A dwell-time based Command Governor approach for constrained switchedsystems

Giuseppe Franze`, Walter Lucia and Francesco Tedesco

Abstract In this paper a switched control architecture forconstrained control systems is presented. The strategy is basedon Command Governor (CG) ideas that are here specializedin order to jointly take into account switching events on theplant dynamics and time-varying constraints. The significanceof the method relies in its capability to avoid constraint violationand loss of stability regardless of any configuration changein the plant/constraint structure by commuting the systemconfiguration (model plant+CG) with a more adequate one.To this end the concept of model transition dwell time is usedwithin the proposed control framework to formally define theminimum time necessary to enable a switching event underguaranteed conditions on the overall stability and constraintfulfillment.

I. INTRODUCTION

Switched systems belong to the class of dynamical systemsdefined by a finite number of subsystems (modes of thesystem) and a logical rule that orchestrates switching amongthese subsystems. The main properties of these modelshave been analyzed in depth in past decades because theyhave numerous applications in the control of mechanicaland automotive systems, power systems, aircraft and trafficcontrol and so on, see for a detailed review [11].

For this class of systems, a first important issue relies onthe fact that the switching amongst stable dynamical systemsmay give rise instability, therefore adequate conditions on theswitching signal behaviour must be imposed. This naturallyleads to the concept of dwell-time, that is the minimum timeinterval in which the system is forced to stay, see e.g. [12].Rapid progress in this field have lead to the introductionof more flexible concepts than the simple dwell-time: theaverage dwell-time proposed by Morse and Hespanha [8]and, more recently, the transition dwell-time which imposesthe minimum permanence time on the current mode beforeswitching to a specific new one and leads to the definitionof the dwell time graph [3].

A second aspect intensely investigated in the switchingsystems literature is the problem of designing switchingcontrollers for plants switching amongst a (finite) collectionof given system configurations. The main reason of the latteris that, even in the case of a unique plant, advantages interms of performance can be achieved by properly switchingamongst the given controllers. Examples of this property canbe found in adaptive schemes [1], [10], supervisory control[12] and robust synthesis [14].

Giuseppe Franze` Walter Lucia and Francesco Tedesco are with DIMES,Universita` della Calabria, Via Pietro Bucci, Cubo 42-C, Rende (CS), 87036,ITALY {franze,wlucia,ftedesco}@dimes.unical.it

Moving from these considerations, in this paper we de-velop a switching control architecture based on the low-computational demanding predictive scheme known in lit-erature as the command governor (see [2], [6]) for theregulation of constrained switched systems subject time-varying constraint paradigms so that the basic CG propertiesare preserved.Here this scheme will be extended in order to adequately takecare during the on-line operations of system configurationswitching occurrences that become necessary in order tosatisfy prescribed control tasks and, possibly, the overallcontrol performance are improved.To this end the definition of guaranteed conditions, un-der which safe switching amongst the family of systemmodes can be performed, represents the key aspect to beinvestigated. In the proposed framework this is achievedby jointly resorting to viability CG arguments and to theconcept of the transition dwell time between two systemconfigurations. Specifically, an efficient way to compute theweighted digraph arising when transition dwell-time labelsis provided and viability/asymptomatic closed-loop stabilityand tracking properties are formally proved.As one of its main merits, the proposed strategy for-mally allows to a-priori discriminate amongst system modes(weighted graph) in order to determine the minimum timepath satisfying control prescriptions.

Finally, a solid numerical example is instrumental to showmerits and effectiveness of the proposed strategy. In fact,simulations dealing with the attitude control problem of alight utility aircraft, namely the Cessna 182, subject to angleof attack and surface deflection constraints are provided.

