Automatica 50 (2014) 1466–1472

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Automatica

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Brief paper

Reduced order extended command governor

Uroš V. Kalabić, Ilya V. Kolmanovsky 1, Elmer G. GilbertDepartment of Aerospace Engineering, University of Michigan, 1320 Beal Avenue, Ann Arbor, MI, 48109, United States

a r t i c l e i n f o

Article history:Received 6 November 2012Received in revised form19 November 2013Accepted 21 February 2014Available online 16 April 2014

Keywords:Hard constraintsExtended command governorPositively invariant setsConstrained systems with disturbancesModel order reduction

a b s t r a c t

Extended command governors (ECGs) are add-on schemes that modify set-point commands as necessaryto ensure that imposed state and control constraints are not violated by closed-loop systems designedfor set-point tracking. In this paper, we propose a reduced order ECG for systems with dynamicsdecomposable into slow and fast state variables. We demonstrate that ECG implementation can bebased on slow states only, thus reducing the computational complexity. This is achieved by introducingadditional constraints, and by slightly tightening the original constraints. We demonstrate that theproposed ECG maintains the response properties of the conventional ECG, including the convergence tothe nearest feasible command in finite time in the case of constant reference commands. The results arealso shown to apply to conventional command governors. For the case when the reduced order state isnot directly measured, a formulation of the result in the presence of a state observer is developed.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Reference governors (RGs), command governors (CGs), and ex-tended command governors (ECGs) are control schemes that areappended to asymptotically stable closed-loop systems to enforcepointwise-in-time state and control constraints. All three gover-nors take the form shown in Fig. 1. Whenever it is possible to setv(t) = r(t) subject to the constraints, this is done. Otherwise, v(t)is determined by a specific rule that assures constraint satisfaction.The rules employmaximumconstraint admissible sets for the stateof the closed-loop systemwith constant reference commands. Un-der reasonable assumptions, both the RG (Bemporad, 1998; Gilbert& Kolmanovsky, 1999; Gilbert, Kolmanovsky, & Tan, 1995) andCG (Bemporad, Casavola, &Mosca, 1997; Casavola,Mosca, &Angeli,2000; Casavola, Mosca, & Papini, 2004) exhibit properties of recur-sive constraint feasibility, finite-settling time for constant or nearlyconstant reference commands, and convergence to an attractor set,when applied to systems with set-bounded disturbances.

The ECG, introduced in Gilbert and Ong (2011), determinesif a command is constraint-admissible by testing whether this

This work was supported in part by the National Science Foundation, AwardNumber 1130160. The material in this paper was not presented at any conference.This paper was recommended for publication in revised form by Associate EditorMartin Guay under the direction of Editor Frank Allgöwer.

E-mail addresses: kalabic@umich.edu (U.V. Kalabić), ilya@umich.edu(I.V. Kolmanovsky), elmerg@umich.edu (E.G. Gilbert).1 Tel.: +1 734 615 9655; fax: +1 734 763 0578.

http://dx.doi.org/10.1016/j.automatica.2014.03.0120005-1098/© 2014 Elsevier Ltd. All rights reserved.

command, combined with the output of an asymptotically stableauxiliary system, does not cause subsequent constraint violation.The state of the auxiliary system and the command to the closed-loop system are determined by solving a quadratic programmingproblem. The constraints in this problem are induced by themaximumconstraint admissible set for the systemextended by thestate of the auxiliary dynamics. Compared to the RG or the CG, theECG enlarges the maximal constrained domain of attraction, whileretaining the key response properties of the RG and the CG.

This paper contributes a method for reducing the computa-tional complexity of the ECG. The reduced order ECG method usesmodel order reduction by exploiting decomposition based on fastand slow dynamics. The ideas are similar to those used in the re-duced order RG of Kalabić, Kolmanovsky, Buckland, and Gilbert(2012), butmore complex because of the need to consider the stateof the auxiliary system. Because of these auxiliary states, model or-der reduction is even more important for making the ECG compu-tationally tractable; since the auxiliary dynamics are appended tothe already present system dynamics, reduction to a lower orderallows their design to be based on a system with fewer variablesand therefore simpler. Model order reduction directly contributesto lower complexity by decreasing the number of state variablesneeded for the implementation of the ECG. In the case of theRG, we have shown a three-fold reduction in computational com-plexity (measured in memory allocation) in applying the RG to apractical turbocharged gasoline engine control problem (Kalabićet al., 2012), and we have also demonstrated handling of infinitedimensional models based on reduced order RG theory. The ap-proach in Garone and Tedesco (2011) is another example of order

U.V. Kalabić et al. / Automatica 50 (2014) 1466–1472 1467

Fig. 1. A schematic of the ECG as applied within a control algorithm.

reduction in predictive control, but is distinctly different from theone here. Our approach decomposes the system into two subsys-tems and allows us to decrease the order of the ECG by using onlythe state of the first subsystem to develop the ECG algorithm. Thereis a trade-off in the order reduction. The errors in the system ap-proximation must be suitably controlled and this is done by tight-ening constraints.

