Symbolic Control for UnderactuatedDifferentially Flat Systems

Adriano Fagiolini, Luca Greco, Antonio BicchiInterdepartmental Research Center E. PiaggioFaculty of Engineering, University of Pisa, Italy

Email: {a.fagiolini, bicchi}@ing.unipi.it,luca.greco@dsea.unipi.it

Benedetto Piccoli, Alessia MarigoC.N.R. - Istituto per le Applicazionidel Calcolo E. Picone, Rome, Italy

Email: {b.piccoli, marigo}@iac.rm.cnr.it

Abstract In this paper we address the problem of generatinginput plans to steer complex dynamical systems in an obstacle-free environment. Plans considered admit a finite descriptionlength and are constructed by words on an alphabet of inputsymbols, which could be e.g. transmitted through a limitedcapacity channel to a remote system, where they can be decodedin suitable control actions.

We show that, by suitable choice of the control encoding,finite plans can be efficiently built for a wide class of dynamicalsystems, computing arbitrarily close approximations of a desiredequilibrium in polynomial time. Moreover, we illustrate by sim-ulations the power of the proposed method, solving the steeringproblem for an example in the class of underactuated systems,which have attracted wide attention in the recent literature.

I. INTRODUCTION

The problem of steering complex physical plants requiresthe development of planning techniques capable of tacklingboth kinematic and dynamic constraints. Classical techniquesusually try to face this problem by solving a kinematic motionplanning problem and then looking for a trajectory and acontrol accounting for dynamic constraints. In recent years,several approaches inspired to the kinodynamic paradigm [1],[2], consisting in trying to solve these problems simultane-ously, have been presented.

Moreover, dealing with physical systems and complexcontrol frameworks, such as those based on hierachicallyabstracted levels of decision, usually involves additional issuesrelated to limited communication and storage resources. Con-sider for instance the case where a robotic agent receives itsmotion plans from a remote high-level control center through afinite capacity communication channel. Or consider a scenarioproviding that plans are exchanged in a networked system of alarge number of simple semi-autonomous agents, cooperatingto achieve a common task, e.g. in a formation control operationor collaborative map building and object tracking.

In this vein, we address in this paper the problem ofplanning inputs to steer a controllable dynamical system ofthe type

x = f(x, u), x X IRn, u U IRr (1)between neighborhoods of given initial and final states. As asolution, we seek a finite plan, i.e. an input function whichadmits a finite description. We are interested in plans with

short description length (measured in bits) and low computa-tional complexity. Particular attention is given to plans amongequilibrium states, regarded as nominal functional conditions.

Several important contributions have appeared addressingdifferent instances of symbolic control problems, e.g. [3][5].In particular, [6] shows that feedback can substantially reducethe specification complexity (i.e., the description length of theshortest admissible plan) to reach a certain goal state.

The main contribution of this paper is to show that, bysuitable use of feedback, finite plans can be efficiently foundfor a wide class of systems. More precisely, we use a symbolicencoding ensuring that a control language is obtained whoseaction on the system has the desirable properties of additivegroups, i.e. the actions of control words are invertible andcommute. Furthermore, under the action of words in thislanguage, the reachable set becomes a lattice. Finitelengthplans to steer the system from any initial state to any final statewithin a given region can thus be computed in polynomialtime. The contribution of this paper can be regarded as anextension of planning techniques in [7] (only applicable so farto driftless nonlinear systems in so-called chained-form), toa much wider class of systems, most notably systems withdrift.

By virtue of feedback encoding, complex nonlinear systems indeed, the same class of differentially flat systems [8]considered in [9] can be transformed (at least locally) intoa linear system. Planning for flat systems can then be achievedin a linear setting, hence projected back on the original systemsby feedback decoding. This process is thoroughly illustrated inthe paper by application to an interesting MIMO underactuatedmechanical system: a simple model of a helicopter.

II. SYMBOLIC CONTROL

Symbolic control is inherently related to the definitionof elementary control events, or quanta, whose combinationallows the specification of complex control actions. A finiteor countable set U of control quanta can be encoded byassociating its elements with symbols in a finite set ={1, 2, . . .}. Furthermore, letters from the alphabet can beemployed to build words of arbitrary length. Let be the setof such strings, including the empty one.