PRELIMINARIES AND NOTATIONSDefinition 1: Given a set S IRn and a point x IRn, the

point-set distance between x and S is defined as:dist(x,S) = inf

sSx s,

where is any relevant norm. 2Definition 2: A directed graph, digraph, is an ordered pair

G = (V,E) such that V is the vertex set; E is a subset of ordered pairs of V known as the edge

set, i.e. E {{u,v} V} 2Definition 3: Let G = (V,E) a graph. A directed path of

length k, k ZZ+, from va to vb is a sequence of k vertices,Vpath = {v1, . . . ,vk} such that

v1 = va, vk = vb, {vi,vi+1} E, i = 1, . . . ,k1.2

2015 American Control ConferencePalmer House HiltonJuly 1-3, 2015. Chicago, IL, USA

978-1-4799-8684-2/$31.00 2015 AACC 1077

Definition 4: Let G=(V,E) and va,vb V be a graph andtwo vertices, respectively. The vertex vb is reachable fromva if there exists a path connecting va to vb. Moreover, wedefine with R va the set of vertices reachable starting fromva. 2

Definition 5: A graph G = (V,E) is an edge-labelledgraph if each edge of G has associated a label. If theselabels are members of an ordered set (e.g. real numbers),G is known as weighted graph and the numerical values asweights. 2

Definition 6: Let G = (V,E) be a weighted digraph, wecall the shortest path problem or the single-pair shortest pathproblem the problem of finding a path V va,vbpath from va E tovb E such that the sum of the weights of its constituentedges is minimized. We call all-pairs shortest path problemthe problem in which we have to find shortest paths V allpathbetween every pair of vertices {va,vb} EE. 2

Definition 7: For given sets A ,E IRn AE := {a+ e :a A ,e E} is Minkowski Set Sum [15]. 2Let H IRpq, we shall denote with row(H) the th rowof H.

II. PROBLEM STATEMENTConsider the following discrete-time linear switched sys-

tem{xp(t+1) = A(t)xp(t)+B(t)u(t)+Bdd(t)

y(t) = Cxp(t)(1)

where xp(t) IRnp denotes the state, u(t) IRnu the input,y(t) IRm the output and d(t) IRd the process disturbance.It is assumed that d(t)D IRd , t ZZ+ := {0,1, ...}, withD a compact set with 0d D. The signal : ZZ+ I :={1, . . . ,L} is a piecewise constant sequence that orchestratesswitchings between the so-called L modes of the system (1).Moreover the switched system is subject to the followingset-membership state and input constraints:

u(t) Ui, t 0, i = 1, . . . ,L, (2)

xp(t) X i, t 0, i = 1, . . . ,L, (3)with Ui, X i compact and convex subsets of IRnu and IRnp ,respectively.The problem statement is:Constrained Tracking (CT) Problem - Given the con-strained switched system (1)-(3) and a reference signal r()IRm on the output vector. Determine a control strategy

u() = f (xp(),r()),such that the closed-loop system is asymptotically stable, theconstraints are always satisfied and y(t) r(t),t 0. 2The problem will be addressed by using the CommandGovernor approach [6], [2], [7] which has to be properlyadapted in order to comply with the switched systemsrationale. Specifically the idea we want to develop is thefollowing:

1) for each i th mode: first a controller (known as theprimal controller), eventually equipped with an integralaction, is computed with the only aim to impose theasymptotic stability property; then a CG unit is off-line

designed by considering the i th set of constraints(2)-(3);

2) a switching i j, i, j I , is enabled if the j thmode is considered better than the current i th toaccomplish the tracking task;

3) a switching (logic) rule amongst the L model config-urations is conceived so that closed-loop stability andconstraint satisfaction are always ensured.

III. BASIC COMMAND GOVERNOR (CG) DESIGNConsider the following discrete-time linear time-invariant

system: x(t+1) =x(t)+Gg(t)+Gdd(t)y(t) = Hyx(t)c(t) = Hcx(t)+Lg(t)+Ldd(t) (4)where x(t) IRn is the augmented state including both plantand primal controller components, g(t) the CG action, i.e. asuitably modified version of the reference signal r(t) IRm,y(t) IRm the plant output which is required to track r(t)and c(t) IRnc the constrained output vector

c(t) C , t ZZ+ (5)with C a specified convex and compact set. In the sequel,the following assumptions are made:Assumption 1 :A1. is a Schur matrixA2. The system (4) is offset-free, i.e. Hy(In)1G = Im.