The paper is organized into 6 sections of which this is the first.Section 2 reviews the theory of the ECG. Section 3 introduces theresults of the reduced order version of the ECG. Section 4 states themain theoremwhich is proven in the Appendix. For the casewherenot all slow states aremeasured, the treatment of observer errors isconsidered in Section 5. Concluding remarks aremade in Section 6.

Standardmathematical notation is used throughout.R is the setof real numbers and Z+ is the set of non-negative integers. Thematrix AT

∈ Rm×n is the transpose of A ∈ Rn×m; In ∈ Rn×n isthe identity matrix; Q ≻ 0 denotes a symmetric positive definitematrix; Bn = x ∈ Rn

: ∥x∥ ≤ 1 is the unit ball corresponding toa norm, ∥·∥. ForQ ≻ 0, let ∥x∥2

Q = xTQx. The sets intU and bdU arerespectively the interior and boundary of U ⊂ Rn. For Q ∈ Rm×m,QU := QU : u ∈ U. For V ∈ Rn, the sets U ⊕ V := u + v : u ∈

U, v ∈ V ⊂ Rn and U ∼ V := z ∈ Rn: z + v ∈ U, ∀v ∈ V are

respectively the Minkowski sum and Minkowski (or Pontryagin)difference. The notation x(t + k|t) denotes the predicted value attime t + k assuming the prediction is made at time t .

2. Extended command governor

The ECG, like the RG and CG, is applied to asymptotically stableclosed-loop systems to prevent them from violating specifiedpointwise-in-time constraints. Let the closed-loop system and itshard constraints be represented by,

x(t + 1) = Ax(t) + Bv(t) + Bww(t), (1)y(t) = Cx(t) + Dv(t) + Dww(t) ∈ Y , t ∈ Z+, (2)

where x(t) ∈ Rn, v(t) ∈ Rm, w(t) ∈ Rℓ, y(t) ∈ Rp, A ∈ Rn×n isasymptotically stable, and (C, A) is an observable pair. Disturbancesequences are represented byw(·) ∈ W where its elements satisfy,for all t ∈ Z+, the condition w(t) ∈ W . It is assumed that 0 ∈ Wand W is compact. The hard constraints are y(t) ∈ Y and mustbe satisfied for all t ∈ Z+ and w(·) ∈ W . The set Y ⊂ Rp isa polyhedron. We note that the subsequent theory only requiresthat Y is convex and closed. However, we assume Y is polyhedralbecause it allows explicit computational procedures.

The output of the ECG is given by,

v(t) = U(x(t), r(t)), (3)

where the function U : Rn× Rm

→ Rm is evaluated algorithmi-cally. Specifically, at the current time t , v(t) is based on the auxil-iary system,

x(t + 1) = Ax(t), (4)

v(t) = C x(t) + ρ(t), (5)

where A is chosen to be asymptotically stable and (C, A) is observ-able, x(t) ∈ Rn is the auxiliary state and ρ(t) ∈ Rm is the steady-state offset. Note that the output of the auxiliary system (4)–(5) isthe constraint admissible control and to see how it is exploited re-quires additional definitions and assumptions. Combining (1)–(2)and (4)–(5) and assuming ρ(t) ≡ ρ,x(t + 1)ρ(t + 1)

=

A B0 Im

x(t)ρ(t)

+

Bw

0

w(t), (6)

y(t) =C D

x(t)ρ(t)

+ Dww(t) ∈ Y , (7)

where,

x(t) =

x(t)x(t)

, A =

A BC0 A

,

B =

B0

, Bw =

Bw

0

, C =

C DC

.

The maximal constraint admissible set for (6)–(7) is,

Oaug∞

:= (x(0), x(0), ρ(0)) : (6)–(7) are satisfied

for all t ∈ Z+ and w(·) ∈ W. (8)

Define,

Π(x) := (x, ρ) : (x, x, ρ) ∈ Oaug∞

. (9)

Under appropriate conditions, both Oaug∞ andΠ(x) exist and can be

determined algorithmically. See Gilbert and Ong (2011) for details.Roughly stated, the appropriate conditions correspond to slightlytightening the constraint y(t) ∈ Y in steady-state. Since Oaug

∞ ispolyhedral,

Oaug∞

= (x, x, ρ) : Hxx + Hxx + Hrρ ≤ h, (10)

Π(x) = (x, ρ) : Hxx + Hrρ ≤ h − Hxx. (11)

