Controllability Issues in Nonlinear State-Dependent

Riccati Equation Control�

Kelly D. Hammetty, Christopher D. Hallz, and D. Brett Ridgelyx

GraduateSchool of Engineering

Ai r Force Instituteof Technology

Wright-Patterson Ai r ForceBase, Ohio 45433-7765

�This paper is declared a work of the U. S. Government and is not subject to copyright protection in the

United States. Thiswork is unclassified and cleared for public release. Sponsored by AFOSR/NM.yCaptain, U.S. Ai r Force; Graduate Student, Department of Aeronautics and Astronautics; currently Con-

trols Engineer, Ai r Force Research Laboratory, Kirtland Air Force Base, New Mexico 87117. E-mail: ham-

mettk@plk.af.milzAssistant Professor of Aerospaceand SystemsEngineering, Department of Aeronauticsand Astronautics;

currently Assistant Professor, Department of Aerospaceand Ocean Engineering, VirginiaPolytechnic Institute

and StateUniversity, Blacksburg, Virginia24061-0203. E-mail: chall@aoe.vt.eduxAssociate Professor of Aerospace Engineering, Department of Aeronautics and Astronautics; currently

Senior Engineer, HughesMissile Systems, Tucson, AZ 85734. E-mail: dbridgely@ccgate.hac.com

1

Abstract

In the last few years, algorithms using state-dependent Riccati equations (SDREs)

have been proposed for solving nonlinear control problems. Under state feedback, point-

wise solutions of an SDRE must be obtained along the system trajectory. To ensure

the control is well-defined, global controllability and observability of state-dependent

system factorizations are commonly assumed. This paper rigorously establishes con-

nections between controllability of the state-dependent factorizations and true system

controllability. It is shown that a local equivalence always holds for the class of systems

considered, and special cases which imply global equivalence are also given. Addition-

ally, a notion of nonlinear stabilizability is introduced, which is a necessary condition

for global closed loop stability. The theory is illustrated by application to a five-state

nonlinear model of a dual-spin spacecraft.

1 Introduction

Since the early 1960’s, a number of researchers have proposed nonlinear control algorithms

which involve application of linear design methods to linear-like ‘factored’ representations

of a nonlinear system [1, 2, 3, 4, 5, 6, 7]. For continuous time, state feedback, input-affine,

autonomous nonlinear dynamic systems

_x = a(x) + b(x)u; a(0) = 0

z =

26664

h(x)

�R(x)u

37775 ; h(0) = 0 (1)

2

with state vectorx 2 Rn, control vectoru 2 Rm, penalized variablez 2 Rs, and non-

singular (for allx) control penalty matrix function�R(x), it is assumed that one can obtain

linear-looking factored representations of the form

a(x) = A(x)x; b(x) = B(x); h(x) = H(x)x (2)

This concept has alternatively been called ‘apparent linearization’ [3], ‘extended lineariza-

tion’ [5], or most recently ‘state-dependent coefficient factorization’ [6] of (1).

The earliest known proposal for using state feedback Riccati-based linear control meth-

ods on nonlinear systems appears in [1], motivated by finite-time suboptimal regulation. In

[1], the steady state stabilizing solution to a state and time-dependent Riccati differential

equation (RDE) is obtained analytically for low-order example systems to provide a basis

for a suboptimal control algorithm. Global asymptotic stability for each example is proven,

and suboptimal controller performance is shown to compare quite favorably with that of the

corresponding optimal controllers. Variations on this approach are proposed in [2] and [3],

which also involve solving an algebraic state-dependent Riccati equation (SDRE) for any

location traversed in the state space. In [4], the same basic idea is revisited, and conditions

relating the suboptimal solution to the optimal solution are derived. In both [3] and [4], the

state and control weights�R andH are assumed to be constant matrices, so that the regula-

tion problem is indeed quadratic, while in [1] and [2],�R andH are arbitrary time-varying

matrices.

More recently, in [5] application of any linear control algorithm to (2) is suggested, but no

theoretical justification for such an approach is given. In [6] both state and output feedback

3

SDRE approaches to regulation and nonlinearH1 control problems are proposed, where

weighting matrices are not restricted to be constants or functions of time, but may instead

be functions ofx. Local stability is proven for suboptimal state feedback versions of the

above, and an additional necessary condition which must be satisfied for optimality of the

state feedback regulator is given. In [7] a Lyapunov function is proposed for establishing

global stability of the suboptimal SDRE state feedback regulator, based on a restricted class

of weighting matrix functions.

While some progress has been made in theoretically justifying such ‘linearized’ meth-

ods, much remains to be done. In this paper we develop some of the needed theory, by

examining controllability issues in the context of the state feedback regulator problem. We

derive conditions under which global factored and true nonlinear controllability hold, and

show how each type of controllability is separately important to successful application of the

SDRE approach. These issues have not been addressed in the literature, and have signifi-

cant implications for both factorization selection and global stability of SDRE-based control

algorithms.