The analysis of the action of a generic on the state space

can be quite hard, and the structure of the reachable set undergeneric quantized controls can be very intricated (even forlinear systems: see e.g. [7], [10], [11]). However, a suitablechoice of the set U, namely a suitable control quantization,and of the control encoding can make symbolic control apowerful tool. This motivates the investigation of encodingsachieving simple composition rules for the action of words ina sublanguage , so as to obtain that the global action ofa command string is independent from the order of applicationof each control symbols in . Under this hypothesis, a choicefor always exists such that the reachable set from any pointin X under the concatenation of words in can be describedas a lattice which we assume henceforth. Hence, in suitablestate and input coordinates, the system takes on the form

z+ = z + H, H IRnn, ZZn. (2)Definition 1: A control system x = f(x, u) is additively

(or lattice) approachable if, for every > 0, there exist acontrol quantization U and an encoding E : U , suchthat: i) actions of commute and are invertible, and ii) forevery x0, xf X , there exists x in the orbit of x0 withx xf < .

Remark 1: The reachable set being a lattice under quanti-zation does not imply additive approachability. For instance,consider the example used in [2] to illustrate kinodynamicplanning methods [12][14]. This consists of a double inte-grator q = u with piecewise constant encoding U = {u0 =0, u1 = 1, u2 = 1} on intervals of fixed length T = 1. Thesampled system reads[

qq

]+=[

1 10 1

] [qq

]+[

121

]u, (3)

hence

q(N) = q(0) + Nq(0) +N

i=12(Ni)+1

2 u(i)q(N) = q(0) +

Ni=1 u(i).

The reachable set from q(0) = q(0) = 0 is

R(U , 0) ={[

qq

]=[

12 00 1

], ZZ2

}.

The quantization thus induces a lattice structure on thereachable set. The lattice mesh can be reduced to any desired resolution by scaling U or T . However, the actions ofcontrol quanta do not commute. Indeed, being (u) thesolution of system (3) under the control sequence u, we have(u1u2) = (u2u1) (for instance, (u1u2)(0, 0) = (1, 0),while (u2u1)(0, 0) = (1, 0).

Motivations for planning on lattices are mainly due to thefollowing theorem [15]:

Theorem 1: For an additively approachable system, a spec-ification of control inputs steering the system from any initialstate to any final state, can be given in polynomial time.

III. FEEDBACK ENCODING

A few examples of possible control encoding schemes ofincreasing generality were considered in [15]. Some of themare pictorially described in fig. 1. The most attractive one isthe feedback encoding which consists in associating to eachsymbol a control input u that depends on the symbol itself, onthe current state of the system, and on its structure. Feedbackencoding was exploited for planning the trajectory of a carwith n trailers ( [16], [17]), whose kinematic model is locallyfeedback equivalent to chained form [18] and hence, additivelyapproachable, thus demonstrating that finite plans to steer anynonlinear driftless system can be computed in polynomial timeby theorem 1.

We now turn our attention to the much broader class ofsystems with drift, i.e. systems which possess an autonomousdynamics independent of applied inputs. More precisely, con-sider again system (1)

x = f(x, u), x X IRn, u U IRr

and the associate equilibrium equation f(x, u) = 0. Let theequilibrium set be E = {x X |u U, f(x, u) = 0}. Wesay that system (1) has drift if E has lower dimension than X .

Among systems with drift, linear systems are the simplest,yet their analysis encompasses the key features and difficultiesof planning. Indeed, our strategy to attack the general case con-sists of reducing to planning for linear systems via feedbackencoding. To achieve this, we introduce a further generalizedencoder, i.e. the nested feedback encoding described in fig. 1-c.In this case, an inner continuous (possibly dynamic) feedbackloop and an outer discrete-time loop both embedded onthe remote system are used to achieve richer encodingof transmitted symbols. Additive approachability for linearsystems, by discrete-time feedback encoding (see fig. 1-b), isproved in theorem 6 below. By using the nested feedback en-coding, all feedback linearizable systems are hence additivelyapproachable. Therefore, by resorting to dynamic feedbackencoding ( [8], [19], [20]), we can state the following theorem:

Theorem 2: Every differentially flat system is locally addi-tively approachable.