To determine U(x, r), let,

∥(x, ρ)∥2:= ∥x∥2

S + ∥ρ∥2S , (12)

where S ∈ Rn×n and S ∈ Rm×m satisfy the conditions: S ≻ 0,S ≻ 0, ATSA − S ≺ 0. Since A is asymptotically stable, there existsan S satisfying the Lyapunov-like condition. Let the pair,

(xop, ρop) = argmin(x,ρ)∈Π(x)

∥(x, ρ − r)∥2. (13)

Then,

U(x, r) := C xop + ρop. (14)

It is now possible to state the main results of Gilbert and Ong(2011); to do this we need some standard definitions and assump-tions. The main assumptions have already been stated: A and A areasymptotically stable, (C, A) is an observable pair, and S and S arepositively definitematrices such that S and A satisfy the Lyapunov-like condition above; furthermore, W is compact and contains 0,and Y is closed and convex. Let,

F∞(rs) = Γ rs ⊕ F∞, Γ := (In − A)−1B,

F∞ = limt→∞

Ft , Ft =

t−1i=0

AiBwW , F0 = 0.(15)

The set F∞(rs) is the attractor set for (1) when v(t) ≡ rs (Gilbert& Kolmanovsky, 1999; Gilbert & Ong, 2011; Kerrigan, 2000; Kol-manovsky & Gilbert, 1998). Specifically, F∞(rs) is compact and

1468 U.V. Kalabić et al. / Automatica 50 (2014) 1466–1472

for all x(0) and all ε > 0, there exists a t ∈ Z+ such thatx(t) ∈ F∞(rs) ⊕ εBn for all t ≥ t . It follows from (1)–(2) thatwhen v(t) ≡ ρ, we must restrict ρ to a set Ω ⊂ intΩd, where,

Ωd = r ∈ Rm: (CΓ + D)r ∈ Y∞, (16)

Y∞ = Y ∼ DwW ∼ CF∞. (17)

Details are provided in Gilbert and Ong (2011). Finally let,

X := x : ∃(x, ρ) s.t. (x, x, ρ) ∈ Oaug∞

. (18)

This set is the projection of Oaug∞ ⊂ Rn

× Rn× Rm onto Rn. The

following theorem is the main result of Gilbert and Ong (2011).

Theorem 1. Consider the system (1)–(3)with r(t) ∈ Rm, w(·) ∈ W ,U(·) defined by (14), and x(0) ∈ X. Then: (i) x(t), v(t), and y(t) aredefined 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 r∗

s = argminr∈Ω ∥r − rs∥2S . (iii) There exists a tf ∈ Z+ such

that v(t) = r∗s for all t ≥ tf . (iv) Given ε > 0, there exists a tε ∈ Z+

such that x(t) ∈ F∞(r∗s ) ⊕ εBn for all t ≥ tε .

3. Reduced order ECG

This section presents a model order reduction that is basedon the decomposition of the system (1)–(2) into normal modes.Specifically, the system is represented by fast and slow states,where the ECG is based on the slow state only. A more generaldecomposition (Skelton, 1988) is possible and its development issimilar, but because it does not lead to a reduction in model order,it is not explicitly considered here.

We begin by transforming the system via an appropriatecoordinate transformation so that (1)–(2) are split into fast andslow subsystems. Consider an invertible coordinate transformationP : Rn

→ Rn such that,

P−1x(t) =

x2(t)x1(t)

, P−1AP =

A2 00 A1

,

P−1B =

B2B1

, P−1Bw =

Bw,2Bw,1

, CP =

C2 C1

,

where A1 ∈ Rn1×n1 and A2 ∈ Rn2×n2 are, respectively, matrices ofthe fast and slow dynamics; by this, we mean that the magnitudesof all eigenvalues of A1 are small when compared to the magni-tudes of all eigenvalues of A2. Assuming that such a transformationcan be constructed, our approach is to design the ECGbased only onthe dynamics of x2(t) and develop conditions that bound the errorthat is introduced by the deviation of x1(t) from steady-state.

Let P = diag(P, In) so that,

P−1AP =

A2 0 B2C00 A

, (19)

wherein,

A =

A1 B1C0 A

, B =

B10

, (20a)

Γ =

Γ10

, Bw =

Bw,10

, C =

C1 0

, (20b)

and Γ1 := (In1 − A1)−1B1. Because x1(t) represents the fast state,

subsequent developments are guided by the approximation that,

x1(t) ≈ Γ1ρ(t), (21)

and we introduce a modified approximate output,

y2(t) = C2x2(t) + C1Γ1ρ(t) + Dv(t) + Dww(t). (22)

To maintain the true and modified outputs close to each other, weintroduce an artificial output error set, Ey ⊂ Rp, and subsequentlyuse it as an artificial constraint on their difference, y(t) := y(t) −

y2(t),

y(t) ∈ Ey. (23)