The outline of the paper is as follows. In Section 2 we briefly review the SDRE nonlinear

regulation control algorithm. We then establish connections between factored and true non-

linear controllability in Section 3. In Section 4, we formalize our results, and illustrate the

given theorems with simple examples. We then demonstrate the significance of our results

by analyzing the impact of both types of controllability issues on SDRE nonlinear regulation

of a more realistic dynamic model in Section 5. Finally, we summarize and conclude in

4

Section 6.

2 Control Algorithm

We assume thata andh in (1) are real-valuedC1 functions ofx on Rn. Under this as-

sumption the system (1) can be written [8] (nonuniquely) in the so-called state-dependent

coefficient (SDC) form (2), whereA andH are chosen to be (at least) continuous. The con-

trol algorithm may is as follows. Consider any initial conditionx0 2 Rn. The objective is

near-optimal regulation, i.e., drive the state to zero while simultaneously keeping the cost

function

J =Z1

0

zT z dt =Z1

0

xTQ(x)x + uTR(x)u dt (3)

‘close’ to its optimal value, where we have definedR(x) = �RT (x) �R(x) > 0 andQ(x) =

HT (x)H(x) � 0 8 x. This may be accomplished locally [6, 7] by setting

u(x) = �R�1(x)BT (x)P (x)x (4)

whereP (x) is the maximal, stabilizing solution to the algebraic continuous time SDRE

AT (x)P (x) + P (x)A(x)� P (x)B(x)R�1(x)BT (x)P (x) +Q(x) = 0 (5)

Now, to ensure the desired solution of (5) exists for allx, one may assume the pairsfA(x); B(x)g

andfH(x); A(x)g are controllable and observable, respectively, for allx, where we employ

the common definitions of controllability and observability from linear systems theory [9].

Of course, less restrictive assumptions such as stabilizability and detectability would also be

5

sufficient. However, since stabilizability is defined based on controllability, in this paper we

study the relationship between the ‘factored’ controllability assumed in order to guarantee

existence of solutions of (5), and the true nonlinear controllability of the system (1).

3 Factored versus True Controllability

In previous studies of controllability, e.g. [11], researchers have established connections

between factored and true system controllability in only the linear time-invariant (LTI) case,

and, it appears, any other possible connections between the two have been left undeveloped.

In this section we review the known results linking the two types of controllability, and then

proceed to develop additional cases in which the two are linked.

Comparison of factored versus true nonlinear controllability is facilitated by consider-

ing controllability in terms of the dimension of invariant locally reachable and unreach-

able spaces. For controllable systems, the dimension of the locally reachable space must

equal the dimension of the state space. Thus, for LTI systems, controllability is estab-

lished [10] by verifying that the rank of the controllability matrix,Mcl, equalsn, where

Mcl = [B AB A2B � � � An�1B]. For the SDC factored system (2), this test generalizes

to a rank test on the factored controllability matrix function

rank [Mcf(x)] = rank [B(x) A(x)B(x) A2(x)B(x) � � � An�1(x)B(x)] = n 8 x (6)

whereas for the original system, local controllability is characterized [11] at eachx in terms

of the dimension of the span of the smallest nonsingular and involutive distribution,�c(x),

6

containing the columnsbi of B(x); 1 � i � m, and invariant undera and thebi. This

distribution assigns to eachx 2 Rn a vector space, an open subset of which is reachable

from the given point by using piecewise constant inputs. A sufficient condition [11] for the

system (1) to be locally controllable at the pointx is thus

rank[�c(x)] = n (7)

and the system is said to beweakly controllable(on S) if (7) holds for all x (2 S). The

following recursive algorithm for generating�c is given in [11]:

1. Let�0 = span(B) = span(bi)

2. Let�1 = �0+ [a; bi] + [bj; bi]; 1 � i � m; 1 � j � m where[a; g] is the Lie bracket

of a andg, i.e., [a; g] = (@g=@x)a � (@a=@x)g and the+ indicates the sum of the

spans.

3. Let�k = �k�1 + [a; dj] + [bi; dj]; 1 � i � m; 1 � j � n wherefdjg is a basis for

�k�1

4. Terminate when�k+1 = �k

For LTI systems, the above procedure gives�c = Mcl [11]. Also,�n�1 always equals�c

on an open and dense subset ofRn, and if each�k is nonsingular, then�n�1 = �c for all x.

If a system fails the appropriate controllability rank test, then the uncontrollable sub-

space may be determined [11] by finding the annihilator of the appropriate matrix, i.e., the

left nullspace ofMcl; Mcf or�c. Also, the controllable and uncontrollable subspaces as de-

termined fromMcl and�c are guaranteed to be invariant subspaces, whereas, the respective

7

subspaces determined fromMcf hold only for the singlex value being considered. Thus,

invariance of these pointwise sets is not ensured.

The above concepts allow controllable/uncontrollable system decompositions [10, 11].