IV. LINEAR SYSTEMS

In this section we consider linear systems of type

x = Fx + Gu (4)

with x IRn, u U = IRr and rank G = r. We start by somepreliminary results characterizing the equilibrium set E .A. Preliminaries

Let us recall from [15] the following lemmas:Lemma 3: For a controllable linear system (4), dim E = r.Application to (4) of piecewise constant encoding of sym-

bolic inputs (scheme a in fig.1) with durations Ti = T, i,generates the discrete-time linear system

x+ = Ax + Bu, (5)

P.-C. DecoderRemote System

P.-C. DecoderPhysical Plant

),( uxfx =& xDccbaPhysical Plant

),( uxfx =& xDccba

a)

D.T. Feedback DecoderRemote System

Physical Plant

),( uxfx =& xDdcba ZOH

Sampled System

b)

),( uxfx =& xDcDdcba ZOHPhysical Plant

C.T. Feedback DecoderSampled System

D.T. Feedback DecoderRemote System

c)

Fig. 1. Three examples of symbolic encoding of control. Symbols transmittedthrough the finite-capacity channel are represented by letters in the leftmostblocks. From the top: a) piecewise constant encoding; b) discrete-time feed-back encoding; c) nested discrete-time continuous-time feedback encoding.

with

A = eFT , B =

( T0

e(Ts)Fds

)G.

Lemma 4: The equilibrium manifold of a controllable linearcontinuous-time system is invariant under discrete-time feed-back encoding, for almost all sampling times T .

A crucial observation concerning systems with drift iscontained in the following lemma.

Lemma 5: For a linear system (4), it is impossible to steerthe state among points in E while remaining in E , except forspecial cases.

As we want E to be a lattice, the previous observationsdrive towards policies for periodic steering of systems amongequilibria. In particular, we are interested in seeking symbolicinput encodings that periodically achieve additive approacha-bility on E . Within this perspective, we recall the followingtheorem also from [15]:

Theorem 6: For a controllable linear discrete-time systemx+ = Ax + Bu, there exist an integer > 1 and a linearfeedback encoding

E : U ,i Kx + wi

with constant K IRnn and wi W , W IRr a quantizedcontrol set, such that, for all subsequences of period Textracted from x(), the reachable set is a lattice of arbitrarilyfine mesh. In other terms, for the undersampled system z(k) =x( + k), , k IN, it holds

z+ = z + H, H IRnn, ZZn

and there exists a choice of a finite W such that H < .

We recall preliminarily a result which can be deriveddirectly from [7].

Lemma 7: The reachable set of the scalar discrete timelinear system + = + v, IR, v W def= W with > 0 and W = {0,w1, . . . ,wm}, wi IN with at leasttwo elements wi wj coprime, is a lattice of mesh size .

Proof of Theorem 6: For the controllable pair (A,B),let S, V , and K0 be matrices such that (S1(A + BK0)S,S1BV ) is in Brunovsky form (see e.g. [21]). Denote withi, i = 1, . . . , r the Kronecker control-invariant indices. In thenew coordinates = S1x we have

+ = S1(A + BK0)S + S1BV v = A + Bv.

Let v = K1 + v, where: v W = 1 1W r rW , with kW ={0,kw1, . . . ,kwmk}, kwj IN k = 1, . . . , r, j =1, . . . ,mk, each kW including at least two coprimeelements kwi kwj ;

K1 IRrn such that its ith row (denoted K1i) containsall zeroes except for the element in the (i1 + 1)thcolumn which is equal to one (recall that by definition0 = 0).

Denoting with (Ai , Bi) the i-th sub-system in Brunovskyform, it can be easily observed that (Ai +BiK1i)i = Ii ,the i i identity matrix. Hence, if we let

= l.c.m. {i : i = 1, ..., r},we get

[S1((A + BK0)S + BV K1)

] = In.Let i IRi denote the ith component of the state vector

relative to the pair (Ai , Bi). For any IN we have

i( + i) = i() +

vi()...vi( + i 1)

(6)On the longer period of T , we have

i( + ) = i() +

i 1

k=0 vi( + ki)... i 1

k=0 vi( + i 1 + ki)

def= i() + vi(),

hence

( + ) = () +

v1...vr

def= () + vor, in the initial coordinates,

x( + ) = x() + Sv.

In conclusion, by the linear discretetime feedback encoding

E : U ,i (K0 + V K1S1)x + V vi

with vi W , for all -periodic subsequences z(k) = x( +k), it holds

z+ = z + S, ZZn

with = diag (1I1 , , rIr ) .

It is also clear that, for any , it is possible to choose suchthat z can be driven in a finite number of steps (multiple of) to within an -neighborhood of any point in IRn.