Note that,

y(t) = C1(x1(t) − Γ1ρ(t)). (24)

The choice of Ey must satisfy conditions that are determined by themodel decomposition of system (1)–(2): Ey is compact, convex, andsatisfies,

Ey ⊂ int Y∞,2, (25)

where,

Y∞,2 := Y ∼ DwW ∼ [0 C2]F∞ ⊃ Y∞. (26)

Finally, the modified output must be constrained to a tightenedform of Y ,

y2(t) ∈ Y ∼ Ey, (27)

where (23) and (27) together imply that y(t) = y2(t) + y(t) ∈

(Y ∼ Ey) ⊕ Ey ⊂ Y .Thus to guarantee y(t) ∈ Y for all t ∈ Z+, the ECG can be ap-

plied to a modified system based on the dynamics of A2 and con-straints on the modified output (27) with the additional constraintin (23).

An approach to satisfying (23) is based on translating it to aset of sufficient conditions on v(t). We define a state error set,Ex ⊂ Rn1 × Rn, satisfying,

AEx ⊕ BwW ⊂ int Ex, (28)

CEx ⊂ Ey. (29)

The condition in (28) implies that Ex is robustly invariant and con-tractive with respect to A and is used to recursively guarantee con-straint admissibility of the error dynamics and also to guaranteeconvergence. The condition in (29) relates Ex to Ey and guaranteesconstraint admissibility in the presence of disturbances.

If we define,

x(t) := (x1(t) − Γ1ρ(t), x(t)) (30)

then from (29), 0 ∈ W and the definition of C , it follows thatx(t) ∈ Ex =⇒ y(t) = C1(x1(t) − Γ1ρ(t)) = C x(t) ∈ CEx ⊂ Ey.

The operation of the ECG based on (13)–(14) assures thatthe predicted response with selected (x(t), ρ(t)) satisfies theconstraints for all future time instants. The following propositioncharacterizes the predicted trajectories of the errors x(t) and y(t).The prediction of the trajectory of y(t) is needed in order tosubsequently enforce the constraint (23).

Proposition 2. The dynamics of x(t+k|t) and y(t+k|t) for k ∈ Z+

satisfy,

x(t + k + 1|t) = Ax(t + k|t) + Bww(t + k),

x(t|t) = x(t|t − 1) + ∆x(t),(31)

y(t + k + 1|t) = C x(t + k + 1|t), (32)

U.V. Kalabić et al. / Automatica 50 (2014) 1466–1472 1469

where,

∆x(t) :=

−Γ1∆ρ(t)

∆x(t)

, (33)

∆ρ(t) := ρ(t) − ρ(t − 1),

∆x(t) := x(t) − Ax(t − 1),(34)

x(t|t − 1) =

x1(t) − Γ1ρ(t − 1)

Ax(t − 1)

. (35)

Proof. To show (31), consider some k ≥ 0. Then, x(t + k + 1|t) =

[(x1(t + k + 1|t) − Γ1ρ(t))T x(t + k + 1|t)T]T = A[x1(t +

k|t)T x(t + k|t)T]T + Bww(t + k) + Bρ(t) − Γ ρ(t) = A[x1(t +

k|t)T x(t + k|t)T]T + Bww(t + k) + (In1+n − (In1+n − A)−1)Bρ(t) =

A[x1(t + k|t)T x(t + k|t)T]T + Bww(t + k)− A(In1+n − A)−1Bρ(t) =

Ax(t + k|t) + Bww(t + k).

Also,

x(t|t) =

x1(t|t) − Γ1ρ(t)

x(t|t)

= x(t|t − 1) + ∆x(t), (36)

completing the derivation of (31).The output error (32) follows from the fact that y(t + k +

1|t) = C1(x(t + k + 1|t) − Γ1ρ(t)) = C x(t + k + 1|t) fork ≥ 0.

The following proposition provides conditions that enforceconstraint admissibility of y(t) for all future time.

Proposition 3. Let x(t + k|t) and y(t + k + 1|t) satisfy (31)–(32).Suppose x(t|t − 1) ∈ Ex. If,

A∆x(t) ∈ Ex ∼ AEx ∼ BwW , (37)

then x(t + k + 1|t) ∈ Ex, y(t + k + 1|t) ∈ Ey for all k ∈ Z+.

Proof. The proof is by induction. Suppose k > 1 and assumex(t + k|t) ∈ Ex. From (28) and (29), it follows that x(t + k+ 1|t) =

Ax(t + k|t) + Bww(t + k) ∈ AEx ⊕ BwW ⊂ Ex, which impliesthat y(t + k + 1|t) = C x(t + k + 1|t) ∈ CEx ⊂ Ey. For k = 1,x(t +1|t) = Ax(t|t)+ Bww(t) = Ax(t|t −1)+ A∆x(t)+ Bww(t) ∈

AEx ⊕ (Ex ∼ AEx ∼ BwW ) ⊕ BwW ⊂ Ex.

