IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 8, AUGUST 2014 2271

Computation of Time Optimal FeedbackControl Using Groebner Basis

Deepak U. Patil and Debraj Chakraborty

Abstract—The synthesis of time optimal feedback control of a singleinput continuous time linear time invariant (LTI) system is considered. Ifthe control input u(t) is constrained to obey |u(t)| ≤ 1, then it is knownthat the optimal input switches between the extreme values ±1, accordingto some “switching surfaces” in the state space. It is shown that for systemswith non-zero, distinct and rational eigenvalues, these switching surfacesare semi-algebraic sets and a method to compute them using Groebnerbasis, is proposed. In the process the null-controllable region for suchsystems is characterized and computed. Numerical simulations illustratethe proposed computational methods.

Index Terms— Linear time invariant (LTI), Pontryagin’s maximumprinciple (PMP).

I. INTRODUCTION

The classical time optimal control transfers the states of a lineartime invariant (LTI) system from a given initial state to the origin ofthe state space in the minimum possible time, using inputs from aconstrained set of inputs (e.g., |u| ≤ 1). The Pontryagin’s maximumprinciple (PMP) [1], shows that the open loop solution to this problemis bang-bang and provides necessary conditions characterizing theoptimal input. However, a feedback solution is highly desirable sincea) unlike the open loop solution, a feedback controller would notrequire recalculation of the entire input signal for each distinct initialcondition, and b) a feedback solution would be robust to uncertaintiesand disturbances and drive the system to origin repeatedly even ifthe system is forced away from the optimal trajectory due to externaldisturbances. Examples where time optimal feedback control is em-ployed includes, space-craft attitude control [2], robotic manipulators[3] and pursuit evasion games [4], etc. In spite of several attempts(e.g., [5], [6]), a general method for constructing the feedback solutionseems to be unavailable. In this article, we provide a partial solutionto this synthesis problem, by providing an algorithmic method forconstructing the time-optimal feedback function for controllable linearsystems with non-zero, distinct and rational eigenvalues.

We consider a nth order single input linear time invariant system

x(t) = Ax(t) +Bu(t); x(0) = x0 (1)

where the initial condition x0 ∈ X0 ⊆ Rn, x : R+ → Rn is the statevector and the input u(t) is constrained within a normalized set U ={u(t) : |u(t)| ≤ 1}. It was shown in [1], [7] that the time optimal con-trol switches between the extreme admissible values (±1) according tothe so called “switching surfaces” in the state space. Though, switchingsurfaces were constructed explicitly for some example second-ordersystems in [1], [7], the structure and properties of the switching

Manuscript received September 2, 2012; revised April 26, 2013,November 9, 2013; accepted January 19, 2014. Date of publication January 29,2014; date of current version July 21, 2014. This technical note was supportedin part by Indian Space Research Organization Grant ISRO/RES/STC-IITB/08-09, Bharti Center for Communication, IIT Bombay and Department of Scienceand Technology under Grant SB/S3/EECE/051/2013. Recommended by Asso-ciate Editor D. Hristu-Varsakelis.

The authors are with the Department of Electrical Engineering, Indian Insti-tute of Technology Bombay, India (e-mail: deepakp@ee.iitb.ac.in; dc@ee.iitb.ac.in).

Digital Object Identifier 10.1109/TAC.2014.2303239

surfaces were described much later by [5] and [6]. Subsequently,several articles considered the synthesis of these surfaces for limitedcases e.g., for second-order LTI systems [1], [8]–[10] and n-integratorchains [11]. But in general there is no method to synthesize suchsurfaces for nth-order systems.

In this article, for controllable systems with non-zero, distinct andrational eigenvalues, we show that the resulting switching surfaces canbe described by semi-algebraic sets. Moreover, using a Groebner basisbased implicitization algorithm [12], we derive implicit expressions, interms of polynomials and rational functions, describing the switchingsurfaces. A feedback logic for optimal switching is synthesized usingthese polynomials of the state variables. By extension, we showthat the set of initial conditions transferable to the origin is alsoa semi-algebraic set and algorithms for the computation of this setare provided. A preliminary version of these results was presentedin [13].

The use of Groebner basis for implicitization and other applicationsare described in [12], [14], [15]. Some relatively recent applications ofthis technique in systems theory can be found in [14], [16]–[19]. Allthe Groebner basis computations in this technical note were done with[20], [21].

II. TERMINAL MANIFOLDS AND SWITCHING LOGIC

The theory of terminal manifolds and switching locus [5] is re-viewed briefly in this section. Assume that the A matrix in the LTIsystem described by (1) has non-zero, distinct and rational eigenvalues(λ(A) ⊂ Q− {0}), and the pair {A,B} is controllable. The set ofstates of system (1) that can be steered to origin is characterized byx0 = −

∫ t

0e−AτBu(τ)dτ . If the input u(t) is allowed to vary over

the set of admissible inputs U , then x0 defines all initial conditionswhich can be steered to the origin in time t. Such initial conditions arecalled null-controllable states in time t and they are characterized bythe following set: R(t) = {x : x =

∫ t

0e−AτBu(τ)dτ, ∀u(t) ∈ U}.