By noticing that, under a constant command g(t) t,the disturbance-free steady state solution of (4) is

x := (In)1Gy := Hy(In)1G (6)c := Hc(In)1G+L

and by considering the following Minkowski differencerecursions on the constrained set C

C0 := C LdDCk := Ck1 Hck1GdD (7)

C :=k=k=0

Ck

it has been proved in [2] that at each time instant the CGaction g() is computed according to the following convexoptimization over a finite prediction horizon k0 ZZ+ :

g(t) = arg minV (x(t))

|| r(t)||, =T > 0 (8)

ThenV (x(t)) =

{ W : c(k,x,) Ckk = 0, . . . ,k0

}(9)

with c(k,x,) = Hc

(kx+

k1i=0

ki1G

)+L,

W ={ IRm : c C

}, C =C B (10)

and B a ball of radius centered at the origin, is the set ofall constant virtual commands whose state evolutions startingfrom x satisfies all the constraints also during the transients.Finally the main CG properties are below summarized [2].

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Property 1: Consider system (4) along with the CG se-lection rule (8). Let Assumption 1 be fulfilled and V (x(0))be non-empty. Then:

1) The minimizer in (8) uniquely exists at each t ZZ+ andis obtained by solving a convex constrained optimiza-tion problem, viz. V (x(0)) non-empty implies V (x(t))non-empty along the trajectories generated by the CGcommand (8);

2) The set V (x(t)), x(t) IRn, is finitely determined, viz.there exists an integer k0 such that if c(k,x(t),) Ck,k {0,1, . . . ,k0}, then c(k,x(t),) Ck k ZZ+;

3) The constraints are fulfilled for all t ZZ+;4) The overall system is asymptotically stable. Specifically,

whenever r(t) r, g(t) converges in finite time eitherto r or to its best steady-state admissible approximationr, with

g := r := arg minW

r2, =T > 0 (11)

IV. COMMAND GOVERNOR SWITCHING CONDITIONSFrom now on we shall refer to the generic i I system

configuration: x(t+1) =ix(t)+Gig(t)+Gd,i d(t)y(t) = Hy,i x(t)ci(t) = Hc,i x(t)+Lig(t)+Ld,i d(t) (12)Strictly speaking, a switching between the i and j 6= iconfigurations can occur only if (see [5])

C i C j 6= /0 (13)By considering the output admissible set for the generic ithCG unit

Zi :={[rT , xT ]T IRmIRn |ci(k,x,r)C i , kZZ+

}(14)

and by denoting with X i , the set of all states, which canbe steered to feasible equilibrium points without constraintviolation as

X i :={

x IRn|[x

]Zi for at least one IRm

}(15)

we have that if (13) holds true, the following conditions arealso valid

W i W j 6= /0 and X i X j 6= /0 (16)Then, the switching i j can be enabled if at a certain timeinstant t the following condition is verified

x(t) X i X j (17)Now we are up to clarify the term better between the i thand j th modes with respect to the tracking goal of theproposed CT problem. By considering the graph G= (I ,E),with E including as edges the system configuration pairs{k,h} for which (16) holds true, and by assuming that

r(t) /W i , x(t) X ithen if

j := arg minkR i

dist(r(t),W k ) (18)

the j th mode is evaluated better than the i th one.The consequence of all the above developments is that ifat t = t condition (17) is satisfied then asymptotic stability

d

X j

X i

d

x(0)

Yi,j

t ij* x(tij)*

Fig. 1. Dwell-time characterization

and constraint fulfilment under switching occurrences areguaranteed.However this situation has to be considered a fortuitousscenario during the on-line operations. Usually, the statetrajectory x() must be steered to X i X j , therefore a timeinterval must lasts before the switching event occurs. Thisreasoning leads to the dwell-time concept [8] that will bediscussed and adapted to the proposed framework in the nextsection.

V. TRANSITION DWELL TIMEThe aim of this section is to define and compute lower

bounds on the minimum time intervals existing between anyedge {i, j} E that are necessary in order to ensure thesatisfaction of (17) whatever is the starting state conditionx(t) X i .To this end, we will make use of the following definition:

Definition 8: Let t0, t1, . . . , tk with 0= t0 < t1 < t2 < .. . 0 such that x IRn one has thatti||x|| Miti||x||, t ZZ+;

(Hc,i) and (Gi) the maximum singular values of thematrices Hc,i and Gi respectively;

> 0 an arbitrary small tolerance level on the Capproximation (see [2]): C()C C()+B;

dmax := maxdDd2;

[

T Tzi Tzi]

the support function describing the ball B;see [2].