Thus, for an LTI system with controllable subspace of dimensiond, we may write

_x1 = A11x1 + A12x2 +B1u

_x2 = A22x2 (8)

wherex1 2 Rd is in the controllable subspaceCl, x2 2 Rn�d is in the uncontrollable sub-

spaceUl, andfA11; B1g is a controllable pair. For the nonlinear system (1) with controllable

subspace of dimensiond, we similarly may write

_x1 = a1(x1; x2) + b1(x1; x2)u (9)

_x2 = a2(x2) (10)

wherex1 2 Rd is in the controllable subspaceCnl, x2 2 Rn�d is in the uncontrollable

subspaceUnl, and (9) satisfies rank[�c] = d. Finally, (8) – (10) allow us to discuss the notion

of stabilizability. Conceptually, a system is stabilizable if its uncontrollable part is stable.

For LTI systems, stabilizability is equivalent toA22 in (8) being a Hurwitz matrix, whereas

nonlinear stabilizability requires that the unaffectable subsystem (10) be stable. Clearly, if

a system may be written as (9)-(10), for it to be drivable to the origin, this latternonlinear

stabilizabilitycondition must hold, regardless of the factored controllability properties of the

system.

8

4 Theorems and Examples

With some minor extensions, the discussion of Section 3 allows us to give some basic theo-

rems, the first of which is the lack of a general equivalency between factored and true non-

linear controllability. For purposes of space, proofs are sketched, and the reader is referred

to [12] for details.

Theorem 4.1 Consider the system (1) witha(x) andh(x) assumed to beC1 functions, so

that (1) may be written as in (2). Assume the pairfA(x); B(x)g is controllable for allx, so

that (6) holds. Then the system (1) is not necessarily weakly controllable (onRn).

Proof: The proof is by counterexample. Consider the system

_x1 = x1x2 + x2

_x2 = u (11)

With A(x) chosen to be

A(x) =

26664x2 1

0 0

37775 (12)

It is easily verified thatMcf is globally full rank, but�c as defined in Section 3 is such that

atx1 = �1, rank[�c] = 1, so that (7) fails to hold.

The proof of the previous theorem shows that it is possible for a state dependent factor-

ization to hide the existence of an uncontrollable, invariant set (the set of allx 2 R2 such

thatx1 = �1 for (11)). It is interesting to note that this fact is not mentioned in [6], nor,

it appears, has it historically been well-known when SDRE type methods were previously

9

suggested. In fact, in [3] the factorization (12) is recommended as a better factorization for

(11) than the choice

A(x) =

26664

0 x1 + 1

0 0

37775 (13)

because (12) allows solution for the control at allx, while (13) does not. Thus, the lack of

true controllability of this system was not recognized by the authors of [3]. By definingJ as

the Jacobian ofa, i.e.,J(x) = @a=@x, the following theorem may be proven.

Theorem 4.2 Consider the system (1) written as (2) withn = 2, and letB be a constant

matrix. Also, assume thatA(x) is chosen such thatJ(x)B = kA(x)B 8 x, where0 6= k 2 R.

Then, if the factorization (2) is controllable for allx, the system (1) is weakly controllable

on an open and dense subset ofR2. Conversely, if (1) is weakly controllable onR2, then (2)

is controllable for allx.

Proof: The proof follows by checking the rank ofMcf and�c under the assumptions.

We now give some corollaries to Theorem 4.2, which are easily proven and useful when

considering factorization choices.

Corollary 4.1 The assumptionTA(x)xB = (J � A)B = (k � 1)AB, k 6= 0, is equivalent

to the assumptionJ(x)B = kA(x)B, k 6= 0, whereTA(x) = (@A=@x)

Corollary 4.2 TA(x)xB 6= �AB for all x is necessary for the assumptions of Thereom 4.2

to hold

Corollary 4.3 If B is not rankn, thenAB = 0 iff JB = 0 is a necessary condition for the

contrapositive of Theorem 4.2 to hold

10

We note that the conditions of the Theorem 4.2 are sufficient, but not necessary. It is

possible for bothMcf and�c to be full rank withoutAB being in the range ofJB alone,

as long as bothAB andJB provide the remainder of a spanning set ofR2 not provided

by B. In Theorem 4.2 we restrict attention to 2nd-order systems, because for higher-order

systems, more iterations on�k and thus more Lie bracket calculations will generally be

required. The divergence between succeeding entries inMcf and�c increases asn in-

creases, making useful comparisons between the two more difficult. However, the fact that

TA(x)x + A(x) = J(x) allows us to draw the following conclusion regarding factored and

local nonlinear controllability for anyn.

Theorem 4.3 Consider the system (1) written as (2), and assume (6) holds. Then the system

(1) is weakly controllable on some local neighborhood of the origin.

Proof: The conclusion follows sinceJ(0) = A(0) impliesMcf(0) = Mcl, whereMcl is in

terms of the system linearization, and the assumptions guaranteeMcf (0) is full rank.

The set where the system (1) is weakly controllable is thus the set on which its linearization

dominates, which may be arbitrarily small, as illustrated later.

