG, large unexpected jumps in state may require RG re-initialization. The extended command governor (ECG) considered in this paper retains the key properties of the CG (finite settling time and attractor (6)) but has a MDA, D(rs ) = DECG for all rs ∈ Ω, that is larger than DCG . The basic idea is to append to (1)- (2) fictitious dynamics and a new input v(t): u(t) = C¯ x ¯(t) + v(t), ¯x(t). x ¯(t + 1) = A¯

(9) (10)

Here, x ¯(t) ∈ Rn¯ and A¯ is asymptotically stable, but otherwise A¯ and C¯ are chosen freely. The dynamics are fictitious in that x ¯ does not appear in U (x, r); (9) and (10) are used only to introduce certain sets that lead to the definition of U (x, r) and the proof of the main results. e∞ (ρ) and Pe∞ . For v(t) ≡ ρ ∈ Ω, The set DECG has an explicit characterization that depends on two other sets, O n+¯ n let the state of system (1), (2), (9), (10) be denoted by x e(t) = (x(t), x¯(t)) ∈ R . Then,

and

e∞ (ρ) : = {e O x(0) : (1), (2), (9), (10) are satisfied, v(t) = ρ ∀t ∈ Z + and {w(t)} ∈ W}, e∞ (ρ), Pe∞ : = ∪ρ∈Ω O DECG = {x : ∃ x ¯ s.t. (x, x¯) ∈ Pe∞ }.

(11)

(12)

(13)

Thus, DECG = Projx Pe∞ , the projection of Pe∞ ⊂ Rn × Rn¯ onto Rn . Since putting x¯(t) ≡ 0 in system (1), (2), (9), (10) gives it the form of system (1), (2), it is easy to confirm that DCG ⊂ DECG . Of course, the relative size of ¯ Example applications of the ECG are promising. The DCG compared to DECG depends on the choice of A¯ and C. increase in size of DECG over DCG may be appreciable, competing in some examples with feasibility sets generated by dynamic programming. In Remark 1 of their paper on CGs, Casavola et al. (2000) introduce the idea of virtual input sequences and mention their potential value in expanding the allowed set of initial conditions for the controlled system. They include a ¯ C) ¯ by (46). The contributions of this paper go feedback controller that corresponds, in our approach, to choosing (A, well beyond their brief introduction. They include: generality of the fictitious dynamics, completeness of theoretical results, controller simplification, comparisons with related control schemes, examples, and complete proofs of main results. The paper is organized as follows. This section concludes with basic assumptions and notations. Section 2 introduces needed properties of various sets. The nonlinear function U (x, r) is defined in Section 3 and Section 4 states a theorem on the key properties of the ECG. The number of computational operations associated with the evaluation of the feedback function U (x, r) may be large. This may be a crucial concern in some applications. Section 5 addresses this concern by introducing an alternative feedback controller that for both CG and ECG greatly reduces the number of operations. A second theorem shows that theoretical results of Section 4 are fully preserved with only a moderate reduction in the set of allowed initial conditions. Section 6 treats a particular choice of A¯ and C¯ that has an interesting MPC interpretation. Section 7 considers an example problem and compares, for it, the sets O∞ (rs ), DCG and DECG as n ¯ is varied. Additional comparisons are made with sets obtained in (Chisci, Rossiter & Zappa, 2001) and sets obtained by dynamic programming. Conclusions follow in Section 8. Appendices contain a precise statement of needed results from the prior literature and proofs of theorems. Mathematical notations are familiar. Positive semi-definite (definite) matrices Q ∈ Rn×n are denoted by Q  (≻)0. The n by n identity matrix is In and a prime denotes matrix transpose. For x ∈ Rn and x¯ ∈ Rn¯ , the notation (x, x¯) represents, depending on the context, either the pair x, x¯ or [x′ x ¯′ ]′ ∈ Rn+¯n . When k·k is a norm in Rn , Bn is the corresponding unit ball {x : kxk ≤ 1}. For Q ≻ 0, kxk2Q := x′ Qx. Prefixes cl and int denote respectively set closure and

3

interior. For x ∈ Rn , {x} ⊂ Rn is the corresponding singleton. Let X, Y ⊂ Rn , q ∈ R and Q ∈ Rm×n . Then: qX := {qx : x ∈ X}; QX := {Qx : x ∈ X}; X + Y := {x + y : x ∈ X, y ∈ Y }, the Minkowski sum; X ∼ Y := {z : z + y ∈ X, ∀y ∈ Y }, the Minkowski difference. If X is (bounded)[closed]{convex}<polyhedral> then QX and X ∼ Y , for any Y ⊂ Rn is (bounded)[closed]{convex}<polyhedral>. If both X and Y are (bounded)[closed]{convex}<polyhedral> the same properties apply to X + Y . The main assumptions on the problem data are: (A1) A and A¯ are asymptotically stable, (A2) 0 ∈ W and W is compact, (A3) Y is closed and convex. Additional, somewhat technical, assumptions appear as needed in Sections 2, 3 and 5. Assumptions (A1) and (A2) imply the limit in (7) exists, F∞ is compact and 0 ∈ F∞ (Kolmanovsky & Gilbert, 1998). 2

Preliminaries

To implement the nonlinear feedback function U (x, r) and state the main results for the ECG, it is necessary to define certain sets, describe their properties and indicate how concrete representations of them may be obtained. System (1), (2), (9), (10) with v(t) ≡ ρ ∈ Ω is equivalent to "

where x e(t) =

x e(t + 1) v(t + 1)

"

x(t) x ¯(t)

#

=

"

e B e A

0 Im

#"

x e(t) v(t)

#

+

"

ex E 0

#

w(t),

(14)

ex y(t) = C e(t) + Dv(t) + Ey w(t) ∈ Y, v(0) = ρ ∈ Ω, t ∈ Z + , #

e= , A

"

A B C¯ 0 A¯

#

e= , B

"

B 0

#

ex = , E

"

Ex 0

#

h i e = C DC¯ . , C

(15)

(16)

System (14), (15) has a maximal constraint admissible set, aug O∞ := {(x(0), x¯(0), ρ) : (14), (15) are satisfied ∀t ∈ Z + and {w(t)} ∈ W} ⊂ Rn+¯n+m .

(17)

e∞ (ρ) and Pe∞ This set is important because other sets of interest can be obtained from it. For instance, the sets O aug aug e e }. Also, of Section 1 are given by O∞ (ρ) = {(x, x ¯) : (x, x¯, ρ) ∈ O∞ } and P∞ = {(x, x¯) : ∃ρ ∈ Ω s.t. (x, x¯, ρ) ∈ O∞ DECG = X where aug aug X := {x : ∃(¯ x, ρ) s.t. (x, x¯, ρ) ∈ O∞ } = Projx O∞ = Projx Pe∞ .

(18)

aug can be computed by a recursion (Gilbert & Ong, 2008; Kolmanovsky & Gilbert, 1998) that is easily The set O∞ implemented when Y and Ω are polyhedral. The recursion generates sets Okaug ⊂ Rn+¯n+m for k = 0, 1, 2, · · · such aug aug that Ok+1 = Okaug implies O∞ = Okaug . However, additional assumptions must be imposed on system (14)-(15) so aug has properties needed in subsequent developments. that the recursion terminates finitely and that O∞

Supporting details, taken from the literature and expressed in the nomenclature of (14)-(15), are summarized in aug aug Appendix A. Specifically, conditions (i) - (iv) of the appendix imply: O∞ 6= ∅, the recursion is finite and O∞ is compact and convex. Condition (i) consists of assumptions (A1)-(A3). The remaining three conditions are imposed by assumptions (A4) -(A6). e = G where Condition (iv) is a requirement on Ω. Using expressions of (.1) and (.2) of Appendix A, Ye∞ = Y∞ and G Y∞ := Y ∼ Ey W ∼ CF∞ , G := CΓ + D.

(19)

Thus, condition (iv) is equivalent to assumption (A4): Ω is non-empty, compact, convex and satisfies the inclusion GΩ ⊂ intY∞ . Clearly, (A4) requires intY∞ 6= ∅ and Ω ⊂ Ωd where Ωd := {ρ : Gρ ∈ Y∞ }.