It is interesting to note that, for single-input systems, theencoding considered in theorem 6 is indeed optimal in terms ofminimizing the periodicity by which the lattice is achievable.However, for multi-input systems, the period of (l.c.m.i i)Tused in theorem 6 can be reduced to a minimal of (maxi i)T .This can be achieved by the planning algorithm describedbelow in section IV-B.

As it can be expected, the behavior of the system betweensuch periodic samples is in general not specified, and may turnout to be unacceptable. Indeed, if a goal has to be reached,which is far from the origin, the intersample behavior mayhave a large-span erratic behavior. Nevertheless, the feedbackencoding scheme allows to solve this problem while keepingthe systems evolution arbitrarily close to the equilibriummanifold (see [15]).

Notice finally that, in Brunovsky coordinates, E has aparticularly simple structure. Letting 1i IRi denote avector with all components equal to 1, we have that foreach i-dimensional subsystem, the equilibrium states arei = i1i , i IR, hence

E = {| = diag (1I1 , , rIr )1n}B. Planning algorithm

Based on the above results, we now provide explicitly anefficient method to steer from an arbitrary state x IRn towithin an -neighborhood of a given goal state x + IRn(x and not necessarily in E).

1) Compute the desired displacement in Brunovsky coor-dinates = S1, and let i IRi , i = 1, . . . , rdenote the desired displacement for the ith subsystem;

2) Compute the lattice mesh size in Brunovsky coordinatesi = 2i , where

[1 2 r

]= S

11 0 00 12 0 0 0 1r

;3) Find i, the nearest point to i on the lattice generated

by i iW and centered at i = (S1x)i.4) For each i = 1, . . . , r, let the quantized control set be

iW = {0,iw1, . . . ,iwmi}, iwj IN, and denote byiU the vector [iw0 iw1 iwmi ], where iw0 = 0. Write

i = i iC iU (7)

where iC is a matrix in ZZimi+1 with componentsiCh,j+1 = ich,j, h = 1, . . . , i, j = 0, . . . ,mi. Each

element ich,j of iC describes the number of timesthat the control iwj has to be used to steer the hthcomponent of i.

5) Find integers ich,j, h = 1, . . . , i, j = 1, . . . ,mi solv-ing the system of diophantine equations (7), and find the

smallest integers ich,0 such that, h,mi

j=0 |ich,j|def=

Ni. Nii is thus a number of steps sufficient to steerthe ith subsystem;

6) (Optional) Among all solutions of (7), find the one which

minimizes maxih=1mi

j=1 | ich,j|def= Ni. Notice that

Nii is the minimum length of a string of symbols iniW obtaining the goal. However, no polynomial-timealgorithm is known for such optimization;

7) Let N = maxi Nii, and i the corresponding index.Then, for all i = 1, . . . , r i = i, compute i =(Ai)ri( + i) with ri = N Nii. Repeatsteps 4) and 5) with the new i.

The explicit construction of a procedure to decode planspecifications iC into a string of control inputs iV for theith channel is finally described in Matlab-like code:

C=iC ;iV = [ ] ;whi le (C = 0 )

f o r h =1 :i ,j =1 ;whi le C( h , j ) == 0 , j = j +1 ; endiV = c a t ( iV , s i gn (C( h , j ) ) iwj ) ;C( h , j ) = C( h , j ) s i gn (C( h , j ) ) ;

endend

V. SIMULATIONS

A. Helicopter

In this section, we illustrate the power of the nested feed-back encoding of fig. 1-c, by solving the steering problem foran example in the class of underactuated, differentially flatmechanical systems.

Consider a simplified model for the helicopter depicted infig. 2. At first approximation, we can look at the helicopteras a rigid body, directly actuated by the thrust T of the mainrotor and the torque of the tail rotor. Referring to fig. 2,denote with (x, y, z) the position of its center of mass andwith (, , ) its orientation with respect to the x, y, and zaxes. With this choice, the dynamic model takes on the form

M x = T S ,M y = T C S ,M z = T C C M g ,J = C C ,

where M is the mass of the helicopter, J is its inertia aboutthe z axis, and g is the gravity acceleration. It is worth notingthat we have no direct control over the angles (roll) and (pitch), but only through aileron and fore-aft cyclic controlrespectively (see [9]). Nonetheless, in the remainder of this

Fig. 2. Simplified model of a helicopter. The system is actuated by the thrustT of the main rotor and the torque of the tail rotor. Angles (roll) and (pitch) are indirectly established by means of cyclic control of the aileron andthe fore-aft respectively, and therefore can represent two additional inputs.

section, we will consider them as real control inputs for thesake of simplicity.