Remark 1. Note that Proposition 3 requires that x(t|t − 1) ∈ Ex.This property will always be ensured by the ECG from the previoustime step, provided a feasible solution to ECG optimizationproblem exists at t = 0. With this in mind, based on Proposition 3,if the ECG ensures, through the selection of ρ(t) and x(t), thaty2(t + k|t) ∈ Y ∼ Ey for all k ∈ Z+ and additionally ∆x(t) givenby (33) satisfies (37), then y(t + k|t) ∈ Y for all k ∈ Z+.

Remark 2. The reduced order ECG offers most benefit whenthe reduction is based on a clear separation of the eigenvaluesdetermining the fast and slow dynamics of the closed loop system.When all the eigenvalues ofA1 aremuch smaller inmagnitude thanthose of A2, the underlying approximation in (21) is reasonable andEx ∼ AEx ∼ BwW more closely approximates Ex ∼ BwW , resultingin (37) being less stringent.

We use Remark 2 in order to help with the design procedure ofthe reduced order ECG. We now summarize the steps used to de-sign the reduced order ECG and the online computations involved.As with the regular ECG, the development is split into an offlineand online algorithm; the former corresponds to the constructionof the appropriate constraint sets and the latter corresponds to thereduced order ECG online control law.

Algorithm 1 (Offline). Given a system (1)–(2), find an invertibletransformation, P , such that the state matrix is in a split form as in(19). This decomposition should follow the insight of Remark 2, sothat the eigenvalues of A1 are relatively small inmagnitude.We arenow able to form a subsystem corresponding to slow eigenvalues,which is in the form of (6)–(7),x2(t + 1)x(t + 1)

=

A2 B2C0 A

x2(t)x(t)

+

B20

ρ(t) +

Bw,20

w(t), (38)

y2(t) =C2 DC

x2(t)x(t)

+ (C1Γ1 + D)ρ(t) + Dww(t) ∈ Y ∼ Ey. (39)

This is the reduced order systemwith a tightened output constraintand we develop a regular ECG corresponding to it, i.e., we find(C, A) and Oaug

∞ ⊂ Rn2 × Rn× Rm, along with its corresponding

set Π(x2) ∈ Rn× Rm, to use in the online algorithm.

All that is left is to handle the dynamics of A1, so define thematrices as in Eq. (20) and choose the sets Ex ⊂ Rn1 × Rn andEy ⊂ Rp, such that,

Ey ⊂ int Y∞,2,

AEx ⊕ BwW ⊂ int Ex,

CEx ⊂ Ey,

(40)

and define a new constraint set,

E :=

(∆x, ∆ρ) : A

−Γ1∆ρ

∆x

∈ Ex ∼ AEx ∼ BwW

, (41)

so that we can impose the following condition online,

(∆x(t), ∆ρ(t)) ∈ E, (42)

where the definitions of ∆x(t) and ∆ρ(t) are given in (34). Thiscompletes the specification of the offline procedure.

Algorithm 2 (Online). At each discrete time instant t , updatethe auxiliary system state and offset based on the estimate ormeasured value of the slow state x2(t) by solving the followingminimization problem,

(xop, ρop) = argmin(x,ρ)

∥(x, ρ − r)∥2, (43a)

subject to (x, ρ) ∈ Π(x2),

(x − Ax−1, ρ − ρ−1) ∈ E,(43b)

where the norm in (43) is defined as in (12) and the control lawU2 : Rn2 × Rn

× Rm× Rm

→ Rm is given by,

U2(x2, x−1, ρ−1, r) = C xop + ρop, (44)

so that,

v(t) = U2(x2(t), x−1(t), ρ−1(t), r(t)), (45)

where for t > 0, x−1(t) and ρ−1(t) are the solutions to (43) at theprevious time-step and for t = 0, the variables are freely selectablebut are assumed to satisfy,x1(0) − Γ1ρ−1(0)

Ax−1(0)

∈ Ex. (46)

This completes the specification of the online computation.

1470 U.V. Kalabić et al. / Automatica 50 (2014) 1466–1472

4. Main results

We first note that the following result is immediate byPropositions 2 and 3 and Remark 1.

Proposition 4. If the problem (43) has a feasible solution at timet = 0, then it is recursively feasible, i.e., it has a feasible solution forall t ∈ Z+.