4

(20)

By (A3), (A4) and the properties of Minkowski subtraction, Y∞ is non-empty, closed and convex. Thus, Ωd is nonempty, closed and convex. See Gilbert & Ong (2008) for simple ways of choosing Ω. If Ωd is bounded, (A4) allows Ω to be an arbitrarily good approximation of Ωd . If Ωd is unbounded, approximation is not possible. However, a ˜∞ (ρ) 6= ∅ for all ρ ⊂ Ω. From practical, bounded choice of Ω is usually evident. Result (R3) of Appendix A shows O aug this it is easy to see that Ω = Projρ O∞ . e C) e is an observable pair, (A6) Oaug is bounded for some Conditions (ii) and (iii) of the appendix are: (A5) (A, k + k ∈ Z . Assumption (A5) is not really a restriction. Suppose (A5) is not satisfied. Then the constraint y(t) ∈ Y does not influence unobservable coordinates of system (14), (15). Thus, it is possible, without loss of generality to aug simplify system (14), (15) and have (A5) satisfied. The recursion for O∞ can be used to test for (A6) as it proceeds with increasing k. In the literature, (A6) is generally replaced by the assumption that Y is bounded. While this assumption together with (A5) implies (A6), its use has a disadvantage when, as may happen in the statement of applied problems, Y is unbounded. Then artificial hard constraints must be added to problem constraints to make aug Y bounded, unnecessarily complicating both the computation of O∞ and its characterization. aug Since O∞ is non-empty, compact and convex, X as defined by (18) is non-empty, compact and convex. The set aug Π(x) := {(¯ x, ρ) : (x, x¯, ρ) ∈ O∞ } ⊂ Rn¯ +m ,

(21)

aug appears in the definition of U (x, r). Since it is a section of O∞ , Π(x) is non-empty, convex and compact for all aug x ∈ X . By (21), (17) and Ω = Projρ O∞ , aug ⇒ ρ ∈ Ω. (¯ x, ρ) ∈ Π(x) ⇔ (x, x¯, ρ) ∈ O∞

(22)

aug aug aug is RI: for all w(0) ∈ W . Hence, O∞ implies (x(1), x¯(1), ρ) ∈ O∞ It is evident from (17) that (x(0), x¯(0), ρ) ∈ O∞ aug ¯x, ρ) ∈ Oaug ∀w ∈ W. (x, x¯, ρ) ∈ O∞ ⇒ (Ax + B(C¯ x ¯ + ρ) + Ex w, A¯ ∞

(23)

Further, by (15) evaluated at t = 0, aug (x, x¯, ρ) ∈ O∞ ⇒ Cx + D(C¯ x ¯ + ρ) + Ey w ∈ Y ∀w ∈ W.

(24)

Remark 1 By simple manipulations, the preceding results lead directly to an important conclusion. Suppose U (x, r) has the following property: for each (x, r) ∈ X × Rm there exists a pair (¯ x, ρ) ∈ Π(x) such that U (x, r) = C¯ x ¯ + ρ. Then system (1) - (3) is constraint admissible and X is RI. Specifically, for all w(t) ∈ W, x(t) ∈ X implies x(t+1) ∈ X and y(t) ∈ Y . When Y and Ω are polyhedral so are all sets in the preceding paragraphs. In particular, the recursive procedure for aug O∞ generates matrices Hx ∈ Rq×n , Hx¯ ∈ Rqׯn , Hρ ∈ Rq×m and h ∈ Rq such that aug O∞ = {(x, x ¯, ρ) : Hx x + Hx¯ x¯ + Hρ ρ ≤ h}.

(25)

To avoid unnecessary computations, it is important to minimize q, a requirement which is best achieved by eliminating redundant inequalities from each Okaug as the recursive process proceeds(Kolmanovsky & Gilbert, 1998; Gilbert & Ong, 2008). For each x ∈ Rn , (25) provides an explicit polyhedral expression, Π(x) = {(¯ x, ρ) : Hx¯ x ¯ + Hρ ρ ≤ h − Hx x}.

(26)

aug aug The set X = Projx O∞ is not so blessed. The numerical process for computing Projx O∞ from (25) is both expensive and prone to rounding errors, particularly when n + n ¯ + m is not small. If the purpose of X is to test if a given point x ∈ Rn belongs to X , the difficulty associated with the projection can often be circumvented by using the equivalence of x ∈ X and Π(x) 6= ∅. Since m + n ¯ is usually quite small, testing the condition Π(x) 6= ∅ is easy and numerically straightforward. It is only necessary to apply a linear programming feasibility test to the polyhedron (26).

5

3

Definition of U (x, r)

The nonlinear feedback takes the form U (x, r) = C¯ x ¯ + ρ, where the pair (¯ x, ρ) depends on x ∈ X and r ∈ Rm and is chosen so that it solves an optimization problem: minimize a norm squared,k(¯ x, ρ − r)k2 , subject to (¯ x, ρ) ∈ Π(x). Thus, by Remark 1, X is RI and constraint admissible. Minimizing k(¯ x, ρ − r)k2 has an obvious intuitive rationale: it drives (¯ x, ρ) toward (0, r) so that U (x, r) most closely approximates r. The norm squared is

k(¯ x, ρ)k2 := k¯ xk2S¯ + kρk2S (27) n ¯ ׯ n m×m ′¯¯ ¯ ¯ ¯ ¯ ¯ where S ∈ R and S ∈ R satisfy assumption: (A7) S, S ≻ 0, A S A − S ≺ 0. Since A is asymptotically ¯xk2¯ ≤ k¯ stable, a S¯ satisfying (A7) exists. Assumption (A7) ensures strict convexity of k(¯ x, ρ)k2 and implies kA¯ xk2S¯ S n ¯ for all x ¯∈R . Since Π(x) is non-empty, compact and convex and k(¯ x, ρ − r)k2 is continuous and strictly convex, the minimum of 2 k(¯ x, ρ − r)k on Π(x) exists and is unique. Thus, the pair (¯ xop (x, r), ρop (x, r)) := arg

min

(¯ x,ρ)∈Π(x)

k(¯ x, ρ − r)k2 .

(28)

is defined for all (x, r) ∈ X × Rm . Using (28), U (x, r) := C¯ x¯op (x, r) + ρop (x, r).

(29)

Thus, U : X × Rm → Rm . 4

Main Result

Outcomes of control law (29) are summarized in the following theorem. They make rigorous the properties of the ECG claimed in Section 1 and amplify them in modest but useful ways. Recall that under assumptions (A1)-(A6), X is non-empty, convex and compact. Theorem 1 Consider the system (1)-(3) with {r(t)} ∈ R, {w(t)} ∈ W, U (x, r) defined by (29) and x(0) ∈ X . Suppose assumptions (A1) - (A7) are satisfied. Then: (i) x(t), u(t) and y(t) are defined for all t ∈ Z + . (ii) y(t) ∈ Y and x(t) ∈ X for all t ∈ Z + . Suppose further there exists ts ∈ Z + such that r(t) = rs for all t ≥ ts . Define rs∗ = arg minr∈Ω kr − rs k2S . Then: (iii) There exists a tf ∈ Z + such that u(t) = rs∗ for all t ≥ tf . (iv) Given ǫ > 0, there exists a tǫ ∈ Z + such that x(t) ∈ F∞ (rs∗ ) + ǫBn for all t ≥ tǫ . Connections of the theorem with Section 1 are clear. Identities (13) and (18) and x(t) ∈ X show DECG = X , the MDA for system (1)-(3). Result (iv) defines precisely the convergence of x(t) to F∞ (rs∗ ). Part (iii) shows u(t) has finite settling time to rs∗ , the closest point in Ω to rs as measured by k · kS . Correspondingly, the tracking error e(t) := r(t) − u(t) equals to rs − rs∗ for all t ≥ tf . The proof of Theorem 1 is given in Appendix B. Parts (i)-(ii) are simple consequences of observations made in Sections 2 and 3. Details of proof for parts (iii) and (iv) are non-trivial. Major extensions of Theorem 1 are probably not possible. Perhaps something can be done for command sequences {r(t) ∈ Ω : t ∈ Z + } that vary slowly. Then it is intuitively reasonable to expect e(t) ≈ 0 after initial transients have died down. This intuition is supported by several simulation studies in which {r(t)} varies slowly and e(t) = 0 for t sufficiently large. While Theorem 1 does not require Y and Ω to be polyhedral, the computation of U (x, r) is much more complex for other classes of these sets, such as ellipsoids. Remark 2 With minor changes in setup, the results and proof of Theorem 1 apply to the standard CG. The changes aug a a go as follows. Replace O∞ by O∞ where O∞ is the maximal constraint admissible set for the system x(t + 1) = Ax(t) + Bu(t) + Ex w(t), x(0) = x u(t + 1) = u(t), u(0) = ρ ∈ Ω, y(t) = Cx(t) + Du(t) + Ey w(t) ∈ Y.