The set of all null-controllable states is, X0 =⋃

t∈[0,∞)R(t). The

set X0 can be written in a parametric form by using the followingtheorem [5].

Theorem 1: For any k = 1, . . . , n and a sequence 0 < t1 < . . . <tk < ∞, consider the control u on [0, tk] with values ±1 alternatingin intervals [0, t1), . . . , [tk−1, tk). Then u is an optimal control, sothat the point, x = ±(−

∫ t1

0+∫ t2

t1− . . .+ (−1)k

∫ tk

tk−1)e−AτBdτ

has tk as the least time required to reach origin. Conversely, everyoptimal control on any interval [0, θ] where 0 < θ < ∞ is of thedescribed form.

To characterize all states which can be steered to origin in timetk using a bang-bang input with (k − 1)-switches, we define thefollowing functions:

F+k (t1, . . . , tk) =

⎛⎜⎝−

t1∫0

+ . . .+ (−1)k

tk∫tk−1

⎞⎟⎠ e−AτBdτ

F−k (t1, . . . , tk) = − F+

k (t1, . . . , tk). (2)

Observe that there is a physical binding on ti, i = 1, 2, . . . , k, i.e.,0 ≤ t1 ≤ t2 ≤ . . . ≤ tk. Let, Vk = {(t1, t2, . . . , tk) : 0 ≤ t1 ≤ t2 ≤. . . ≤ tk < ∞} for k = 1, . . . , n. Then the set of states, which canbe steered to origin in less than k-switches with u = +1 to begin with,is denoted by M+

k and can be defined as follows: M+k = {x : x =

F+k (v), ∀v ∈ Vk}. Similarly we can define M−

k = −M+k . Thus the

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2272 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 8, AUGUST 2014

Fig. 1. X0 = M+2 ∪M−

2 for case of example 4.

set of all states which can be steered to origin in n− 1 switches can bedefined as follows: Mn = {x : x = F±

n (v), ∀v ∈ Vn}. The followingresults hold [5]:

Corollary 2: For a system defined by (1), a bang-bang control on aninterval [0,∞) is time-optimal if it has at most n− 1 discontinuities.

Lemma 3: Mn = X0 [5].Lemma 3 is extremely useful for our purposes, since it states that

the null-controllable region with only bang-bang inputs (with at mostn− 1 discontinuities) is actually the entire null controllable region.Hence any state in X0 can be driven to the origin with such a bang-bang control input. Moreover, such an input is always time optimal.

It is further shown in [5], that the set X0 is divided symmetricallyinto two parts M+

n and M−n by Mn−1, which itself is divided into

two parts M+n−1 and M−

n−1 by Mn−2 and so on. In general wecan write, Mk = M+

k ∪M−k ∀k = 1, . . . , n and X0 = M+

n ∪M−n .

Clearly, M0 ⊂ M1 ⊂ . . . ⊂ Mn. We illustrate the structure of Mk fora simple 2-state system:

Example 4: For a second order LTI system x = Ax+Bu withA = diag(1, 2), B = [1 1]T and |u| ≤ 1, the corresponding structureof X0, which is divided in two parts namely M+

2 and M−2 by

M+1 ∪M−

1 is as shown in Fig. 1.Based on this structure of X0 an iterative switching logic can be

defined that drives any state in X0 to the origin in minimum time.For any initial condition x0 ∈ intX0, time-optimal switching shouldideally induce the following set of events. We assume x0 ∈ intM+

n

for illustration. A similar sequence would be valid for x0 ∈ intM−n

with opposite signs:

• Input u = 1 pushes x from M+n to the manifold M−

n−1.• As soon as x ∈ M−

n−1, switch 2 pushes x to M+n−2 and so on.

• Input at (n− 1)th switch pushes x from M+1 (if n is odd) or M−

1

(if n is even) to the origin.

On inspection of this logic, it seems that we need to compute M+k

and M−k (k = 1, . . . , n− 1) to implement feedback based switching.

Before we give an algorithm for this computation in the next section,we note the following:

Remark 5: Recall that A was assumed to have distinct rationaleigenvalues. Without loss of generality, we can further assume A tobe diagonal, for the construction of the switching surfaces Mk (k =1, . . . , n− 1) and for using the switching logic defined above. This isbecause, for any real similarity transformation x = Tx on a system,the corresponding set M+

k = {x = Tx : x ∈ M+k } and similarly

M−k = {x = Tx : x ∈ M−

k } [5]. Thus, computation of M+k and M−

k

for the diagonalized system is enough to compute the correspondingM+

k and M−k for all similar systems.

III. PARAMETRIC REPRESENTATION

In this section, we show, using a simple substitution of vari-ables, that the switching surfaces Mk, k = 1, . . . , n− 1 and thenull-controllable set X0 = Mn can be represented parametri-cally as polynomials. These polynomials are then implicitized inSections IV and V, using Groebner basis techniques to get polynomialequalities and inequalities involving only the state variables. Thesesemi-algebraic sets define both the switching surfaces and the null-controllable region.