Then, the following algorithm results:Dwell-time Computational Procedure (DCP) -

1) Initialize i, j = 0;2) If x(0) Yi, j such that x(i, j) / X j then

2.1) i, j = i, j +1 and goto 2);3) else Stop.Notice that the prescriptions of step 2) are performed

by using the arguments of [6]. In fact, by computing thefollowing quantities

G(k, p,i j) =[

maxxYi, j

rowp(TjHc, j( j)k)x(i, j)]

+rowp(TjRcik )r j,i rowp(b j)

p = 1, . . . ,z, k = 0, . . .k j0one has that step 2) becomes

if G(k, p,i, j) 0, p = 1, . . . ,z, k = 0, . . .k j0, goto 3)else goto 2.1).

VI. SWITCHING COMMAND GOVERNOR SCHEMEThe scheme consists of two phases: an off-line part where

a transition dwell-time weighted digraph G is determined;the on-line phase where admissible switching events amongstthe L system configurations take place in order to satisfy theCF problem requirements.Transition Dwell-time Switching Command Governor(TDS-CG) algorithm -

Off-line (Transition dwell-time weighted digraphcomputation) -

1. Let G= (I ,E= /0) be a weighted digraph. For eachpair i, j I , i 6= j, if (16) holds true then add theedge {i, j} to E;

2. {i, j} E, if there exists at least an equilibriumxri, j such that xri, j Y i, j X j , then determine thereference ri, j by solving (21) and compute i, j byapplying the DCP procedure;

2.1. else set i, j = ;3. Associate weight i, j and label ri, j to the each edge{i, j} E.

4. Determine all-pairs shortest paths Vallpath and allreachable sets R i, i E, by applying the Floyd-Warshall Algorithm [9];

1080

5. Store G, Vallpath and Ri. Initialize kold = i.

On-line -1. Select the system mode j by solving (18);2. If j 6= i then2.1. Pick the shortest path Vi, jpath Vallpath and consider

the vertex k successive to i along Vi, jpath;2.2. If k 6= kold then = 0;2.3. Solve

g(t) = arg minV i(x(t))

|| ri,k||

and apply g(t);2.4. If x(t) Yi,k; then := +1, and kold = k;2.5. If i,k then i k, = 0;

3. else solveg(t) = arg min

V i(x(t))|| ri,k||

and apply g(t).The main properties of the TDS-CG strategy are summarizedin the next Proposition.

Proposition 2: Given the constrained switched system(1)-(3) and assume that A1 and A2 hold. Let G = (I ,E) bethe transition dwell-time switched digraph associated to (12)-(13), then the TDS-CG scheme satisfies the CT prescriptionsregardless of any system mode switching complying with thestructure of G.

Proof - By resorting to standard viability arguments andcollecting the discussions of Sections IV and V. 2

VII. NUMERICAL EXAMPLEIn this section an application of the proposed control

architecture is presented. Specifically, the aim is to deal withthe flight control problem of a Cessna 182 aircraft subjectto asset and command constraints.

The 3-DOF longitudinal model of the aircraft motiondynamics is described by the following equations

v = v2Sw

2m(CD0 +CD+CDq

qc2v+CDe e)+

Tm

cosgsin() (30)

= qv2Sw

2mv(CL0+CL+CLq

qc2v+CLe e) (31)

Tmv

sin+gv

cos()

q =v2Swc

2Iyy(Cm0 +Cm+Cmq

qc2v

+Cme e) (32)

= q (33)

where xp = [v, , q, ]T and u = [T, e]T . The meaning ofthe involved variables and parameters can be found in [13].In this numerical simulation, we have that the samplingtime is Tc = 0.01s, and the physical constraints are belowreported:

C IR6 :

30 v 77 [m/sec]0.26 0.26 [rad]1.75 q 1.75 [rad/sec]0.7 0.7 [rad]