The previous three theorems dealt with an equivalence relationship betweenMcf and

�c involving assumptions onA(x) or on the dimension of the state. The following (trivial)

theorem gives one case in which such assumptions are unnecessary.

Theorem 4.4 Consider the system (1) written as (2) and letm � n. AssumeB(x) has rank

n for all x. Then (2) is controllable for allx, and the system (1) is weakly controllable on

Rn.

11

Proof: The proof follows trivially from constructingMcf and�c, notingB(x) satisfies by

itself the rank requirement.

The importance of Theorem 4.4 is that pointwise controllability issues are not a driving

force in factorization choices fora whenB is globally rankn, as they definitely are when

this is not the case. We now illustrate the above theorems with some examples.

Example 1Consider the system

_x1 = x2

_x2 = x1x2 + u (14)

We havea = [x2 x1x2]T , b = [0 1]T , and sincem < n, Theorem 4.4 may not be used.

However, we have

J =

26664

0 1

x2 x1

37775 ; �c = �1 =

26664

0 1

1 x1

37775 (15)

so that�c has rank2 for all x. Now, choose

A(x) =

26664

0 1

0 x1

37775) A(x)B =

26664

1

x1

37775 = JB 8 x (16)

so that the conditions of Theorem 4.2 are satisfied. Checking the factored controllability

matrix we find

Mcf(x) =

26664

0 1

1 x1

37775 = �c(x) 8 x (17)

12

and we see that the system is weakly controllable onR2, while the factored system is con-

trollable for allx 2 R2 as well. On the other hand, if we choose

A(x) =

26664x2 1� x1

0 x1

37775 (18)

then the factored controllability matrix function

Mcf(x) =

26664

0 1� x1

1 x1

37775 (19)

clearly loses rank atx1 = 1. Given the analysis above, the first choice ofA(x) for this

example is preferable, because it guarantees global existence of the control. Thus, even

when the original system is weakly controllable, care must be taken when choosing the

factorization, as both poor and good choices from an implementation standpoint may exist.

In the next example we illustrate the nonnecessity of theAB = kJB condition as dis-

cussed above.

Example 2Consider the system

_x1 = x2

_x2 = x21 + x22 + u (20)

Theorem 4.4 may not be invoked, but simple computations give

J =

26664

0 1

2x1 2x2

37775 ; �c = �1 =

26664

0 1

1 2x2

37775 (21)

13

so that�c has rank2 for all x. Now, choose

A(x) =

26664

0 1

x1 x2

37775) A(x)B =

26664

1

x2

37775 (22)

so thatAB does not equal a multiple ofJB for any x. However, checking the factored

controllability matrix we find

Mcf (x) =

26664

0 1

1 x2

37775 (23)

which is full rank for allx, so that global weak and factored controllability both hold. Thus,

even though Theorem 4.2 does not apply,[1 2x2]T and[1 x2]T both provide vector functions

which, together withb = [0 1]T , spanR2 for all values ofx.

Finally, we return to the counterexample in the proof of Theorem 4.1, and see how Corol-

lary 4.2 comes into play.

Example 3Consider the system

_x1 = x1x2 + x2

_x2 = u (24)

We have rank(B) = 1 8 x, and again Theorem 4.4 may not be invoked. We find

J =

26664x2 x1 + 1

0 0

37775 ; �c =

26664

0 x1 + 1

1 0

37775 (25)

so that�c has rank1 atx1 = �1 as we saw before. Now, also as before choose

A(x) =

26664x2 1

0 0

37775) A(x)B =

26664

1

0

37775 (26)

14

which gives the globally full rank factored controllability matrix function

Mcf (x) =

26664

0 1

1 0

37775 (27)

However, we see thatkJB does not equalAB for x = �1 for anyk 6= 0. Also, TAxB =

(J � A)B = �AB whenx1 = �1, and we haveAB 6= 0 whenJB = 0, both of which we

noted were necessary conditions for equivalency of the controllability tests ifB were not rank

n. Thus, the factored system is controllable for allx 2 R2, whereas the original system is

not weakly controllable onx1 = �1. Finally, this example illustrates the potential weakness

of Theorem 4.3. Clearly, the controllability equivalence based on linearization does not hold

in this example beyond a ball of radius one centered at the origin. By replacing the first

element ofa with x1x2 + kx2, with k 2 R, and decreasing the absolute value ofk, we can

construct an example for which the conclusion of Theorem 4.3 holds on an arbitrarily small

neighborhood of the origin. Thus, the study of global as opposed to local controllability

equivalence is indeed well-motivated.

To this point we have shown that both factored and true nonlinear controllability are sep-

arately important to the success of the SDRE approach. Factored controllability is important

from a computational standpoint, while true controllability has ramifications for stability.

Although we have shown these concepts are basically different, we have established some

conditions, particularly for low-order systems, when they simultaneously hold, and addi-

tionally have established implications from one form of controllability to the other. We now

extend our study of these relationships by considering controllability issues in a nontrivial

15

design example, showing how the desired properties may be established.