Recall that all x ∈ Mk are characterized by the functions F+k

or F−k defined in (2), which takes t1, . . . , tk as arguments. Since

we can assume A to be diagonal without loss of generality (seeSection II: Remark 5 above), we can alternatively write each com-ponent of state x (xi, i = 1, . . . , n) as some other function, de-noted as f±

ki, which takes arguments e−λit1 , . . . , e−λitk for alli = 1, . . . , n. Here f+

ki corresponds to F+k and f−

ki to F−k . Thus,

xi = f±ki(e

−λit1 , . . . , e−λitk ) ∀i = 1, . . . , n; λi ∈ λ(A). Since theeigenvalues of A are assumed to be rational, the denominator ofλi is denoted by di (i = 1, . . . , n), and the least common multi-ple of denominators di’s by l = lcm(d1, . . . , dn). Substituting zi =e−(ti/l) ∀i = 1, . . . , k, we can get parametric representations of xi’sin terms of zi’s. These representations are either polynomial or rationalfunctions depending on the signs of the eigenvalues of A. It is evidentthat if the eigenvalues of A are either all positive or all negative,xi’s can be expressed as polynomials in zi’s. On the other hand, ifthe eigenvalues (recall that the eigenvalues are rational and non-zero)of A have mixed signs, we get rational parametric representationsof xi’s.

Consider the case wherein some eigenvalues of A are positiveand the remaining are negative: Rearrange the eigenvalues suchthat λi, i = 1, . . . , q are positive and λj , j = q + 1, . . . , n (i.e. n−q eigenvalues) are negative. On substituting zi = e−(ti/l) ∀i =1, . . . , k, we get, xi = f+

ki(zpi1 , . . . , zpik ), i = 1, . . . , q and xi =

f+ki(z

−pi1 , . . . , z−pi

k ), i = q + 1, . . . , n where pi = |lλi| is an integer.While xi’s for i = 1, . . . , q are polynomials in z1, . . . , zk, the xi’sfor i = q + 1, . . . , n can be written only as rational functions in thevariables z1,, . . . , zk. Hence for both sets of xi’s defined above: xi =N+

ki(z1, . . . , zk)/D+ki(z1, . . . , zk), i = 1, . . . , n, where N+

ki and D+ki

are polynomial numerators and denominators of fki respectively andD+

ki = 1 ∀i = 1, . . . , q. This gives the following rational representa-tion for M+

k and M−k :

M+k =

{(x1, . . . , xn) : xi=

N+ki(z1, . . . , zk)

D+ki(z1, . . . , zk)

and (4)

}(3)

0<zk ≤ zk−1 ≤ . . . ≤ z1 ≤ 1

M−k =−M+

k . (4)

Remark 6: Note that by substituting zi = e−(ti/l) (for the casewhere all eigenvalues of A are positive) or zi = eti/l (for the casewhere all eigenvalues of A are positive), we have D+

n−1,i = 1 andN+

n−1,i = f+n−1,i for all i = 1, . . . , n (which means that xi’s can be

expressed as polynomials in zi’s).The above expression for M+

k and M−k [i.e., (3) and (4)] involves

z1, . . . , zk and hence cannot be used for state based switching. Itwould be convenient to eliminate the variables z1, z2, . . . , zk from therepresentation of M+

k and M−k and find an alternate representation

of M+k and M−

k in terms of only the state variables x1, . . . , xn.Such a representation can be used directly for state-feedback basedswitching of the input values between ±1. For this purpose, we willuse an implicitization method based on the construction of Groebnerbases to eliminate z1, . . . , zk and form an implicit representation for

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 8, AUGUST 2014 2273

the set M±k . More details on Groebner basis methods can be found in

[12], [14].

IV. SWITCHING SURFACE AND FEEDBACK LOGIC

From the switching logic outlined at the end of Section II, it seemsthat we need all the nested switching surfaces Mk, k = 1, . . . , n− 1for implementing the switching feedback law. However, in practice, wewill show that the lower dimensional switching surfaces (i.e., M±

k fork < n− 1) are not essential for this purpose. Hence we only describethe implicitization procedure for M±

n−1. If required, M±k , k < n− 1

can be computed from M±n−1 easily (see remark 8).

A. Implicitization

1) Implicitization of (3): Parametric representations of M±n−1 is

described by (3) for k = n− 1. To eliminate zk, k = 1, . . . , n− 1from (3) we will follow the standard implicitization steps (please referto [12]) described next:

a) Form an ideal J+n−1=〈D+

n−1,1x1 −N+n−1,1, . . . ,D

+n−1,nxn −

N+n−1,n, 1−D+

n−1,1D+n−1,2 . . .D

+n−1,ny〉.

b) Compute Groebner basis G+n−1 of J+

n−1 w.r.t. lexicographicordering as y � z1 � z2 � . . . � zn−1 � x1 � . . . � xn.

c) The element g+n−1 ∈ G+n−1 ∩Q[x1, . . . , xn] defines the small-

est variety containing the parametric representation xi = f+n−1,i.

Similarly for M−n−1.