6

(30) (31) (32)

Specifically,

a O∞ := {(x(0), ρ) : (30) − (32) are satisfied ∀t ∈ Z + and {w(t)} ∈ W} (33) aug a and the results of Appendix A apply where in them O∞ is replaced by O∞ and the tildes are removed. Define a X = P∞ = Projx O∞ , U (x, r) = arg min kρ − rk2s , ρ∈Π(x)

(34)

a where Π(x) = {ρ : (x, ρ) ∈ O∞ }. Replace assumptions (A1), (A5), (A6) and (A7) respectively by A is asymptotically stable, (A,C) is an observable pair, Ok∞ is bounded for some k ∈ Z + and S ≻ 0.

5

Reduction of Online Computations

Online computational effort associated with U (x, r) is mostly determined by the quadratic programming problem (28). When Y and Ω are polyhedral, it has a constraint set Π(x), defined by (26). Thus, given x and r, its solution can be obtained by standard iterative procedures or explicitly, in x and r, by parametric programming (Bemporad, Morari, Dua & N., 2002). In both cases, computations generally increase with q, the row dimension of the matrices in (26). In turn the value of q depends on problem data, typically increasing rapidly as n and n ¯ increase and the spectral radius of A approaches 1 (as happens in sampled-data systems with high sample rates). This section describes modifications of ECG and CG that allows sizeable reductions in q. Before proceeding with the modifications, it is worth noting that q depends on the choice of Ω as allowed by assumption (A4). Usually, q decreases as Ω becomes smaller. Reductions in the size of Ω by only a few percent from Ωd often reduce q by a factor of 3 or more. Since Ωd is polyhedral, it is easy to construct a collection of polyhedral sets Ωi such that Ωi+1 ⊂ Ωi ⊂ Ωd and (A4) is satisfied. Thus, the tradeoff between the sizes of q and Ω can be aug . evaluated empirically by multiple computations of O∞ The modifications of ECG and CG are motivated by the idea described for RGs by Gilbert & Kolmanovsky (1999). a aug for CG by simpler, inner approximations. While the procedures for ECG and O∞ It exploits the replacement of O∞ described in (Gilbert & Kolmanovsky, 1999) for obtaining inner polyhedral approximations apply with essentially no change, the feedback strategy required to implement the modifications is very different. The strategy is developed aug by inner approximations here for the ECG and its extension to CG is easy - replace inner approximations of O∞ a and (¯ x, ρ) by ρ. of O∞ ˆ aug ⊂ Oaug . In the definitions of ˆ aug , a compact and convex set, denote an approximation of Oaug with O Let O ∞ ∞ ∞ ∞ aug aug ˆ Sections 2 and 3 replace O∞ by O∞ and use ”hat” accents to denote corresponding sets and functions. For ˆe ˆ ˆ aug }, O ˆ aug }, Pˆe ∞ := {(x, x¯) : ˆ aug , Π(x) := {(¯ x, ρ) : (x, x¯, ρ) ∈ O ¯) : (x, x¯, ρ) ∈ O example, Xˆ := Projx O ∞ (ρ) := {(x, x ∞ ∞ ∞ aug aug ˆ := Projρ O ˆ ˆ s.t. (x, x¯, ρ) ∈ O ˆ }, Ω and ∃ρ ∈ Ω ∞

∞

ˆ (x ¯op (x, r), ρˆop (x, r)) := arg

min

ˆ (¯ x,ρ)∈Π(x)

k(¯ x, ρ − r)k2 .

(35)

ˆ ˆ ⊂ Ω and Π(x) ˆ Clearly, Xˆ ⊂ X , Π(x) ⊂ Π(x), Ω 6= ∅ if and only if x ∈ Xˆ . The simple extension of (29), ˆ u(t) := C¯ x ¯op (x(t), r(t)) + ρˆop (x(t), r(t)),

(36)

ˆ ˆ does not work as a modified control law. If Π(x(t)) 6= ∅ it does follow from Π(x(t)) ⊂ Π(x(t)) and Remark 1 that ˆ y(t) ∈ Y and x(t + 1) ∈ X . But there is no guarantee that x(t + 1) ∈ Xˆ . Thus, Π(x(t + 1)) = ∅ is a possibility which results in u(t + 1) being undefined. ˆ A valid feedback control strategy must have a backup procedure that applies when Π(x(t)) = ∅. Let t > 0 and suppose ¯x(t − 1), ρ(t − 1)) ∈ (¯ x(t − 1), ρ(t − 1)) ∈ Π(x(t − 1)) and u(t − 1) = C¯ x ¯(t − 1) + ρ(t − 1). Then (22) and (23) imply (A¯ Π(x(t)) and by Remark 1, y(t) ∈ Y and x(t + 1) ∈ X . This observation is the basis for a feedback control strategy ˆ¯(t), ρ(t)), with backup. Unlike (3), its implementation becomes a dynamic system with state (x ˆ input (x(t), r(t)) and

7

ˆ : Rn¯ × Rm → Rm defined output u(t). To see this, introduce functions: Fˆ : Rn × Rm × Rn¯ × Rm → Rn¯ × Rm and U by ˆ ˆ¯op (x, r), ρˆop (x, r)) if Π(x) Fˆ (x, r, x¯, ρ) : = (x 6= ∅ ˆ = (¯ x, ρ) if Π(x) = ∅, ˆ (¯ U x, ρ) : = C¯ x ¯+ρ

(37) (38)

Then, the feedback control strategy with backup is given by ˆ (x ¯(t), ρ(t)) ˆ = Fˆ (x(t), r(t), A¯xˆ¯(t − 1), ρˆ(t − 1)) ˆ (x u(t) = U ¯ˆ(t), ρ(t)) ˆ

(39)

ˆ¯op (x(0), r(0)), ρˆop (x(0), r(0)), ˆ ¯(0), ρ(0)) ˆ = (x (x

(41)

(40)

With x(0) ∈ Xˆ and

it follows from the above observation that closed-loop system (1),(2),(39),(40) has a well-defined solution satisfying the conditions y(t) ∈ Y and x(t) ∈ X for all t ∈ Z + and {w(t)} ∈ W. Specifically, results (i) and (ii) of Theorem 1 are obtained. Unfortunately, results (iii) and (iv) are not achievable with (37)-(40). To obtain them, it is necessary to redefine ˆ Fˆ . This requires, for Π(x) 6= ∅, introduction of additional sets, Ψ(x, r) and H + (x, r) ⊂ Rn¯ × Rm . See Figure 1 ˆ¯op (x, r)k2¯ + kρˆop (x, r)k2S } for their geometrical interpretation. The sets Ψ(x, r) := {(¯ x, ρ) : k¯ xk2S¯ + kρ − rk2S ≤ kx S ˆ ˆ ˆ¯op (x, r), ρˆop (x, r)) whose normal is the gradient of and Π(x) are separated by the support hyperplane to Π(x) at (x v(¯ x, ρ) = k¯ xk2S¯ + kρ − rk2S evaluated at the same point. Thus, ¯ x−x ˆ ˆ¯op (x, r)) + (ˆ H + (x, r) = {(¯ x, ρ) : x ¯op (x, r)′ S(¯ ρop (x, r) − r)′ S(ρ − ρˆop (x, r)) ≥ 0}

(42)

ˆ is the corresponding ”upper” half space that contains Π(x). Let Θ(x, r, x ¯, ρ) : = 1 =0 =0

ˆ if Π(x) 6= ∅ and (¯ x, ρ) ∈ H + (x, r) ˆ if Π(x) 6= ∅ and (¯ x, ρ) ∈ / H + (x, r) ˆ if Π(x) = ∅.