5 Design Problem

The chosen problem involves angular momentum control of an axial dual-spin spacecraft,

and exhibits highly nonlinear dynamics and limited controllability. Thus, it well serves to

illustrate the theory. In the sequel we first describe the design problem, and then give the

equations of motion and design objectives. Following this we present a brief controllability

analysis of the open loop system, from both the factored and true nonlinear perspectives.

Simulation results demonstrate successful application of the method.

5.1 Problem Description

A dual-spin satellite consists of two bodies capable of relative rotation, with one body

spinning relatively fast (the rotor) to provide stabilization, and one body (the platform)

spinning relatively slowly in order to perform mission requirements (i.e., to remain earth-

pointing) [13]. Typical deployment scenarios result in both bodies initially spinning at nearly

the same rate about a single axis (the so-called all-spun condition), so that some type of

spinup maneuver is required to despin the platform. Spinup maneuvers have been investi-

gated by numerous researchers as described in [14] and [15]. Typical maneuvers employ a

small, constant internal torque applied to the rotor. In this paper we apply the SDRE nonlin-

ear regulation technique to the so-called transverse spinup maneuver. This maneuver, also

known as the dual-spin turn, has initial conditions such that the spacecraft spins about a prin-

16

cipal axis nearly perpendicular to the rotor spin axis, and desired final condition such that all

the spacecraft’s angular momentum is contained in the rotor.

The model used in [13, 14] is an axial gyrostat, comprised of a rigid asymmetric platform

and a rigid axisymmetric rotor constrained to relative rotation about its symmetry axis. The

rotational dynamics equations for this model may be written in terms of four states: the three

components of the system angular momentum and the axial component of the rotor angular

momentum. Simple analysis of the equations of motion reveals that the numerical SDRE

regulation approach may not be used to stabilize this system. This is due to the fact that the

(unique) system linearization about the desired equilibrium point is not stabilizable in the

linear sense, and therefore stabilizing solutions to the SDRE do not exist near the origin.

Remark: These results point out a limitation of numerical SDRE methods: for systems

with linearly uncontrollable imaginary open loop eigenvalues in their linearizations, it may

be possible to nonlinearly stabilize the corresponding modes (as controllability analysis of

the single-rotor gyrostat would indicate). The numerical SDRE regulator, however, reverts

to standard Linear Quadratic Regulation (LQR) near the origin, and so will not in general

be stabilizing for such systems.

In light of the above, we investigate a two-rotor gyrostat model where the second (un-

controlled) rotor is subject to an internal viscous damping torque. As shown in the sequel,

the addition of this second rotor results in a stabilizable system linearization at the origin.

17

5.2 Equations of Motion

A general version of the two-rotor gyrostat model used here is developed in detail in [15].

Referring to Fig. 1, we define the vector,x 2 R3, as the gyrostat angular momentum vector

expressed in the body-fixed principal reference frame (e1; e2; e3), noting thate1 is the axis

about which the platform and controlled rotor may have relative rotation (the spin axis).

We neglect external torques so thatkxk is constant, assumed nonzero, and scaled so that

kxk = 1. We define the scalar� to be the controlled rotor axial angular momentum (about

thee1 axis), and the control inputu to be the associated applied rotor torque. For the specific

choice of the off-axis rotor spin axis being in thee2 direction, we let the associated rotor

axial angular momentum be�2. We also define dimensionless inertia parametersij via

ij = 1 � Ip=Ij; j = 1; 2; 3, whereIp = I1 � Is, Is is the axial moment of inertia of

the controlled rotor, andIj are the principal moments of inertia of the gyrostat. Using these

definitions, the equations of motion may be written

_x1 = (i2 � i3)x2x3 + �2�2x3

_x2 = (i3x1 � �)x3

_x3 = �(i2x1 � �)x2 � �2�2x1

_�2 = e2x2 � d2�2

_� = u (28)

where

�2 = 1=I2; e2 = �2Is2 ; d2 = 1 + e2 (29)

18

andIs2 is the off-axis rotor principal moment of inertia with respect to thee2 axis.

Note thati1 does not appear in the system equations, but does impact the system via

initial conditions. This can be seen from the fact that in the all-spun condition, the initial

condition for� is � = i1x1. The relationship between the other two inertia parameters

determines whether the spacecraft is oblate or prolate [14]. Finally, the cone or nutation

angle� (the angle between thee1 axis andx) is defined via� = arccos x1.

The design goal of performing transverse spinup maneuvers may be related to the state

variables in (28) in the following way. In the transverse spinup maneuver, we start with

x1 � 0, � = i1x1, and we desire to drive the state to(x1; �) = (1; 1), noting that if we

achieve this objective, thenx2 = x3 = 0. The control problem, as posed, is actually a con-

strained nonzero setpoint problem. We therefore transform the problem into a form more

suitable for SDRE regulation. In the next section we show how factorization controllability

considerations motivate removal of thex1 state equation, which we do to obtain an uncon-

strained problem. We address the nonzero setpoint problem of driving� to one by the change

of coordinates� = � � 1. This part of the control problem then reduces to regulation of�,

which fits readily into the SDRE framework. Finally, by recognizingkxk = 1 always holds,

we attempt only the regulation of(x2; x3) to (0; 0), using the constraint to makejx1j = 1.