Remark 7: In general, the variety obtained by Groebner basis basedelimination i.e., {(x1, . . . , xn) : g

+n−1(x1, . . . , xn) = 0}, might be

larger than {(x1, . . . , xn) : xi=(N+n−1,i(z1, . . . , zn−1)/ D+

n−1,i(z1,

. . . , zn−1)), i = 1, . . . , n} and in particular, than M+n−1. However,

by existence and uniqueness of time-optimal control, the set ofsolutions of the equation g+n−1(x1, . . . , xn) = 0, for which thereexist z1, . . . , zn−1 satisfying the inequality (4) is exactly equalto M+

n−1 i.e., M+n−1 = {(x1, . . . , xn) : g

+n−1(x1, x2, . . . , xn) = 0,

for 0 < z+n−1 ≤ z+n−2 ≤ . . . ≤ z+1 ≤ 1}.2) Implicitization of (4): For M+

n−1, we denote the respective zkby z+k and similarly for M−

n−1. To describe M+n−1 completely and

ensure that x1, . . . , xn ∈ X0, we need to impose condition (4) inaddition to g+n−1 = 0. For that we need to rewrite inequality (4) interms of x1, x2, . . . , xn. This is accomplished by solving each zkin terms of x1, x2, . . . , xn and then imposing inequality (4). Thetechnique described below for this purpose, works identically foreach z+k , k = 1, . . . , n− 1. Let G+

zkbe the Groebner basis obtained

by using ordering . . . � . . . � . . . � z+k � x1 � . . . � xn. Considerthe set G+

zk∩Q[zk, x1, . . . , xn]: let there be m elements in this set

and let us denote the elements by g+1 , g+2 , . . . , g+m, each of whichis a polynomial in the variables (zk, x1, . . . , xn). Now, it might bepossible to explicitly solve at least one among g+i = 0 (i = 1, . . . ,m),for z+k (i.e., z+k can be expressed as a closed form expression of(x1, . . . , xn)). This happens, for example, when in any one polyno-mial among g+i (i = 1, . . . ,m), all the monomials containing z+k havethe same degree of z+k . It turns out that such a situation occurs in allthe numerical problems described in this technical note.

However, in general, it is possible that all the polynomials g+i = 0(i = 1, . . . ,m) contain monomials with varying degree in z+k andconsequently an explicit (closed form) solution for z+k is not possible.In that case, whether the current state vector (x1, . . . , xn) satisfies(4), can be checked based on the following real time computation.We substitute (x1, . . . , xn) in to any one (usually the simplest) ofg+i = 0 (i = 1, . . . ,m), which then becomes a polynomial of thesingle variable z+k . This polynomial is then solved numerically for

z+k . Finally we need to check whether any one set of solutions z+k(k = 1, . . . , n− 1) satisfies the corresponding inequality (4).

Remark 8: Though unnecessary in this scheme, it should be notedthat all the lower dimensional switching surfaces Mn−k can be com-puted using Mn−1 itself by M+

n−k = {(x1, . . . , xn) : g+n−1(x1, x2,

. . . , xn) = 0, for 0 < z+n−1 ≤ z+n−2≤ . . .≤z+k−1 = . . . = z+1 = 1}.Remark 9: Further note that, the computationally intensive

Groebner basis based implicitization is done off line, whereas the onlyon line computation required to implement the feedback switchinglogic involves evaluation of multivariate polynomials and in the worstcase, solution of univariate polynomial equations.

B. Switching Algorithm

Using the above expression for Mn−1, any point on the statespace (x1, . . . , xn) can be tested to check whether it belongs to theswitching surface Mn−1. Now recall the first step of the switchinglogic described at the end of Section II, which predicts that anoptimal control for state trajectory starting from Mn should changeits sign on intersecting Mn−1. Since we now have a description ofMn−1, this logic can now be implemented directly. However, an idealimplementation of the switching logic would require that the inputshould switch sign as soon as x(t) ∈ Mn−1. The trajectory, which isnow in Mn−1, will travel along Mn−1 until it intersects Mn−2. Thissituation should repeat for all the nested lower dimensional surfaces.Hence, clearly, for the ideal realization of optimal switching, we needto check x(t) ∈ Mn−1 instantaneously.

In any actual realization of the switching logic (either in simulationor in practice) due to unavoidable delay in computation (say δ sec-onds), the trajectory overshoots Mn−1 at the first switch. The state-trajectory, which ideally should have stayed in Mn−1 after hittingMn−1, overshoots and enters Mn slightly before the first switchoccurs. Since the state is now in Mn the switching surface Mn−1

(instead of Mn−2 required for ideal switching) can be used again forthe second switch. This situation repeats for all successive switchesand hence the entire switching law can be summarized as Algorithm 1below. The initial input u(0) can be determined by checking whetherinitial condition x0 is in M+

n or M−n by using the algorithm described

in Section V. Moreover, it is evident that since Mn−1 is a set ofmeasure zero in Mn, generically, x0 ∈ Mn\Mn−1.

Algorithm 1 Switching logic

Require: current x(t) and u(t)if x(t) ∈ Mn−1 thenu(t+ δ) = −u(t)

elseu(t+ δ) = u(t)

end if

We need to guarantee that the trajectory generated by such delayedswitching remains close to the actual time optimal trajectory andhence reaches arbitrarily close to the origin. This is shown below inLemma 10, for asymptotically decreasing switching delays.