Fig. 1. Schematic representation of Rn¯ × Rm and the half space H + (x, r).

8

(43)

Then the redefined function Fˆ is ˆ Fˆ (x, r, x¯, ρ) : = (x ¯op (x, r), ρˆop (x, r)) if Θ(x, r, x¯, ρ) = 1 = (¯ x, ρ) if Θ(x, r, x ¯, ρ) = 0.

(44)

ˆ ˆ¯op (x, r), ρˆop (x, r)), the computational expense Given a test for emptiness of Π(x) and a procedure for computing (x for evaluating Fˆ (x, r, x¯, ρ) is small. To complete the statement of the results corresponding to those in Theorem 1 requires an additional assumption: aug aug ˆ∞ (A8) O ⊂ O∞ is compact, convex and, for some ǫ > 0, satisfies the condition, ˆe e + Fe∞ + ǫBn+¯n ⊂ O ˆ {Γρ} ∞ (ρ), ∀ρ ∈ Ω.

(45)

e and Fe∞ . See Appendix A for the definitions of Γ

ˆ defined by (38) and Fˆ defined by (44). Suppose x(0) ∈ Xˆ , Theorem 2 Consider system (1), (2), (39)-(41) with U {r(t)} ∈ R, {w(t)} ∈ W, assumptions (A1)-(A8) are satisfied and rˆs∗ = arg minr∈Ωˆ kr − rs k2S . Then, results (i)-(iv) of Theorem 1 are satisfied when rs∗ is replaced by rˆs∗ . ˆ aug is a reasonably good approximation of Oaug and Theorem 2 is proved in Appendix C. In the usual situation, O ∞ ∞ F∞ is small in the sense that inclusion (R3) of Appendix A is satisfied conservatively. This makes it likely that ˆ aug is a good condition (45) is satisfied. See Appendix D for a specific procedure that tests condition (45). When O ∞ aug approximation of O∞ , Θ(x(t), r(t), xˆ ¯(t), ρ(t)) ˆ = 0 infrequently and system (1), (2), (39)-(41) has a response similar to that of system (1)-(3). Remark 3 As can be seen from the proof of Theorem 2, it is possible to replace (41) by the weaker condition that ˆ While this condition increases the allowed set of initial states beyond x(0) ∈ Xˆ , ˆ (x ¯(0), ρˆ(0)) ∈ Π(x(0)) and ρˆ(0) ∈ Ω. an explicit characterization of Π(x(0)) is required. The set Π(x(t)) is not needed for t > 0. ˆ aug well inside Oaug has an advantage. The larger gap between Remark 4 Although it makes Xˆ smaller, choosing O ∞ ∞ aug aug ˆ O∞ and O∞ gives the modified ECG robustness to changes generated by modelling errors in system (1), (2). ˆ aug , ˆ aug to be polyhedral. Other characterizations of O Remark 5 The hypotheses of Theorem 2 do not require O ∞ ∞ such as ellipsoids, deserve investigation. ˆ aug is conceptually The Gilbert & Kolmanovsky (1999) procedure for obtaining polyhedral approximations of O ∞ aug aug ˇ simple. It exploits two basic steps. The first step generates outer approximations O∞ ⊃ O∞ by selectively removing rows from the vector inequalities in (25). The selection of rows is based on successively finding rows that do the ˇ aug ≈ Oaug . Each eliminated row is determined by solving a sequence of linear least damage to the approximation O ∞ ∞ programming (LP) problems. The objective is to eliminate as many rows as possible while retaining a reasonably aug aug ˇ∞ good outer approximation. The second step shrinks O about a central point z in the interior of O∞ . Specifically, aug aug ˇ α is maximized subject to α(O∞ − {z}) + {z} ⊂ O∞ . The resulting αmax is determined by solving a LP. If ˇ aug − {z}) + {z} with 0 < α < αmax is a candidate for O ˆ aug . Finally, condition (45) must αmax > 0, the set α(O ∞ ∞ be checked. For additional details on the overall nature of the computations see Gilbert & Kolmanovsky (1999), aug ˆ = Ω. Note the simplifications that occur in Theorem 1 and including the feasibility of constructing O∞ so that Ω ˆ = Ω. Remark 3 when Ω The potential effectiveness of the modified ECG in reducing computational complexity is illustrated by results obtained in Kolmanovsky, Gilbert & Tseng (2009). This paper concerns nonlinear feedback control for the lateral dynamics of a 4-wheel vehicle, where u(t) is the front wheel angle and y(t) ∈ R is a normalized measure of differential loading on the rear wheels. Both wheels remain on the ground if −1 < y(t) < 1. The linearized disturbance-free model has the form (1),(2) where n = 4, m = 1, p = 1, Y = [−0.95, 0.95]. The control scheme was implemented using the MPC interpretation of ECG described in the next section with N = n ¯ = 3. Complexity of the resulting aug O∞ ⊂ R8 was initially extremely high. Since the corresponding set X ⊂ R4 included values of x(t) much larger than

9

those physically possible, additional variables were added to the specification of y(t) and Y so as to bound x(t). This aug decreased the complexity to q = 208. An interior approximation of O∞ for this more strongly constrained system aug aug ˆ∞ was then obtained. Although O was smaller than O∞ by only a few percent, the reduction in complexity was ˆ aug . Parametric programming, applied to quadratic significant: only 36 linear inequalities were required to define O ∞ ˆ ˆ programming problem with Π(x) defining U (x, r), was also considered. See Kolmanovsky et al. (2009) for details. 6

A MPC Interpretation

The ECG has a nice MPC interpretation that preserves the results of Theorem 1. It is obtained by setting 

 Im 0 · · · .. .. ..   h i . . .   ∈ RmN ×mN and C¯ = Im 0 · · · 0 ∈ Rm×mN . · · · 0 Im   0 ··· ··· 0

0 . . . A¯ =  0 

(46)

These choices of A¯ and C¯ together with v(t) ≡ ρ ∈ Ω, and x ¯=x ¯(0) = (u0 , · · · , uN −1 ), generate a control sequence of the form u(t) = ut + ρ, 0 ≤ t < N = ρ, t ≥ N.

(47)

aug It is easy to see (x, x ¯, ρ) ∈ O∞ if and only if (x, x¯, ρ) satisfies, for all {w(t)} ∈ W, the conditions:

x(σ + 1) = Ax(σ) + B(uσ + ρ) + Ex w(σ), x(0) = x, y(σ) = Cx(σ) + D(uσ + ρ) + Ey w(σ) ∈ Y, 0 ≤ σ ≤ N − 1, a (x(N ), ρ) ∈ O∞

(48) (49) (50)

a is defined by (30)-(33). and O∞

Constraints (48) - (49) are similar to those that appear in MPC formulations where the length of the horizon is N and (50) imposes a terminal state constraint. Various approaches exist for the complication introduced by w(σ). See, for example (Chisci et al., 2001; Mayne, Seron & Rakovi´c, 2005; Wang, Ong & Sim, 2009) and references therein. Here the approach is to consider the set of all outputs, y(σ) generated by (48) and (49) with x(0) = 0. Using (7), this set is given by Ey W + CFσ . Thus, the effect of {w(t)} ∈ W in (48) and (49) can be represented by replacing Y by Yσ := Y ∼ Ey W ∼ CFσ .