The fact thatx1 remains free to take on either of the values plus or minus one is a disadvan-

tage of this approach. Simulation results verify success of the method, however, in that if we

start withx1 small but positive, we drivex1 to positive one as desired.

19

5.3 Open Loop/Controllability Analysis

Before developing the control design, we study the open loop model (28) and its relevant

controllability properties. Performing on (28) the iterative procedure for determining the

Control Lie Algebra described in Section 3, one can identify the uncontrollable coordinate

�(x) in the state space

� = c(x21 + x22 + x23) (30)

wherec is an arbitrary constant. This uncontrollable coordinate corresponds to conservation

of angular momentum, so thatx trajectories remain on the unit momentum sphere centered

at (x1; x2; x3) = (0; 0; 0). This uncontrollable coordinate, by itself, does not prevent us from

reaching the desired equilibrium state since it lies on the sphere, and thus, does not imply that

the system is not nonlinearly stabilizable. However, whenx2 = x3 = 0, the uncontrollable

space has dimension four, and, in fact, consists of the(x1; x2; x3; �2) space. This means that

the equilibrium surfaces(�1; 0; 0; 0; �) have unchangingx components, regardless of how

we select the control. The implication is that, if we seek to drive the system tox1 = 1, we

should avoid trajectories passing near the equilibrium atx1 = �1, and vice versa. Complete

analysis of the Control Lie Algebra [12] reveals no other invariant, uncontrollable manifolds

with which we need to be concerned.

Since we have identified one globally uncontrollable coordinate, we seek to eliminate it

and proceed with control design on the reduced system. To this end we define the coordinate

transformationq1 = x3, q2 = x2, q3 = �, with � defined as in (30) withc = 0:5, and

20

eliminatex1 as an independent variable by writing

x1(q) = �q1� q21 � q22 (31)

Since the Jacobian of this mapping is

J =

2666666664

0 0 1

0 1 0

x1 x2 x3

3777777775

(32)

we see by the Inverse Function Theorem that the mapping is not one-to-one in a neighbor-

hood ofx1 = 0. Thus, we need to know which hemisphere of the momentum sphere we are

in to complete the mapping, so as to choose the appropriate sign in (31). With this coordinate

change the equations of motion become

_q1 = (�� i2x1)q2 � �2x1�2

_q2 = (i3x1 � �)q1

_�2 = e2q2 � d2�2

_� = u (33)

wherex1 is defined as in (31), and we have eliminated the trivial state equation_q3 = 0.

The nonlinear controllability procedure for this system indicates lack of controllability of

thex states only at the desired equilibrium(q1; q2; �2) = (0; 0; 0), providedx remains in the

positive momentum sphere (x1 > 0). Recall, however, that we actually need to shift the state

to make the desired closed loop equilibrium point the origin, as discussed in Section 5.2. To

21

do so we define

� = �� 1 (34)

leading to the system equations

_q1 = (� � i2x1)q2 + q2 � �2x1�2

_q2 = (i3x1 � �)q1 � q1

_�2 = e2q2 � d2�2

_� = u (35)

For (35) we find the Jacobian at zero to be

J(0) =

266666666666664

0 1� i2 ��2 0

i3 � 1 0 0 0

0 e2 �d2 0

0 0 0 0

377777777777775

(36)

so that stabilizability of the linearization requires that the eigenvalues of the upper left3� 3

subblock of (36) have negative real parts. In this paper we have nondimensionalized the

spacecraft moments of inertia by assumingIp = 1. This gives the simple equalities1� i2 =

�2 andi3 � 1 = �1=I3 � ��3, so that the characteristic equation of this subblock is

�3 + d2�2 + �2�3�+ �2�3 = 0 (37)

Using Routh’s criterion [17] it is easy to show that the linearization of (35) is stabilizable.

Linearized detectability is also guaranteed, provided� has an independent, globally positive

22

definite penalty, i.e.,H is of the form

H =

266666666666664

� � � 0

� � � 0

� � � 0

0 0 0 �(x)

377777777777775

(38)

where�(x) is a positive definite function.