Let a state trajectory start from an initial condition (x0, t0) withx0 ∈ int(X0 \Mn−1) (This is generically true since Mn−1 is alower dimensional surface) and switch according to Algorithm 1, withdelays δ1, δ2, . . . , δn at the first n switchings. Let the correspondingswitchings occur for state values x1, x2, . . . , xn at the switchinginstants t1, t2, . . . , tn. We compare this trajectory to the correspondingoptimal trajectory starting from x0. Let the optimal trajectory switchat the values x′

1, x′2, . . . , x

′n at times t′1, t

′2, . . . , t

′n. Clearly, x′

n = 0.

2274 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 8, AUGUST 2014

Because of the switching delays (say δi), the optimal trajectory shouldswitch before the actual trajectory switches, such that t1 = t′1 + δ1.Denote the optimal trajectory by xo(t) and the trajectory generated byAlgorithm 1 by xd(t). The following result holds:

Lemma 10: For any x0 ∈ int(X0 \Mn−1) and any ε > 0, thereexist switching delays δ1, δ2, . . . , δn with δi > 0 ∀i = 1, . . . , n suchthat ‖xo(t)− xd(t)‖ < ε ∀t ∈ [t0, tn] and ‖xn‖ → 0 as max(δi) →0, i = 1, . . . , n.

Proof: Let x0 ∈ int(M+n ) (the following proof holds similarly

for x0 ∈ int(M−n )). Then, the first switch occurs for xo(t) when it

reaches M−n−1, at (x′

1, t′1), where x′

1 ∈ M−n−1. The xd(t), on the other

hand, switches at (x1, t1) where x1 ∈ M−n and t1 = t′1 + δ1. Since

x(t) is a continuous function of t (see [5]), for any ε > 0, ∃δ1 suchthat ‖x1 − x′

1‖ < ε. The second portion of xo(t) starts from (x′1, t

′1)

and ends on M+n−2 at (x′

2, t′2), where x′

2 ∈ M+n−2. The second portion

of xd(t) starts from (x1, t1) on M−n , intersects M+

n−1 at (x′′2 , t

′′2 ),

overshoots due to the delay δ2 and switches at (x2, t2), where x2 ∈M+

n and t2 = t′′2 + δ2. Using similar arguments as above it is possibleto choose δ2 such that x2 ≈ x′′

2 .Next we claim that x′

2 ≈ x′′2 . To prove this we consider two time

optimal trajectories starting from x1 and x′1 respectively. Since x1 ∈

M−n and x′

1 ∈ M−n−1, the first portions of both these trajectories use

the same initial input u = −1 and these trajectories will first switch at(x′′

2 , t′′2 ) and (x′

2, t′2) respectively. Evidently, the switching instants for

these two trajectories will be the corresponding solutions of x1 = F−n

and x′1 = F−

n , which can be converted in to (see Section IV-A2 andV) polynomials in (x1; z

′′2 ) and (x′

1; z′2) respectively (where z′′2 and

z′2 are the transformed t′′2 and t′2 respectively). Now, it is well knownthat roots of univariate polynomials are continuous functions of itscoefficients. Hence, since x1 ≈ x′

1, it follows that z′′2 ≈ z′2, t′2 ≈ t′′2and (t′2 − t′1) ≈ (t′′2 − t1). Then from continuity of the vector field(corresponding to u = −1), it follows that x′

2 ≈ x′′2 . Hence, there ex-

ists δ2 such that x2 ≈ x′2. This argument extends to subsequent switch-

ing instants i = 2, . . . , n, showing that there exists δ1, δ2, . . . , δi suchthat xi ≈ x′

i for all i = 1, . . . , n. Therefore, we conclude that for anyε > 0, there exist switching delays δ1, δ2, . . . , δn with δi > 0 ∀i =1, . . . , n such that ‖xo(t)− xd(t)‖ < ε ∀t ∈ [t0, tn]. Moreover, asmax(δi) → 0, we have xi → x′

i for all i = 1, . . . , n. In particularxn → x′

n. But, we have x′n = 0. Thus ‖xn‖ → 0 as max(δi) → 0 for

all i = 1, . . . , n. �Remark 11: In practical implementation with a fixed sampling

frequency, the trajectory may intersect Mn−1 in between two samplinginstants, whereas at the sampling instants g±n−1 �= 0. However, bycontinuity of the (real parts of) solutions of polynomial equations,the corresponding inequalities (4) are valid in small neighborhoodsof Mn−1. Hence these inequalities are checked first at each sam-pling instant, and once any one set of inequalities are valid, g±n−1

is checked for changes of sign between two consecutive samplinginstants. A change of sign indicates that the trajectory has justcrossed Mn−1.