(51)

aug Specifically, (x, x¯, ρ) ∈ O∞ if and only if (x, x¯, ρ) satisfies the disturbance-free conditions:

x(σ + 1) = Ax(σ) + B(uσ + ρ), x(0) = x, y(σ) = Cx(σ) + D(uσ + ρ) ∈ Yσ , ∀ 0 ≤ σ ≤ N − 1, a (x(N ), ρ) ∈ O∞ ∼ (FN × {0}), ρ ∈ Ω.

(52) (53) (54)

The MPC formulation is completed by introducing the cost function used in (28). Set S¯ = diag S˘k where S˘k ⊂ Rm×m , k = 0, 1, · · · , N − 1 and consider the assumption: (A7′ ) S˘k , S ≻ 0, k = 0, 1, · · · , N − 1. Then by (46), (A7′ ) implies (A7) and (xop (x, r), ρop (x, r)) is defined by (u0,op (x, r), · · · , uN −1,op (x, r), ρop (x, r)) = arg

10

min

(u0 ,··· ,uN −1 ,ρ)

N −1 X k=0

u′k S˘k uk + (ρ − r)′ S(ρ − r)

(55)

where (u0 , · · · , uN −1 , ρ) satisfies (52) − (54).

(56)

¯ (3) and (29), From the definition of C, u(t) = U (x(t), r(t)) = u0,op (x(t), r(t)) + ρop (x(t), r(t)).

(57)

Since (52)-(57) implements a MPC strategy equivalent to ECG, Theorem 1 provides the following result. Theorem 3 Consider system (1),(2) with {r(t)} ∈ R, {w(t)} ∈ W and u(t) defined by (57). Let X := {x : ∃u0 , · · · , uN −1 , ρ such that (52)-(54) are satisfied} and assume x(0) ∈ X . Suppose assumptions (A1)-(A6) and (A7′ ) are satisfied. Then the results (i)-(iv) of Theorem 1 apply to the MPC strategy described by (52)-(57). Remark 6 The incorporation of ρ in (52)-(54) is atypical of formulations found in the MPC literature. It allows the MPC formulation of (55)-(57) to explicitly handle varying {r(t)} ∈ R. In addition, as Theorem 3 shows, the use of ρ leads to a large MDA, DECG = X . Remark 7 For the special case where r(t) ≡ 0, Chisci et al. (2001) describe a MPC scheme related to (52)-(57). In our notations, it requires the following changes: set ρ = 0 in (52) - (53), replace (54) by x(N ) ∈ O∞ (0). It follows that their domain of attraction, DCRZ , is a proper subset of DECG . By means of an example, they compare DCRZ with DCG . They also propose a supplementary control strategy that allows variable set points. However, the set of allowable set points is significantly smaller than Ω and they do not use a cost function, such as the one in (55), to achieve the results like those in Theorem 3. a Remark 8 When Y is polyhedral, the sets Yσ and O∞ ∼ (FN × {0}) are polyhedral. By using the support function of W they are easily determined without computing Fσ . See Gilbert & Ong (2008) and Kolmanovsky & Gilbert (1998).

¯ C¯ given by (46) it is not clear which method of online implementation is most computationally Remark 9 For A, efficient: feedback (3) with U (x, r) defined by (29) or the MPC formulation (52)-(57). The MPC formulation has the advantage that it can be implemented with available MPC software. On the other hand, the solution of quadratic program (28) can be obtained explicitly in x and r and it is possible to apply the important computational simplifications described in Section 5. 7

An Example

By means of an example problem, this section illustrates the relative extent of the domains of attraction: O∞ := aug . In addition, these sets are compared O∞ (0), DCG = ∪ρ∈Ω O∞ (ρ), DCRZ (Remark 7) and DECG = X = Projx O∞ with robust, constraint-admissible feasibility sets obtained by dynamic programming (DP). Unlike the domains of attraction, DP feasibility sets do not require the hard constraints to be satisfied for all sequences {w(t)} ∈ W. They take into full account the explicit evolution of {w(t)} and are the largest possible feasibility sets. Since DP is only used to construct feasibility sets, it is unnecessary to address here the various complex issues connected with the implementation of the optimal feedback policies (Mayne, 2001). The example problem is the one considered in (Chisci et al., 2001). It starts with an unstable plant to which a stabilizing feedback is applied. With the stabilizing feedback in place, the data for system (1) and (2) are: n = 2, m = 1, A = [0.3566 − 0.0922; −0.7434 0.2078], B = Ex = [1.0 1.0]′ , C = [−0.7434 − 1.0922], D = [1], Ey = [0], W = {w : |w| ≤ 0.12}, Y =: {y : |y| ≤ 1}. The matrices A¯ and C¯ are given by (46) so that the MPC interpretation of Section 6 applies with n ¯ = N . Assumptions (A1)-(A3), (A5)-(A6) are satisfied by the preceding data. Assumption (A7) is satisfied by choosing S = [1] and S¯ = IN . The set Ωd = {ρ|ρ ≤ 0.5887} and (A4) is satisfied by setting Ω = {ρ|ρ ≤ 0.5884}. Recursive computation yields characterization (25) with q = 28 when n ¯ = 12. In what follows, a superscript N is added to DECG and DCRZ to display their dependence on the length of a finite horizon. The j-step DP feasibility set, for a terminal set T ⊂ Rn , is defined by the recursion Qj+1 (T ) = Q+ (Qj (T )), Q0 (T ) = T where, for Ψ ⊂ Rn , Q+ (Ψ) = {x ∈ Rn : ∃u ∈ Rm s.t. Ax + Bu + Ex w ∈ Ψ and Cx + Du + Ey w ∈ Y ∀w ∈ W }.

11

Thus, QN (T ) is the set of states that can be robustly transferred to T in N steps or less by a state-dependent control sequence that satisfies (1) and (2). Figures 2 through 4 summarize the comparisons of the various sets. N Figure 2 shows DCG and DECG for increasing values of N . The MDA (O∞ ) for system (1) and (2) with u(t) = r(t) ≡ 0 is also included for comparison. It is much smaller than DCG , confirming the significant advantage of the CG and N RG feedback strategies. The set DECG ⊃ DCG displays the additional coverage provided by the ECG. 8 8 Figure 3 shows again the sets O∞ , DCG and DECG . They are compared with DCRZ , Q8 (O∞ ), Q20 (O∞ ) and Q8 (DCG ). 8 8 The set DCRZ is quite different from DCG , overlapping it but being far less wide. Intuitively, DECG can be viewed 8 as an appropriate combination of DCG with DCRZ . This interpretation is discussed in (Chisci et al., 2001), but a control strategy for implementing it is not given. Based on the MPC literature the most natural terminal set for a 8 DP feasibility set is O∞ . Surprisingly, Q8 (O∞ ) is only slightly larger than DCRZ . Only when N = 20 does QN (O∞ ) 8 become competitive with DECG . An alternative to using larger N is to combine DP with the CG by using a dualmode approach (Mayne, 2001) where the terminal set is DCG . This makes Q8 (DCG ) competitive with Q20 (O∞ ), 8 but not very different from DECG . These comparisons suggest that with respect to the size of domain of attraction, the ECG may compete quite well with DP feasibility sets. Since implementation of DP feedback controllers involves many complex issues this result is indeed encouraging. 3 Finally, Figure 4 compares DECG with Q3 (DCG ) for different disturbance sets. In particular, W in system (1) - (2) is replaced successively by {0}, W, 2W , and 3W . In the disturbance free case where W = {0}, it is easy to confirm 3 3 from the choices of A¯ and C¯ that DECG = Q3 (DCG ). For the disturbance set W , the difference between DECG and 3 Q3 (DCG ) is quite small and is within reason for 2W and 3W . For a disturbance set 4W , both DECG and Q3 (DCG ) are empty.

3

N = 12 N =8 N =4

2

N =2

x2

1

DCG

0

O∞ −1

−2

−3 −20

−15

−10

−5

0

x1

5

10

15

20

N Fig. 2. The sets of O∞ ,DCG and DECG for N = 2, 4, 6, 8, 12.