The above discussion shows that adding the off-axis damping rotor guarantees stabiliz-

ability and detectability in a neighborhood of the origin, foranyparametrizationA, and for

a suitable parametrizationH. We are interested, however, in performing transverse spinup

maneuvers which start far away from the origin. We thus propose a factorizationA which

guarantees pointwise stabilizability and detectability everywhere in the positive momentum

sphere (x1 > 0). Since� is completely controllable, heuristically we want the factorization

to show strong pointwise linear controllability ofq1 andq2 through�. We therefore choose

A(x) =

266666666666664

0 1� i2x1 ��2x1 q2

i3x1 � 1 0 0 �q1

0 e2 �d2 0

0 0 0 0

377777777777775

(39)

This factorization has the factored controllability matrix function

Mcf(x) =

266666666666664

0 q2 ��2q1 �1�2q2 + �2e2x1q1

0 �q1 �1q2 ��1�2q1

0 0 �e2q1 e2(�1q2 + d2q1)

1 0 0 0

377777777777775

(40)

23

where

�1 = i3x1 � 1; �2 = 1� i2x1 (41)

The matrix function (40) has determinantdet[Mcf ] = �e2[�21q

32 + d2�1q

22q1 � �1�2q2q

21 +

(e2�2x1 � d2�2)q31] and obviously loses rank for nontrivial values ofq1 andq2, so that (39)

does not yield a globally controllable parametrization. However, (39) does yield guaranteed

stabilizability forx1 > 0, as can be shown using Routh’s criterion and some simple analysis.

For stabilizability offA;Bg, recall that we must haveyTB 6= 0 for all � andy such that

yTA = yT� andRe� � 0 [9]. The characteristic equation of (39) is

�(�3 + d2�2 � �1�2�+ �1�3) = 0 (42)

where�1; �2 are as in (41) and�3 = �2e2x1 � �2d2. We thus have a single zero eigenvalue,

plus three more eigenvalues determined by the roots of the term in parentheses in (42). For

the zero eigenvalue, the corresponding left eigenvector isyT = [0 0 0 1]. SinceyTB = 1 6=

0, this zero eigenvalue is stabilizable. We now show that the remaining three eigenvalues

have negative real parts under some slight additional assumptions. Recall that a necessary

condition for only left half plane roots of a polynomial is that all the coefficients have the

same sign [17]. By definition,d2 > 0, and sincex1 � 1 and i2; i3 < 1, we also have

from (41) that�1 < 0 and�2 > 0, so that��1�2 > 0. The final necessary condition thus

becomes�3 < 0. To complete the analysis, we assume without loss of generality a prolate

spacecraft [18], for whichi2 > i3 > 0. From (41) we see that, forx1 > 0, we have

�3 < �2e2 + (i2 � 1)d2 = �2e2 � �2d2 = �2(e2 � d2) = ��2 < 0 (43)

24

Thus,�3 < 0 and we satisfy the necessary condition for left half plane eigenvalues regardless

of the values of�2 andIs2. We still need to verify a sufficient condition for stability of these

eigenvalues, however. Routhian analysis leads to the additional condition

��1(d2�2 + �3) > 0 (44)

Since�1 < 0, using the definition of�3 we find that (44) becomes

d2�2 + �2x1e2 � �2d2 = �2x1e2 > 0 (45)

which holds as long asx1 > 0, as assumed throughout. Thus, (39) gives a pointwise stabi-

lizable parametrization, as long as trajectories remain in the positive (x1) momentum sphere.

Remark: The existence of this stabilizable parametrization is interesting, since a mani-

fold of open loop equilibrium points with unstabilizable local linearizations is known to exist

[16] in the positive hemisphere for the prolate case. If a trajectory were to pass through this

manifold, then a control algorithm based on Riccati equations and local linearizations such

as LMI-based techniques would fail to yield computable controllers at such points, whereas

the SDRE control is still well-defined.

Note that (39) corresponds to the choices ofc1 = c2 = 0 in the general factorization

A(x) =

266666666666664

0 1� i2x1 + c1� ��2x1 (1� c1)q2

i3x1 � 1� c2� 0 0 �(1� c2)q1

0 e2 �d2 0

0 0 0 0

377777777777775

(46)

25

and observe that, if we pickc1 = c2 = 1, then theq dynamics appear to be pointwise

unaffectable from� for all x. In this case the uncontrollable space is all of the(q1; q2; �2)

space, as opposed to the much smaller uncontrollable space of (39). Stabilizability of the

factorization (46) withc1 = c2 = 1 thus requires that all three nonzero eigenvalues of (46)

be stable, which was sufficient but not necessary for stabilizability of (39). Computing the

characteristic equation of (46) (withc1 = c2 = 1) we find

�[�3 + d2�2 + (� � �1)(�2 + �)�+ (�1 � �)(e2�2x1 � d2(�2 + �))] = 0 (47)

and we see that (47) differs from (42) only in the first and zeroth-order coefficients of�. This

difference is significant because as we near the desired valuex1 � 1, there exist achievable

values of� which render the first-order coefficient negative. For example, leti2 = 0:5 and

i3 = 0:3. Then the first-order coefficient becomes(� + 0:7)(� + 0:5), so that for all values

of � 2 (�0:5;�0:7), the coefficient is negative, and the factorization is not stabilizable.