C. Examples of (Near) Time Optimal Feedback

Example 12: Consider a system defined as (1) with A =diag(1, 2, 3) and B = [1 1 1]T . The points on the switching sur-face are described by M+

2 and M−2 . The equations x = F+

2 (z1, z2)and F−

2 are calculated using (3) and (4). The polynomial equationsx = F+

2 , along with the inequalities 0 < z2 ≤ z1 ≤ 1 describe thesurface M+

2 [see (3) and (4)]. To eliminate z1 and z2 from thedescription of M+

2 we follow the implicitization procedure describedin Section IV. We form the following ideal from the polynomialsin F+

2 : J = 〈x1 − 2z1 + z2 + 1, x2 − z21 + (1/2)z22 + (1/2), x3 −(2/3)z31 + (1/3)z32 + (1/3)〉. A successful elimination of z1 and z2

Fig. 2. Points on the switching surface (example 12).

would give an equation of the form g+2 (x1, x2, x3) = 0. For thispurpose we calculate the Groebner basis G of the ideal J with theordering as z1 � z2 � x1 � x2 � x3. Only relevant elements of thebasis are shown below:

s1=x61 + 6x5

1 − 6x41x2 + 12x4

1 − 24x31x2 − 24x3

1x3

+ 36x21x

22 − 72x2

1x3 − 18x21 + 72x1x

22 + 72x1x2x3

+ 72x1x2 − 36x1x3 − 72x32 − 72x2

2 + 72x2x3 − 18x23

s5=3z2(x21+6x1−6x2)+x3

1+3x21−6x1x2−6x2+6x3. (5)

Clearly s1 does not involve z1 or z2; thus g+2 (x1, x2, x3) = s1. Alongwith conditions 0 < z2 ≤ z1 ≤ 1 on z1 and z2, g+2 (x1, x2, x3) = 0describes M+

2 . We observe s5 contains only degree one terms in z2.Thus, we can write an expression for z2 as, z2 = (−(x3

1 + 3x21 −

6x1x2 − 6x2 + 6x3))/(3x21 + 6x1 − 6x2). Similar expression for z1

can be found from the Groebner basis of the ideal J with orderingz2 � z1 � x1 � x2 � x3, which, after similar calculations turn out tobe z1 = (−(−x3

1 − 3x21 − 3x1 + 3x3))/(3x

21 + 6x1 − 6x2). There-

fore, M+2 = {(x1, x2, x3) : s1 = 0, for 0 < z2 ≤ z1 ≤ 1}. One can

obtain M−2 just by replacing (x1, x2, x3) as (−x1,−x2,−x3) in M+

2 .Fig. 2 shows points on M+

2 and M−2 . The feedback logic can be

implemented according to Algorithm 1.Next we compare, for a sample initial condition, the trajectories gen-

erated according to Algorithm 1, with those generated by optimal openloop input computed according to PMP. For initial condition x0 =[0.2 0.1 0.1] we have u0 = −1. The respective calculated open-loopswitching instants (according to PMP) are t1 = 0.293, t2 = 0.906,and at t3 = 2.15 the system states reach the origin. We implementthe proposed Algorithm 1 for this value of x0. It can be observed fromFig. 3, that the open-loop switching instants and closed loop switchinginstants are matching.

Example 13: Next we consider a similar system to the examplesabove except that, A = diag(1,−2, 3). Fig. 4 shows the switchingsurface. Note that the switching surface is bounded in the unsta-ble directions: x1 and x3, whereas it is unbounded in the stabledirection x2. We have arbitrarily truncated the switching surfacein the x2 direction between −12 ≤ x2 ≤ 12, in Fig. 4 for ease ofrepresentation.

V. REGION OF NULL-CONTROLLABILITY (X0)

According to the discussion from Section II, X0 = M+n ∪M−

n . Re-call from Section III, the different representations of Mn according tothe sign of the eigenvalues of A. It was shown in [22], [23] that if A =diag(A1, A2), with A1 ∈ R(n−m)×(n−m) is with positive eigenvaluesand A2 ∈ Rm×m is with negative eigenvalues and B is partitionedaccordingly as B = [BT

1 BT2 ]

T then, X0 = X ′0 × Rm where X ′

0 is

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 8, AUGUST 2014 2275

Fig. 3. State against time for x0 = (0.2, 0.1, 0.1) (example 12).

Fig. 4. Points on the switching surface (example 13).

the null-controllable region of the system x1 = A1x1 +B1u. Thus, itis enough to consider only the case in which all eigenvalues of A arepositive.

Given any arbitrary x ∈ Rn, we would like to check whetherx ∈ X0. We know, M+

n = {xi = f+ni(z

pi1 , . . . , zpin ) ∀i = 1, . . . , n :