8

Conclusions

This paper considers the ECG feedback control strategy for linear systems with hard constraints, set bounded disturbances, and variable command inputs. The strategy yields a constraint admissible MDA significantly larger than those associated with traditional CGs and RGs. When its input commands settle to a constant value, the ECG has finite settling time and system state converges to an appropriate attractor set. Modified versions for both the CG and the ECG are introduced. For polyhedral sets Y they have the potential for producing large reduction in the number of online feedback computations. While the resulting set of initial conditions is somewhat smaller, the properties of constraint admissibility and finite settling time are preserved. The modified

12

3

Q8 (DCG ) 8 DECG

2

DCG 1

x2

O∞ 0

8 DCRZ

Q8 (O∞ )

−1

Q20(O∞ ) −2

−3 −20

−15

−10

−5

0

5

x1

10

15

20

8 8 Fig. 3. The sets O∞ , DCG , DECG , DCRZ , Q8 (O∞ ), Q20 (O∞ ) and Q8 (DCG )

. 4

3

2

{0}

x2

1

W

3W

0

2W

−1

−2

−3

−4 −25

−20

−15

−10

−5

0

x1

5

10

15

20

25

3 Fig. 4. The sets DECG (solid line) and Q3 (DCG ) (dashed line) for W replaced by {0}, W, 2W and 3W .

feedback controllers are dynamic systems whose characterizations are based on inner, non-RI approximations of the aug set O∞ . A MPC interpretation of ECG is given that proves useful in making connections with prior literature. In an example problem, DECG is significantly larger than both DCG and DCRZ , the MDA for the control strategy proposed in (Chisci et al., 2001). Further, DECG closely approximates Q(DCG ), the DP feasibility set with terminal set DCG . The main theoretical results are Theorems 1 and 2. Self-contained proofs are given for both. Proof of Theorem 2 depends crucially on the ideas used in the proof of Theorem 1. aug Appendix A: Results on O∞

13

This appendix reviews key properties of sets defined in Section 2. For system (14),(15), they are consequence of material found in section 7 of Kolmanovsky & Gilbert (1998) or more directly in Gilbert & Ong (2008). P ek E ex W ⊂ e := (In+¯n − A) e −1 B, e G e := C eΓ e + D, Fe∞ := ∞ A Using notations introduced in sections 1 and 2 define: Γ k=0 aug i n+¯ n e e Fek , O eA e (e e + (C eΓ e + D)ρ ∈ eFe∞ Yek = Y ∼ Ey W ∼ C := {(e x, ρ) : ρ ∈ Ω, C x − Γρ) R , Y∞ := Y ∼ Ey W ∼ C k e Yi , i = 0, · · · , k}.

e C) e is observable, (iii) Oaug is bounded for some k ∈ Z + , Suppose : (i) assumptions (A1)-(A3) are satisfied, (ii) (A, k e ⊂ intYe∞ . Then: (R1) there exists kˇ ∈ Z + such (iv) Ω is non-empty, compact, convex and satisfies the condition GΩ aug aug e + Fe∞ + e that O∞ = Okaug 6= ∅, (R2) O∞ is compact and convex, (R3) there exists e ǫ > 0 such that {Γρ} ǫBn+¯n ⊂ ˇ aug e O∞ (ρ) := {e x : (e x, ρ) ∈ O } for all ρ ∈ Ω. ∞

Simple algebraic manipulations based on (16) show

e = CΓ + D, Ye = Y ∼ Ey W ∼ CF∞ , G e = [Γ 0]′ , Fe∞ = F∞ × {0} ⊂ Rn × Rm . Γ

(.1) (.2)

Appendix B : Proof of Theorem 1 By (¯ xop (x, r), ρop (x, r)) ∈ Π(x), (29), (1)- (3) and Remark 1, x(0) ∈ X implies x(t) ∈ X and y(t) ∈ Y for all t ∈ Z + and {w(t)} ∈ W. Thus, (¯ xop (x(t), r(t)), ρop (x(t), r(t)) and u(t) are defined for all t ∈ Z + and the proof of results (i) and (ii) is complete. Since r(t) = rs for all t ≥ ts and x(ts ) ∈ X , the proof of (iii) and (iv) can be done by assuming r(t) = rs for all t ∈ Z + . To simplify subsequent notations, let x ¯a (t) := x ¯op (x(t), r(t)),

ρa (t) := ρop (x(t), r(t)).

(.1)

To begin the proof of (iii), let V (t) := k(¯ xa (t), ρa (t)) − (0, rs )k2 = k¯ xa (t)k2S¯ + kρa (t) − rs k2S .

(.2)

¯xa (t − 1), ρa (t − 1)) ∈ Π(x(t)). This and (28) Because (¯ xa (t − 1), ρa (t − 1)) ∈ Π(x(t − 1)), (22) and (23) imply (A¯ ¯xa (t − 1)k2¯ + kρa (t − 1) − rs k2 . Assumption (A7) implies kA¯ ¯xa (t − 1)kS¯ ≤ k¯ show that V (t) ≤ kA¯ xa (t − 1)k2S . Thus, S S V (t) ≤ V (t − 1) and there exists a Vm ≥ 0 such that V (t) → Vm . It will now be shown that ¯xa (t − 1) − x kA¯ ¯a (t)k2S¯ + kρa (t − 1) − ρa (t)k2S ≤ V (t − 1) − V (t).

(.3)

This result is based on the following lemma. Lemma 4 Suppose Z ⊂ Rq is closed and convex, zs ∈ Rq , Q ∈ Rq×q is positive definite and zop = arg minz∈Z (z − zs )′ Q(z − zs ). Then kz − zop k2Q ≤ kz − zs k2Q − kzop − zs k2Q ∀z ∈ Z. (.4) Proof : Because Z is closed, zop exists. Write kz − zs k2Q = kz − zop − (zs − zop )k2Q = kz − zop k2Q − 2(z − zop )′ Q(zs − zop )+ kzs − zop k2Q . Then the necessary condition for the optimality of zop and gradient (z − zs )′ Q(z − zs ) = 2Q(z − zs ) imply −2(z − zop )′ Q(zs − zop ) is non-negative for all z ∈ Z. Thus, kz − zs k2Q ≥ kz − zop k2Q + kzs − zop k2Q for all z ∈ Z, which is equivalent to (.4).

14

¯xop (x(t − 1)), ρop (x(t − 1))), zop = (¯ In Lemma 4 let Z = Π(x(t)), z = (A¯ xop (x(t)), ρop (x(t))), zs = (0, rs ), Q = ¯ S). As noted below (.2), z ∈ Π(x(t)). Also, Q ≻ 0 and Π(x) is convex and compact. Thus, by (.1) diag(S, ¯xa (t − 1), ρa (t − 1)) − (¯ k(A¯ xa (t), ρa (t)k2 ≤ ¯xa (t − 1), ρa (t − 1)) − (0, rs )k2 − k(¯ k(A¯ xa (t), ρa (t)) − (0, rs )k2 .

(.5)

¯xk2¯ ≤ k¯ Using (27), (.2) and kA¯ xk2S¯ in this inequality, completes the proof of (.3). S ¯xa (t − 1) − x Since V (t) − V (t + 1) → 0, (.3) shows A¯ ¯a (t) → 0 and ρa (t − 1) − ρa(t) → 0. By writing the first result as ¯ ¯ it follows that x x ¯a (t) = A¯ xa (t − 1) + η(t − 1) where η(t) → 0 and exploiting the asymptotic stability of A, ¯a (t) → 0. Let ¯ xa (t + 1) − x ∆u(t) := u(t + 1) − u(t) = ρa (t + 1) − ρa (t) + C(¯ ¯a (t)). (.6) Thus, ∆u(t) → 0. Of course, this result alone does not show u(t) has a limit as t → ∞. It does lead to the following result. Given ǫ > 0, there exists a ta (ǫ) ∈ Z + such that x(t) ∈ {Γu(t)} + F∞ + ǫBn ∀t ≥ ta (ǫ).