The following theorem shows that the reduced order ECGexhibits characteristics similar to the full order version. Let Ω2be a set satisfying the properties of Ω along with the inclusionΩ2 ⊂ intΩd,2 where,

Ωd,2 := r : (CΓ + D)r ∈ Y∞,2 ∼ Ey ⊂ Ωd. (47)

Furthermore define a set X2 ⊂ Rn2 for the system (38)–(39)analogously to the definition (18) of X for the system (6)–(7).

Theorem 5. Consider the system (38)–(39), (45) with r(t) ∈ Rm,w(·) ∈ W , U2(·) defined by (44), (46) satisfied, and x2(0) ∈ X2.Then: (i) x(t), v(t), and y(t) are defined for all t ∈ Z+. (ii) y(t) ∈ Yand x2(t) ∈ X2 for all t ∈ Z+. Suppose further that there existsa ts ∈ Z+ such that r(t) = rs for all t ≥ ts. Define r∗

s :=

argminr∈Ω2 ∥r−rs∥2S . (iii) There exists a tf ∈ Z+ such that v(t) = r∗

sfor all t ≥ tf . (iv) Given ε > 0, there exists a tϵ ∈ Z+ such thatx(t) ∈ F∞(r∗

s ) ⊕ εBn for all t ≥ tϵ .

The proof is given in the Appendix.Comparing Theorems 1 and 5, we note thatmuch of the original

intuition is preserved. The convergence properties apply to thereduced order system described by (38)–(39), but the constraintsare satisfied for the overall system, i.e. y(t) ∈ Y for all present andfuture time instants.

The assumptions are also restricted to a minimum. Thedifferences from the full order ECG is that the error part of theinitial condition is assumed to already be bounded, i.e. x(0) ∈ Ex.Furthermore, the set of final admissible references is changed fromΩ to Ω2, because of the reduction in constraint set from Y toY ∼ Ey.

Furthermore, as in Remark 2 of Gilbert and Ong (2011), theresult can easily be extended to the theory of the CG, i.e., theconstraint (42) can be used in the CG algorithm under the sameassumptions by making x(t) empty and removing its dynamicsfrom consideration. The result for the RG is analogous and isavailable in Kalabić et al. (2012), where a couple of numericalexamples are also presented.

5. Accounting for observer error

If we do not measure all the components of the state x2(t),then we can design an observer to generate their estimates. Theobserver errors can be accounted for by the ECG in an analogousmanner to fast state deviations from steady state.

Consider that a reduced order ECG has been developed for thesystem (38)–(39) as in Algorithm 1. Let xo(t) be the output of theobserver for x2(t) with gain L,

xo(t + 1) = A2xo(t) + B2v(t) + L(y(t) − yo(t)), (48)yo(t) = C2xo(t) + Dv(t) + C1Γ1ρ(t) (49)

where, without loss of generality, we assume that yo(t) is both ameasured and constrained output. In Algorithm 2 at time t , thestate x2(t) is set to xo(t).

Let x(t) = (x2(t) − xo(t), x1(t) − Γ1ρ(t), x(t)) and y(t) =

y(t)− y2(t). The following proposition characterizes the predictedtrajectories of x(t) and y(t).

Proposition 6. The dynamics of x(t+k|t) and y(t+k|t) for k ∈ Z+

satisfy,

x(t + k + 1|t) = Ax(t + k|t) + Bww(t + k),x(t|t) = x(t|t − 1) + ∆x(t),

(50)

y(t + k + 1|t) = C x(t + k + 1|t), (51)

where,

A =

A2 − LC2 −LC1 00 A1 B1C0 0 A

,

Bw =

Bw,2 − LDw

Bw,10

, C =

C2 C1 0

,

and,

∆x(t) :=

0−Γ1∆ρ(t)

∆x(t)

, (52)

x(t|t − 1) =

x2(t) − xo(t|t − 1)x1(t) − Γ1ρ(t − 1)

Ax(t − 1)

. (53)

Proof. For k ≥ 0, x2(t + k+ 1|t) − xo(t + k+ 1|t) = A2x2(t + k|t)+B2v(t+k|t)+Bw,2w(t+k)−A2xo(t+k|t)−B2v(t+k|t)−L(y(t+k|t) − yo(t + k|t)) = A2(x2(t + k|t) − xo(t + k|t)) + Bw,2w(t +

k) − L(C2(x2(t + k|t) − xo(t + k|t)) + C1(x1(t + k|t) − Γ1ρ(t))+Dww(t +k)) = (A2 −LC2)(x2(t +k|t)−xo(t +k|t))−LC1(x1(t +k|t) − Γ1ρ(t)) + (Bw,2 − LDw)w(t + k).

The rest of (50) is proven by the fact that x2(t|t) − xo(t|t) =

x2(t) − xo(t) and Proposition 2.Finally, (51) follows from the choice of the initialization of

x2(t|t) to xo(t|t − 1).