0 T 2357.2 [N]0.48 e 0.48 [rad]

(34)

0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Pitch angle Reference

M

o

d

e

2

M

o

d

e

14

2-->5 5-->8 8-->11 11-->14

2,51.89

5,83.76

8,115.64

11,145.66

* * **

Fig. 2. Pith angle vs Reference

0 5 10 15 20 2538

39

40

41

42

43

44

45

vconstraints

M

o

d

e

2

M

o

d

e

14

2-->5 5-->8 8-->11 11-->14

2,51.89

5,83.76

8,115.64

11,145.66

* * **

Fig. 3. Velocity v

Then, the flight scenario used for simulation purposes is:The reference pitch attitude angle is initially set to re f =0.085rad. Then at t = 1s the reference is shifted to thefinal set-point re f = 0.217rad. The initial state conditionis chosen as xp(0) = [41.262, 0.126, 0.000, 0.085]T .In order to comply with the problem statement, a set ofequilibria, (xipeq ,u

ieq), i = 1, . . . ,30, for (30)-(33) has been

chosen by using the following rationale: the region definedby constraints (34) has been partitioned so that for eachequilibrium different constraints, Ci C, i= 1, . . . ,30, result.The main reason of the latter is to impose that each linearizedmodel around (xipeq ,u

ieq) is valid within Ci.

The relevant results are collected in next Figs. 2-8. First,it is worth to underline how the TDS-CG algorithm iscapable to ensure safe switching occurrences along the all-pairs shortest paths obtained in the Off-line phase. In fact,all the prescribed constraints are always satisfied (Figs. 3-8) and tracking requirements accomplished (Fig. 2). Dy-namical state behaviours depicted in Figs. 3-6 put in lightthe switching phenomenon. Specifically, starting from thesystem configuration i= 2, at t = 1s formula (18) prescribesa switching to j = 14 and the shortest path is selected:V 2,14path = {2,5,8,11,14} with 2,5 = 1.89, 5,8 = 3.76, 8,11 =5.64, 11,14 = 5.66. Therefore, any switching from a modeto the successive along V 2,14path is subject to the correspondingtransition dwell time limitation. Finally, it is also interestingto analyze the CG output behavior as shown in Fig. 2. Therein fact it is possible to observe that within the transition dwelltime interval the reference is set to a constant value, namelyri j, as indicated in Step 2.3: r2,5 = 0.085, r5,8 = 0.097, r8,11 =0.109, r11,14 = 0.121.

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0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

constraints

M

o

d

e

2

M

o

d

e

14

2-->5 5-->8 8-->11 11-->14

2,51.89

5,83.76

8,115.64

11,145.66

* * **

Fig. 4. Angle of attack

0 5 10 15 20 25

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

qconstraints

M

o

d

e

2

M

o

d

e

14

2-->5 5-->8 8-->11 11-->14

2,51.89

5,83.76

8,115.64

11,145.66

* * **

Fig. 5. Pitch attitude rate q

0 5 10 15 20 250.05

0

0.05

0.1

0.15

0.2

0.25

0.3

constraints

M

o

d

e

2

M

o

d

e

14

2-->5 5-->8 8-->11 11-->14

2,51.89

5,83.76

8,115.64

11,145.66

* * **

Fig. 6. Pitch angle

0 5 10 15 20 250.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

e

constraints

M

o

d

e

2

M

o

d

e

14

2-->5 5-->8 8-->11 11-->14

2,51.89

5,83.76

8,115.64

11,145.66

* * **

Fig. 7. Deflection angle e

0 5 10 15 20 250

500

1000

1500

2000

2500

Tconstraints

M

o

d

e

2

M

o

d

e

14

2-->5 5-->8 8-->11 11-->14

2,51.89

5,83.76

8,115.64

11,145.66

* * **

Fig. 8. Trust T

VIII. CONCLUSIONSA switching command governor strategy for constrained

switched systems has been presented. By resorting to theconcept of transition dwell-time, the edge-labelled weightedgraph of all admissible switchings amongst the system modesis formally defined. As one of its main merits, the proposedscheme is capable to take care of time-varying constraints byon-line exploiting the CG viability properties and transitiondwell-times.

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