(.7)

To confirm (.7) observe that the solution of (1) can be written as x(t) = xu (t) + xw (t), where xw (t) is the solution of (1) with u(t) ≡ 0 and xw (0) = 0 and xu (t) is the solution of (1) with w(t) ≡ 0 and xu (0) = x(0). It follows from (7) that xw (t) ∈ Ft ⊂ F∞ for all t ∈ Z + . Let ∆xu (t) = xu (t + 1) − xu (t). Then ∆xu (t + 1) = A∆xu (t) + B∆u(t) and ∆xu (0) = (A − In )x(0) + Bu(0). Since u(0) = C¯ x ¯(0) + ρ(0) and Π(x(0)) and X are compact, ∆xu (0) is bounded. Thus, ∆u(t) → 0 implies ∆xu (t) → 0. Since xu (t + 1) = Axu (t) + Bu(t) = xu (t) + ∆xu (t), xu (t) = Γu(t) − (I − A)−1 ∆xu (t) → Γu(t). This, together with xw (t) ∈ F∞ proves (.7). Recall (22) and (28). Thus, ρa (t) ∈ Ω. This, (.2) and the definition of rs∗ imply Vm ≥ V ∗ := krs∗ − rs k2S .

(.8)

In fact Vm = V ∗ . This key result is proved by contradiction. Specifically, for Vm > V ∗ it is possible to construct a sequence, {¯ xb (t), ρb (t)}, for t ∈ Z + , such that for sufficiently large t, (¯ xb (t), ρb (t)) ∈ Π(x(t)) and k(¯ xb (t), ρb (t)) − (0, rs )k2 < k(¯ xa (t), ρa (t) − rs )k2 = k¯ xop (x(t), r(t)), ρop (x(t), r(t)) − rs k2 .

(.9)

This result violates definition (28) and yields the contradiction. To construct the sequence, let x ea (t) := (¯ xa (t), ρa (t)) and x eb (t) = (¯ xb (t), ρb (t)). Define ℓ(t, λ) := (1 − λ)e xa (t) + λ(0, rs∗ ), Vb (t, λ) := kℓ(t, λ) − (0, rs )k2 .

(.10) (.11)

Thus, for 0 ≤ λ ≤ 1, ℓ(t, λ) determines points along the line segment with end points x ea (t) and (0, rs∗ ) and Vb (t, λ) + is the corresponding cost of the points. For each t ∈ Z , Vb (t, λ) is a convex quadratic function of λ, 0 ≤ λ ≤ 1 with an upper bound (1 − λ)V (t) + λV ∗ . Let λm (t) be determined by (1 − λm (t))V (t) + λm (t)V ∗ = Vm . See Figure .1. Since V (t) ≥ Vm > V ∗ , it is easy to confirm that 0 ≤ λm (t) ≤ 1 and λm (t) → 0 as t → ∞. There is no guarantee that λm (t) > 0 for all t ∈ Z + , it is possible that there exists a t∗ such that V (t∗ ) = Vm and λm (t∗ ) = 0. To complete the remaining arguments, it is necessary to have a λb (t) such that λb (t) > 0 for all t ∈ Z + and λb (t) → 0. Thus, let λb (t) := λm (t) + δ(t) where {δ(t)} is a sequence such that δ(t) > 0 for all t ∈ Z + and δ(t) → 0. Next define (t,λ) x eb (t) = ℓ(t, λb (t)). The slope of the upper bound is V ∗ − V (t) < V ∗ − Vm < 0. Thus, dVbdλ |λ=0 < 0 and there exists a tm such that Vb (t, λb (t)) = ke xb (t) − (0, rs )k2 < Vm , ∀t ≥ tm .

(.12)

Also, from (.10), x ¯a (t) − x ¯b (t) → 0 and ρa (t) − ρb (t) → 0. This, x¯a (t) → 0 and u(t) = ρa (t) + C¯ x ¯a (t) imply u(t) − ρb (t) → 0. Further, λb (t) → 0, ρa (t) ∈ Ω, rs∗ ∈ Ω and ρb (t) = (1 − λb (t))ρa (t) + λb (t)rs∗ imply the existence of tm such that ρb (t) ∈ Ω for all t ≥ tm .

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Fig. .1. Dependence of Vb (t, λ) (solid line) and its upper bound (dash line) on λ.

Result (.7), together with u(t)− ρb (t) → 0 shows there exists tb (ǫ) such that ǫ > 0 implies x(t) ∈ {Γρb (t)} + F∞ + ǫBn for all t ≥ tb (ǫ). From this and (.2) of Appendix A, it follows that e b (t)} + Fe∞ + ǫBn¯ +n ∀t ≥ tb (ǫ). (x(t), x¯b (t)) ∈ {Γρ

(.13)

e∞ (ρb (t∗ )). (x(t∗ ), x ¯b (t∗ )) ∈ O

(.14)

Set t∗ = max{tm , tb (ǫ)}. Then ρb (t∗ ) ∈ Ω and by result (R3) of Appendix A,

aug e∞ (ρ) = {(x, x¯) : (x, x¯, ρ) ∈ Oaug } show (x(t∗ ), x This and O ¯b (t∗ ), ρb (t∗ )) ∈ O∞ , which implies (¯ xb (t∗ ), ρb (t∗ )) ∈ ∞ Π(x(t∗ )). Finally, (.12) shows k(¯ xb (t∗ ), ρb (t∗ ) − rs )k2 < Vm ≤ V (t) = k¯ xa (t)kS¯ + kρa (t) − rs k2S . This inequality is equivalent to (.9) and the proof of Vm = V ∗ is complete.

Parts (iii) and (iv) are a direct consequence of Vm = V ∗ . To confirm this, let z = (¯ xa (t), ρa (t)), Z = Rn¯ × Ω, ∗ ∗ 2 zs = (0, rs ), zop = (0, rs ) in Lemma 4. Then (.4) implies k(¯ xa (t), ρa (t)) − (0, rs )k ≤ V (t) − V ∗ . Thus, x¯a (t) → 0 and ∗ ∗ ρa (t) → rs imply u(t) → rs as t → ∞. From this result and (.7), part (iv) is proved. Since rs∗ ∈ Ω and x¯a (t) → 0, it e s∗ + Fe∞ + ǫBn+¯n for all t ≥ e follows from part (iv) that there exists a e t(ǫ) such that (x(t), 0) ∈ Γr t(ǫ). Thus by (R3) of ∗ aug ˜ Appendix A and the definition of O∞ (ρ), (x(t), 0, rs ) ∈ O∞ for all t sufficiently large. This shows (0, rs∗ ) ∈ Π(x(t)) for all t sufficiently large. Hence, the definitions of (xop (x, r), ρop (x, r)) and rs∗ complete the proof of (iii) and (iv). Appendix C : Proof of Theorem 2 ˆ¯(t), ρ(t)) ˆ¯op (x(t), r(t)), Consider system (1), (2), (39),(40) with Fˆ (x, r, x¯, ρ) defined by (44). Then (x ˆ is given either by (x ˆ ˆ ˆ¯(t), ρ(t)) ρˆop (x(t), r(t))) or by (x ¯(t−1), ρˆ(t−1)). For the first alternative, (x ˆ ∈ Π(x(t)) ⊂ Π(x(t)). Thus, (x(t), xˆ¯(t), ρ(t)) ˆ ∈ aug aug ˆ ˆ ˆ ˆ ˆ O∞ and Ω = Projρ O∞ implies ρˆ(t) ∈ Ω. For the second alternative ρˆ(t) = ρˆ(t − 1) ∈ Ω and, as argued in Section ˆ¯(t), ρ(t)) 5, (x ˆ ∈ Π(x(t)). Hence, the condition, ˆ ˆ (x ¯(0), ρˆ(0)) ∈ Π(x(0)) and ρˆ(0) ∈ Ω,

(.1)

ˆ for all t ∈ Z + . The first inclusion together with (38) and Remark 1 show implies (x ¯ˆ(t), ρ(t)) ˆ ∈ Π(x(t)) and ρˆ(t) ∈ Ω results (i) and (ii) of Theorem 2 are satisfied. Clearly, x(0) ∈ Xˆ and (41) imply condition (.1). Define ZOP := {t ∈ Z + : t > 0, Θ(x(t), r(t), xˆ¯(t), ρ(t)) ˆ = 1}, + ˆ ZBU := {t ∈ Z : t > 0, Θ(x(t), r(t), x¯(t), ρ(t)) ˆ = 0}.