Propositions 6 and 3 imply that the reduced order ECG with anobserver can be developed by defining new constraint sets Ex ⊂

Rn2 × Rn1 × Rn and Ey ⊂ Rp to replace Ex and Ey, respectively.These sets satisfy,

Ey ⊂ int Y∞,2,

AEx ⊕ BwW ⊂ int Ex,

C Ex ⊂ Ey.

(54)

The constraint in (41) is replaced by the following constraint,−LC1 0A1 B1C0 A

−Γ1∆ρ(t)

∆x(t)

∈ Ex ∼ AEx ∼ BwW . (55)

We can use the constraint in (55) to restrict changes in x(t) andρ(t) analogously to the constraint in (37). In this way, we ensurethat the observer error, in addition to the fast state deviation, doesnot cause constraint violation.

Response properties in Proposition 4 and Theorem 5 hold withappropriate notational modifications.

6. Conclusion

This paper has presented the theory of the reduced order ex-tended command governor. It applies the ordinary, full order, ref-erence governor theory to a reduced order system with tightenedconstraints and appends a new set of constraints that ensure con-straint admissibility of the dynamics which are not included in thereduced order system.

U.V. Kalabić et al. / Automatica 50 (2014) 1466–1472 1471

The reducedorder ECG shows similar properties to the full orderversion, including recursive feasibility and finite settling time, butwith smaller constrained domain of attraction.

Acknowledgment

The authors gratefully acknowledge Dr. Julia Buckland of FordMotor Company for collaborating with them on engine controlproblems that stimulated this work.

Appendix. Proof of Theorem 5

The Appendix contains a sketch of the proof of the maintheorem for the reduced order ECG, Theorem 5, which closelyfollows the proof of Theorem 1, available in Appendix B of Gilbertand Ong (2011).

Parts (i) and (ii) are proven with the help of parts (i) and (ii)of Theorem 1 and Proposition 3 of Section 3. Part (i) is implied bythe former and the definition of E in (41). For (ii), the regular ECGguarantees that y2(t) ∈ Y ∼ Ey for all t ∈ Z+. Proposition 3 (seealso Remark 1) implies that y(t + 1) ∈ Ey for all t ∈ Z+. Sincey(0) ∈ Ey by assumption (29) and (46), then y(t) = y2(t) + y(t) ∈

Y ∼ Ey ⊕ Ey ⊂ Y for all t ∈ Z+ and the proof of (i) and (ii) iscomplete.

We prove the rest of the theorem by defining,

V (t) := ∥(x(t), ρ(t) − rs)∥2≥ 0, (A.1)

Because (x(t − 1), ρ(t − 1)) ∈ Π(x2(t − 1)), at the next timestep, (Ax(t − 1), ρ(t − 1)) ∈ Π(x2(t)). Due to (44), this impliesthat V (t) ≤ ∥(Ax(t − 1), ρ(t − 1) − rs)∥2, where the norm isdefined in (12). According to the Lyapunov-like condition on S andA, ∥Ax(t − 1)∥S ≤ ∥x(t − 1)∥S , and therefore, V (t) ≤ V (t − 1),implying that there exists a Vm ≥ 0 such that V (t) → Vm.

We now prove the following,

∥(∆x(t), ∆ρ(t))∥2≤ V (t − 1) − V (t). (A.2)

First we state the following lemma.

Lemma 7. Suppose Z, ∆Z ⊂ Rq are closed and convex and zr ∈ Z,0 ∈ int∆Z, zs ∈ Rq, 0 ≺ Q ∈ Rq×q, and zop = zop(zr , zs) =

argminz∈Z, z−zr∈∆Z (z − zs)TQ (z − zs). Then,

∥zr − zop∥2Q ≤ ∥zr − zs∥2

Q − ∥zop − zs∥2Q . (A.3)

Proof. Because Z and ∆Z are closed, zr ∈ Z and 0 ∈ ∆Z , zop exists.Now, ∥zr − zs∥2

Q = ∥zr − zop − (zs − zop)∥2Q = ∥zr − zop∥2

Q −2(zr −

zop)TQ (zs − zop)+∥zs − zop∥2Q . Necessary conditions for optimality

on zop and grad(z − zs)TQ (z − zs) imply that −2(zr − zop)TQ (zs −

zop) ≥ 0, yielding ∥zr − zs∥2Q ≥ ∥zr − zop∥2

Q + ∥zs − zop∥2Q .