0 < zn ≤ zn−1 ≤ . . . ≤ z1 ≤ 1}. Since each xi is expressed as apolynomial in zi, i = 1, . . . , n, we can systematically eliminate zi’s byusing the Groebner basis based elimination technique. We first form anideal J+

n = 〈x1 − f+n1, . . . , xn − f+

nn〉 ⊂ Q[z1, . . . , zn, x1, . . . , xn],and by using lexicographic ordering z1 � z2 � . . . � zn � x1 �. . . � xn, compute Groebner basis G+

n for J+n . Then select the el-

ement in G+n which contains none of the variables z1, z2, . . . , zn−1

i.e., g+n ∈ G+n ∩Q[x1, . . . , xn, zn] and solve for zn. Since we know

the test point x, the solution of g+n = 0 for zn involves, solv-ing a univariate polynomial equation. Now, select an element inG+

n which contains none of the variables z1, z2, . . . , zn−2 i.e.,g+n−1 ∈ G+

n ∩Q[x1, . . . , xn, zn, zn−1]. Using the values of zn ob-tained in the previous step solve for zn−1 and so on. Finally,select an element in G+

n which contains all variables z1, . . . , zni.e., g+1 ∈ G+

n ∩Q[x1, . . . , xn, zn, zn−1, . . . , z1]. Using the values ofzn, zn−1, . . . , z2 obtained in the previous steps solve for z1. Now,if there exist any one combination of zn, zn−1, . . . z1 satisfying theinequality 0 < zn ≤ zn−1 ≤ . . . ≤ z1 ≤ 1 then x ∈ M+

n . A similarprocedure holds for M−

n . However, the above procedure involvessolving more than n univariate polynomial equations in z1, . . . , znand verification of n-inequalities. This can be numerically difficult forlarge system orders. Hence we propose a simpler sufficient conditionto test whether x ∈ X0.

Consider g+n (x1, . . . , xn, zn) ∈ G+n ∩Q[x1, . . . , xn, zn] defined

above. By the very nature of the parametrization, tn → ∞ implies

zn = e−tn/l → 0 (where l = lcm(d1, d2, . . . , dn) is the least com-mon multiple of the numbers d1, d2, . . . , dn). Substituting zn = 0in the equation g+n = 0 identifies exactly those (x1, . . . , xn) fromwhere it takes infinite time to reach origin. Hence by puttingzn = 0, we can select a surface which contains the boundaryof set M+

n . A similar procedure holds for M−n . Hence the sets

{(x1, . . . , xn) ∈ Rn : g+n (x1, . . . , xn, 0) = 0} and {(x1, . . . , xn) ∈Rn : g−n (x1, . . . , xn, 0) = 0} together contains the boundary of null-controllable region. The following theorem 14 uses these implicitexpressions for the boundary of X0 and gives a sufficient conditionto check whether a given point x ∈ X0:

Theorem 14: If for all μ ∈ [0, 1], g+n (μx, 0) �= 0 and g−n (μx, 0) �=0 then x ∈ X0.

Proof: By lemma 3, X0 = Mn. Also, 0 ∈ Mn and Mn = M+n ∪

M−n . The boundary of M+

n is contained in the set {(x1, . . . , xn) ∈Rn : g+n (x1, . . . , xn, 0) = 0} and similarly boundary of M−

n is con-tained in {(x1, . . . , xn) ∈ Rn : g−n (x1, . . . , xn, 0) = 0}. Now, for allx ∈ X0, the line segment joining x and origin should not inter-sect boundary of X0. Thus, for all μ ∈ [0, 1], if g+n (μx, 0) �= 0 andg−n (μx, 0) �= 0 then x ∈ X0. �

However, this condition is only sufficient since parts of the surfacesdefined by g+n = 0 and g−n = 0 can lie in the interior of X0. Theneither g+n (μx, 0) = 0 or g−n (μx, 0) = 0 can hold for some μ even if aparticular x ∈ X0. In such a situation, we check whether μx ∈ X0 byusing the inequalities described above.

Example 15: We use the system from example 12. The set X0 isdivided into two parts: M+

3 and M−3 . The functions F+

3 and F−3 are

computed using (2) and converted into polynomials using (3). Next weform an ideal I = 〈2z1 − 2z2 + z3 − x1 − 1, z21 − z22 + (1/2)z23−x2−(1/2), (2/3)z31−(2/3)z32+(1/3)z33−x3−(1/3)〉. We computeGroebner basis G of ideal I w.r.t. lexicographic ordering z1 � z2 �z3 � x1 � x2 � x3. We select an element in G ∩Q[z3, x1, x2, x3]which is as follows: s = 3z43 − 12z33x1 − 12z33 − 6z23x

21 − 12z23x1+

48z23x2 + 18z23 + 4z3x31 + 12z3x

21 + 12z3x1−48z3x3−4z3−x4

1 −4x3

1−2x21+48x1x3+12x1−48x2

2 − 48x2 + 48x3 + 1. Thus g+3 =sand the set of points satisfying g+3 (x1, x2, x3, 0) = 0 andg+3 (−x1,−x2,−x3, 0) = 0 contains the boundary of set X0. For thisexample discontinuities −x4

1 − 4x31−6x2

1+48x1x3+12x1− 48x22−

48x2 + 48x3 + 3 = 0 and −x41 + 4x3

1 − 6x21 + 48x1x3 − 12x1 −

48x22+ 48x2 − 48x3 + 3 = 0 form the boundary of X0.

Next we check whether the point x = [0.2, 0.1, 0.1] lies in X0 byapplying theorem 14. For a given x, we get g+3 (μx, 0) = μ4 − 20μ3 −150μ2 + 1500μ− 1875. Roots of this polynomial are −23.79, 10.32,−5.00 and −1.527. Similarly the roots of the polynomial g−3 (μx, 0) =g+3 (−μx, 0), are 23.79, −10.32, 5.00 and 1.527. None of these rootslie in the interval [0,1] which proves that x ∈ X0.