The helicopters model is dynamically feedback equivalentto a linear system, as we demonstrate in the following. Takeas system outputs y1 = x, y2 = y, y3 = z, y4 = .Differentiating twice these outputs, yields

y1 = T SM ,y2 = T C SM ,y3 =

T C CM g ,

y4 = C C

J ,

(8)

where the inputs are nonlinearly coupled. As a firststep, we add one integrator on each input channel, andextend the system state by defining the auxiliary variablesT = u1, = u2, = u3, and = u4. Then, we differentiateonce more the equations in (8). Thus, we obtain:

x(3)

y(3)

z(3)

(3)

=

SM

T CM 0 0

C SM T S SM T C CM 0C C

M TSCM T C SM 00 S CJ C SJ C CJ

u1u2u3u4

== D(T, , , )u .

Under the hypothesis that T = 0, = 2 , = 2 ,and = arcsin

(C2S

), matrix D is nonsingular, and the

system can be exactly linearized by the choice u = D1v.With respect to the new input v, the systems dynamics isindeed composed of four chains of integrators, i.e. y(3)1 = v

1,

y(3)2 = v

2, y

(3)3 = v

3, y

(3)4 = v

4. Due to its linearity, the

system can be steered by means of the method describedin section (IV-B). First of all, observe that its equilibriummanifold is

E = {x IR12|x = (x, 0, 0, y, 0, 0, z, 0, 0, , 0, 0)} .Now, apply the discrete-time feeedback encoding of fig. 1-cwith unit sampling time t = 1s, and compute matrices S, V ,and K as in theorem 6. In the new coordinates, the equiliriummanifold is given by

E = {(x13, y13, z13, 13)} ,where x = 1 x, y = 2 y , z = 3 z , = 4 , and1 = 2 = 3 = 4 = = 1.

When building a lattice for the system, it is reasonable toask for a tolerance 1 on the x, y, and z coordinates which aremeasured in meters, and a different one, 2, on the variablewhich is instead measured in radiants. Take e.g. as numericalvalues 1 = 1m and 2 = 0.01rad, hence it holds 1 = 2 =3 = 2 and 4 = 0.02. Assume that all points in a hypercubeof sizes 32m32m32m0.32rad have to be reached, usingcontrol sets with m = 4. From [15], we know that the optimalchoice is 1W = 2W = 3W = 4W = {0,3,6,7,8}and every point can be reached within N = 2 steps. Similarlyto the previous example, the actual execution of the plan takesn = 3 times N sampling instants, because of the maximumdimension of the blocks in the Brunovsky form.

As for the helicopters motion, the following task is spec-ified: lift up of a relative altitude of 6m from the actualposition, rotate of an angle = /4 rad while hovering, travelhorizontally of a relative displacement (20m, 20m), and finallygo down to the initial altitude. Plans for steering the systemaccording to the task are computed as in theorem 6. Such plansand the corresponding state evolution are reported in fig. 3. Infig. 4 the helicopters trajectory and shots of the helicoptersposition and attitude are finally shown.

VI. CONCLUSIONS

In this paper, we have described methods for steeringcomplex dynamical systems by signals with finite-length de-scriptions. Systems tractable by symbolic control under encod-ing include all controllable linear systems, nilpotent driftlessnonlinear systems and (dynamically) feedback-linearizablesystems. Many other open problems remain open in order tofully exploit the potential of symbolic control. A limitation ofour current approach is that we assume that a flat, linearizingoutput to be available, as well as state measurements. Con-nections to state observers in planning are unexplored at thisstage. Future work will first investigate application of thesemethods to nondifferentially flat systems, and will enquire inthe additional possibilities which might be offered by usingdifferent feedback laws at different time instants. Finally,effective finite planning in presence of obstacles has not beenconsidered yet and would need thorough investigation.

VII. ACKNOWLEDGMENTS

This work was partially supported by EC through theNetwork of Excellence contract IST-2004-511368 HYCON- HYbrid CONtrol: Taming Heterogeneity and Complexity ofNetworked Embedded Systems, and Integrated Project con-tract IST-2004-004536 RUNES - Reconfigurable UbiquitousNetworked Embedded Systems.

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