In Lemma 7, let Z = Π(x2(t)), ∆Z = E , zr = (Ax(t − 1), ρ(t −

1)), zop = (x(t), ρ(t)), zs = (0, rs), and Q = diag(S, S). Therefore,

∥(∆x(t), ∆ρ(t))∥2 (A.4)

= ∥(Ax(t − 1) − x(t), ρ(t − 1) − ρ(t))∥2 (A.5)

≤ ∥(Ax(t − 1), ρ(t − 1) − rs)∥2− ∥(x(t), ρ(t) − rs)∥2 (A.6)

= V (t − 1) − V (t), (A.7)

proving (A.2) and ∆x(t) → 0, ∆ρ(t) → 0.Define∆v(t) := v(t)−v(t −1). From the above, it follows that

∆v(t) → 0. This leads to the result that for any ε > 0, there existsa tε ∈ Z+ such that,

x(t) ∈ Γ v(t) ⊕ F∞ ⊕ εBn, ∀t ≥ tε. (A.8)

To confirm this, decompose x(t) = xv(t) + xw(t), where xv(t) isthe solution of x(t) with w(t) ≡ 0 and xw(t) is the solution ofx(t) with v(t) ≡ 0. It is apparent that xw(t) ∈ Ft ⊂ F∞, forall t ∈ Z+. Now define, ∆xv(t + 1) := xv(t + 1) − xv(t) =

A∆xv(t) + B∆v(t), therefore ∆xv(t) → 0 as ∆v(t) → 0. Sincexv(t + 1) = Axv(t) + Bv(t) = xv(t) + ∆xv(t), then xv(t) =

Γ v(t) − (I − A)−1∆xv(t) → Γ v(t). This leads to the conclusionthat x(t) = xv(t) + xw(t) → Γ v(t) ⊕ F∞.

Because (x(t), ρ(t)) ∈ Π(x2(t)), then ρ(t) ∈ Ω2. This impliesthat,

Vm ≥ V ∗:= ∥r∗

s − rs∥2S . (A.9)

In the proof of Theorem 1, the next step is to show that Vm = V ∗.The only difference between the assumptions in Gilbert and Ong(2011) and the assumptions herein is the addition of the constraintin (42). Since 0 ∈ int E , updates to ∆ρ(t) and ∆x are always non-zero if theywould benon-zerowhennot considering the constraintin (42); this and∆ρ(t) → 0, ∆x(t) → 0 as t → ∞ imply that theconstraint (42) is inactive for all t sufficiently large. Consequently,we obtain the result that Vm = V ∗.

Parts (iii) and (iv) now follow directly. In Lemma 7, let zr =

(x(t), ρ(t)), Z = Rn× Ω2, ∆Z = E , zop = (0, r∗

s ), and zs = (0, rs).Then, ∥(x(t), ρ(t) − r∗

s )∥ ≤ V (t) − V ∗. Therefore x(t) → 0 andρ(t) → r∗

s . This and (A.8) prove part (iv). Furthermore, similarlyto the proof of Theorem 1, they imply that for sufficiently larget , (0, r∗

s ) ∈ Π(x2(t)) and the constraint (∆x(t), ∆ρ(t)) ∈ E isinactive. Therefore, (44) and the definition of r∗

s imply that for all tsufficiently large, ρ(t) = r∗

s , proving part (iii).

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Uroš V. Kalabić received his B.A.Sc. degree from theUniversity of Toronto in 2010 and his M.S.E. degree fromthe University of Michigan in 2011, both in AerospaceEngineering. He is presently a Ph.D. candidate in FlightDynamics and Control at the Department of AerospaceEngineering, University of Michigan, Ann Arbor. Hisresearch interests include predictive control theory and itsapplication to aerospace and automotive systems.

1472 U.V. Kalabić et al. / Automatica 50 (2014) 1466–1472

Ilya V. Kolmanovsky has received his M.S. and Ph.D. de-grees in Aerospace Engineering, and the M.A. degree inMathematics from the University of Michigan, Ann Arbor,in 1993, 1995, and 1995, respectively. He is presently aprofessor in the Department of Aerospace Engineering atthe University of Michigan, with research interests in con-trol theory for systems with state and control constraints,and in control applications to aerospace and automotivesystems. He has previously been with Ford Research andAdvanced Engineering in Dearborn, Michigan, for close to15 years. He is a Fellow of IEEE, a past recipient of the Don-

ald P. Eckman Award of American Automatic Control Council, and of IEEE Transac-tions on Control Systems Technology Outstanding Paper Award.

Elmer G. Gilbert has been in the Department of AerospaceEngineering at the University of Michigan since 1957 andis now Professor Emeritus. He has published numerouspapers and holds eight patents. He received IEEE ControlSystems Field Award in 1994 and the Bellman ControlHeritage Award of the American Automatic ControlCouncil in 1996. He is a member of the Johns HopkinsSociety of Scholars, a Fellow of the American Associationfor the Advancement of Science, a Fellow of the Institute ofElectrical and Electronics Engineers and a member of theNational Academy of Engineering (USA).