VI. CONCLUSION

In this article we use a Groebner basis based implicitization tech-nique to synthesize a feedback based switching law that nearly pro-duces time-optimal switching. This switching feedback law robustlydrives the system to an arbitrarily small neighborhood of origin infinite time. The proposed method is currently limited to controllablediagonalizable systems with non-zero rational eigenvalues. Relaxingsome of these assumptions is the subject of current and future research.

REFERENCES

[1] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze, and E. Mischenko,The Mathematical Theory of Optimal Control. New York: IntersciencePublishers, 1962.

[2] J. Liang, R. Fullmer, and Y. Q. Chen, “Time-optimal magnetic attitudecontrol for small spacecraft,” in Proc. 43rd IEEE Conf. Decision Control(CDC’04), Dec. 2004, vol. 1, pp. 255–260.

2276 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 8, AUGUST 2014

[3] E. Sontag and H. Sussmann, “Time-optimal control of manipulators,” inProc. IEEE Int. Conf. Robot. Autom., 1986, vol. 3, pp. 1692–1697.

[4] S. H. Lim, T. Furukawa, G. Dissanayake, and H. Durrant-Whyte, “A time-optimal control strategy for pursuit-evasion games problems,” in Proc.IEEE Int. Conf. Robot. Autom., 2004, vol. 4, pp. 3962–3967.

[5] O. Hajek, “Terminal manifolds and switching locus,” Theory Comput.Syst., vol. 6, no. 3, pp. 289–301, 1971.

[6] D. S. Yeung, “Time-optimal feedback control,” J. Optim. Theory Appl.,vol. 21, no. 1, pp. 71–82, 1977.

[7] R. V. Gamkrelidze, “On theory of optimal processes in linear systems,”USSR Acad. Sci., vol. 116, no. 1, pp. 9–11, 1957.

[8] E. B. Lee and L. Markus, Foundations of Optimal Control Theory.New York: Wiley, 1967.

[9] E. B. Lee, “Mathematical aspects of the synthesis of linear minimumreponse-time controllers,” IRE Trans. Autom. Control, vol. 5, no. 4,pp. 283–290, Sep. 1960.

[10] M. Athanassiades and O. J. M. Smith, “Theory and design of high-orderbang-bang control system,” IRE Trans. Autom. Control, vol. 6, no. 2,pp. 125–134, May 1961.

[11] U. Walther, T. T. Georgiou, and A. Tannenbaum, “On the computation ofswitching surfaces in optimal control: A gröbner basis approach,” IEEETrans. Autom. Control, vol. 46, no. 4, pp. 534–540, Apr. 2001.

[12] D. Cox, J. Little, and D. O’Shea, Ideals, Varieties and Algorithms: AnIndroduction to Computational Algebraic Geometry and CommutativeAlgebra. New York: Springer, 2007.

[13] D. Patil and D. Chakraborty, “Switching surfaces and null-controllableregion of a class of lti systems using groebner basis,” in Proc. 51st IEEEAnnu. Conf. Decision Control (CDC), Dec. 2012, pp. 1279–1284.

[14] B. Buchberger, “Gröbner bases and systems theory,” MultidimensionalSyst. Signal Process., vol. 12, pp. 223–251, Jul. 2001.

[15] Z. Lin, L. Xu, and N. Bose, “A tutorial on gröbner bases with applicationsin signals and systems,” IEEE Trans. Circuits Syst. I, vol. 55, pp. 445–461,Feb. 2008.

[16] H. Shin and S. Lall, “Decentralized control via groebner bases and vari-able elimination,” IEEE Trans. Autom. Control, vol. 57, no. 4, pp. 1030–1035, 2012.

[17] J. Helton, F. Kronewitter, and M. Stankus, “Noncommutative computeralgebra in the control of singularly perturbed dynamical systems,” in Proc.38th IEEE Conf. Decision Control, 1999, vol. 4, pp. 4086–4091.

[18] N.-P. Ke, “Computer algebra for exact complex stability margin com-putation,” in Proc. 39th IEEE Conf. Decision Control, 2000, vol. 5,pp. 4235–4240.

[19] M. Yagi, T. Hisakado, and K. Okumura, “Algebraic representation oferror bounds for describing function using groebner base [nonlinear cir-cuit analysis example],” in Proc. IEEE Int. Symp. Circuits Syst., 2005,pp. 2831–2834.

[20] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, SINGULAR

3-1-5—A Computer Algebra System for Polynomial Computations,2012. [Online]. Available: http://www.singular.uni-kl.de

[21] W. Stein and D. Joyner, Sage Mathematics Software (Version 4.3), 2009.[Online]. Available: http://www.sagemath.org

[22] T. Hu and Z. Lin, Control Systems With Actuator Saturation: Analysis andDesign. Boston, MA: Birkhäuser, 2001.

[23] O. Hajek, Control Theory in the Plane. New York: Springer, 2008.