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(.2) (.3)

As the solution of system (1),(2),(39),(40) evolves, t belongs to either ZOP or ZBU . It is now shown that ZOP is unbounded. Assume to the contrary that ZOP is bounded and let tBU := 1 + max{t : t ∈ ˆ¯(t), ρ(t)) ˆ¯(tBU ), ρˆ(tBU )) for all τ ∈ Z + . ZOP }. This implies that t = tBU + τ ∈ ZBU for all τ ∈ Z + . Then (x ˆ = (A¯τ x ˆ¯(t) → 0 and u(t) → ρˆ(tBU ) as t → ∞. From this and result (i), x(t) → {Γˆ ρ(tBU )} + F∞ Therefore, xˆ¯(t) → 0 Thus, x ˜ ρ(tBU )} + F˜∞ . Since ρ(t) = ρ(tBU ) ∈ Ω, ˆ assumption (A8) shows and (.2) of Appendix A imply (x(t), xˆ ¯(t)) → {Γˆ ˆ˜ (ˆ ˜ ρ(tBU )} + F˜∞ + ǫBn+¯n ⊂ O ˆ(tˆ)) ∈ {Γˆ there exists tˆ ∈ Z + and ǫˆ > 0 such that (x(tˆ), x¯ ∞ ρ(tBU )). This in turn implies aug + ˆ∞ . Consequently, (x ˆ tˆ)) ⊂ H (x(tˆ), r(tˆ)) and Θ(x(tˆ), r(tˆ), x ˆ¯(tˆ), ρˆ(tˆ)) ∈ O ˆ ˆ¯(tˆ), ρˆ(tˆ)) = 1. Thus, (x(tˆ), x ¯(tˆ), ρˆ(tˆ)) ∈ Π(x( tˆ ∈ ZOP , violating the assumption that tBU + τ ∈ ZBU for all τ ∈ Z + . With ZOP unbounded, the rest of the proof is very similar to the proof of results (iii) and (iv) in Appendix B. Since ˆ Thus, there is no loss of generality if it is assumed ˆ ¯(ts ), ρˆ(ts )) ∈ Π(x(ts )) and ρˆ(ts ) ∈ Ω. r(t) = rs for all t ≥ ts , (x that r(t) = rs for all t ∈ Z + . To make the correspondence to details in Appendix B easier to follow, let ˆ¯(t), ρ(t)) ˆ (¯ xa (t), ρa (t)) := (x

(.4)

¯xa (t − 1), ρa (t − 1)) ∈ H + (x(t), r(t)). From this for all t ∈ Z + . Define V (t) by (.2) of Appendix B. When t ∈ ZOP , (A¯ 2 2 ¯ ˆ¯op (x(t), r(t))k2¯ + kρˆop (x(t), r(t)) − rs k2S . Therefore and Figure 1, it is clear kA¯ xa (t − 1)kS¯ + kρa (t − 1) − rs kS ≥ kx S 2 2 ¯xa (t − 1)k2¯ + kρa (t − 1) − rs k2 , ¯xa (t − 1)k ¯ + kρa (t − 1) − rs k ≤ V (t − 1). When t ∈ ZBU , V (t) = kA¯ V (t) ≤ kA¯ S S S S and again V (t) ≤ V (t − 1). Thus, there exists a Vˆm such that V (t) → Vˆm . ˆ ˆ¯, ρ − rs k2 . For t ∈ ZOP , using this It is easy to confirm (see Figure 1) (x ¯op (x, r), ρˆop (x, r))) = min(xˆ¯,ρ)∈H + (x,r) kx + ¯xa (t − 1) − x result in Lemma 4 with Z = H (x(t), r(t)), zop = (¯ xa (t), ρa (t)), zs = (0, rs ), z = (A¯ ¯a (t − 1)) shows ¯xa (t − 1) − x kA¯ ¯a (t)k2S¯ + kρa (t − 1) − ρa (t)k2s ≤ V (t − 1) − V (t).

(.5)

¯xa (t − 1) − x For t ∈ ZBU it follows that kA¯ ¯a (t)k2S¯ + kρa (t − 1) − ρa (t)k2s = 0. Thus, (.5) applies for all t > 0. Consequently, as in Appendix B, x ¯a (t) → 0 and ρa (t − 1) − ρa (t) → 0 as t → ∞. ˆ Let ∆u(t) be defined by (.6) of Appendix B. Then ∆u(t) → 0 and (.7) of Appendix B is satisfied. Since ρa (t) ∈ Ω, ∗ ∗ ∗ (.2) of Appendix B and the definitions of Vˆ and rˆs imply Vˆm ≥ Vˆ . It follows that Vˆm = Vˆ ∗ . With appropriate changes in notation, the proof of this result is the same as the one given in Appendix B that shows Vm = V ∗ . The argument that leads to the contradiction that Vˆm > Vˆ ∗ is done for t ∈ ZOP rather than for t ∈ Z + . The changes in notation require replacement of Vm , V ∗ , rs∗ and Ω, respectively, by Vˆm , Vˆ ∗ , rˆs∗ ˆ and Ω. With the same changes in notation, the last paragraph of Appendix B remains the same except for its last ˆ˜ (ρ), two sentences. Replace them by the following sentences. Thus, by assumption (A8) and the definition of O ∞ aug ˆ∞ ˆ (x(t), 0, rˆs∗ ) ∈ O for all t sufficiently large. This shows (0, rˆs∗ ) ∈ Π(x(t)) for all t sufficiently large. Hence, (35) and the definition of rˆs∗ completes the proof of (iii) and (iv). Appendix D : An Algorithmic Test for Condition (45) aug ˆ∞ Assume O is a compact polyhedron,

ˆ ˆ aug := {(x, x ˆxx + H ˆ x¯ x ˆ ρ ρ ≤ h}, O ¯, ρ) : H ¯+H ∞

(.1)

ˆ ∈ Rqˆ. Using the definition in (.2) of Appendix A, it follows that ˆ x ∈ Rqˆ×n , H ˆ x¯ ∈ Rqˆ×¯n , H ˆ ρ ∈ Rqˆ×m and h where H ˆ−H ˆxx + H ˆ x¯ x ˆ ρ ρ for all x ∈ {Γρ} + F∞ + ǫBn , (45) is equivalent to condition: there exists an ǫ > 0 such that H ¯≤h

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ˆ This condition is in turn equivalent to the condition: x ¯ ∈ Bn¯ , ρ ∈ Ω. ˆ ˆxΓ + H ˆ ρ )ρ + H ˆ x z∞ < h (H

ˆ ∀ z∞ ∈ F∞ and ρ ∈ Ω.

(.2)

sΩˆ (ν) := maxρ∈Ωˆ ν ′ ρ,

(.3)

For η ∈ Rn and ν ∈ Rn¯ , let sF∞ (η) := maxz∈F∞ η ′ z,

ˆ Then (.2) is equivalent to the condition denote the support functions of F∞ and Ω. ˆ xi′ ) < hi , i = 1, · · · , qˆ. ˆxΓ + H ˆ ρ )i′ ) + sF∞ (H sΩˆ ((H

(.4)

Here the superscript i denotes the ith row of the corresponding matrix. Thus, condition (45) is verified by confirming the qˆ strict inequalities in (.4). ˆ ˆ xx + H ˆ x¯ x¯ + H ˆ ρ ρ ≤ h. The support function sΩˆ (ν) can be obtained by solving the LP: minimize ν ′ ρ subject to H ∗ ∗ ∗ ′ ∗ n Let (x , x ¯ , ρ ) be a solution of the LP. Then sΩˆ (ν) = ν ρ . For µ ∈ R , let sW (µ) be the support function of W . Then the support function sFP ∞ (η) can be approximated to any desired degree of accuracy from the exponentially ′ j ′ convergence series, sF∞ (η) = ∞ j=0 sW ((η A Ex ) ), (Ong & Gilbert, 2006b).

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