This issue is not just of theoretical importance, for simulated attempts at SDRE nonlinear

regulation of the gyrostat using (46) withc1 = c2 = 1 became numerically unstable when the

factorization became unstabilizable. We mention this particular case because it corresponds

to a natural SDC parametrization that arises from considering the gyrostat as a Hamiltonian

system [15]. In this framework, the nonreduced state equations can be written_x = �rH�x,

wherer represents the gradient operator,H is an appropriate Hamiltonian, and the(�)�

notation represents the skew symmetric matrix form of a vector [14]. Thus, selection of an

appropriate SDC factorization can be nontrivial, and pointwise controllability issues should

play a strong role in the process.

26

Finally, we need to consider detectability issues to ensure well-posedness of the control.

It has been shown in [12] that forH of the form

H = diag(k1; k2; k3; k4) (48)

with ki 2 R and k4 6= 0, detectability is guaranteed withA chosen according to (39),

providedx1 > 0. We investigate the effect of using two choices forH in the next section.

5.4 SDRE Nonlinear Regulator Simulation Results

In this section we give typical simulation results for the transverse spinup maneuver using a

sampled data implementation of the SDRE nonlinear regulator (which computes the SDRE

solution at timetk and applies the resulting constant control until the next sampling time

tk+1). For comparison purposes we also simulate the same maneuver using a small, constant

torque. All simulations were performed in Matlab/Simulink using Runge-Kutta, fourth-order

integration. The sampling rate was10 Hz, arbitrarily chosen to be implementable and to

yield smooth-looking trajectories, and the integration step size was0:01 seconds. In all

simulations, we setR = 1. We give results for the five-state gyrostat model with damping.

We choose a prolate spacecraft configuration, with inertia parametersi2 = 0:5, i3 = 0:3,

andi1 = 0:2. The initial condition is(x3; x2; �2; �) = (sin(75�); 0; 0; 0:2 cos(75�)� 1),

corresponding to the all-spun condition with a cone angle of75�. We show state history

results using

H = H1 = diag(0; 0; 0; 0:1) andH = H2 = diag(1; 1; 1; 0:1)

27

in Figures 2 and 3, respectively. The penalty value of0:1 on � was selected to keep the

control magnitude near0:1, a reasonable value for spinup maneuvers [15].

We observe that forH = H1, � is driven to zero fairly quickly, but thex2 andx3 states

have large initial oscillations, whereas forH = H2, the initial oscillations ofx2 andx3 are

comparatively smaller and slower than forH = H1. At the simulation end time, thexi

oscillations are comparatively larger forH = H2, and� takes much longer to go to zero

for the H = H2 case. These behaviors are intuitively satisfying in that they reflect the

appropriate emphasis on controlling either� only (H = H1) or a weighted combination of

all the states (H = H2). Although it is not apparent from Figure 3 that� actually does go to

zero in theH = H2 case, it is shown in [12] that it does. By leavingk4 = 0:1 and decreasing

the weights on the other states, it is possible to trade off performance in terms of obtaining

quicker regulation of� at the cost of larger initial deviations inx2 andx3. For comparison

purposes we show state histories for theu = 0:01 constant torque case in Figure 4. Note that

the large initial oscillations in thex2 andx3 states continue to occur longer in the constant

torque case, and their decay rates appear slower than in either of the SDRE cases. The initial

decay of� in the constant torque case is also slower than in both SDRE cases, but the final

settling time is somewhere in between those of the SDRE cases. Finally, we show control

histories for all three cases in Figure 5. In the figure,uc is the constant torque control, while

us1 andus2 are, respectively, the controls for theH = H1 andH = H2 cases. Significant

features to be noticed from Figure 5 are the the smooth decay ofus1, and the large initial

value ofus2, and its subsequent highly oscillatory nature. The high energy content ofus2

28

is again intuitively satisfying in that we expect large control effort to be needed in order

to actively regulate the penalizedx2 andx3 states, since they are only indirectly affectable

through�.

6 Summary and Conclusions

Nonlinear controllability of input-affine systems and controllability of state-dependent fac-

torizations in terms of linear tests are not generally equivalent. Global pointwise control-

lability of a state-dependent factorization is sufficient to guarantee weak controllability on

a neighborhood of the origin, and global controllability of both types holds when the input

matrix functionB(x) has rank equal to the dimension of the state space,n, for all x. We have

given sufficient conditions for controllability equivalency for second-order systems with con-

stantB matrices, which may be extended to necessary conditions whenB is not rankn. For

higher-order systems, conditions guaranteeing equivalency become increasingly complex,

due to differences between the Lie brackets which characterize nonlinear controllability and

theAjB products in the factored controllability test.

While factored controllability (plus observability) for allx is sufficient to guarantee

global well-posedness of state-dependent Riccati equation (SDRE)-based control algorithms,

it is in general not sufficient to guarantee true controllability outside some possibly small

neighborhood of the origin, a fact which has significant ramifications for global stability.

In fact, regardless of how well the SDRE algorithm operates, it can only affect the part of

the state which is nonlinearly controllable, leading to a nonlinear stabilizability necessary

29

condition for stability.

Successful transverse spinup maneuvers of an axial gyrostat were performed using SDRE

nonlinear regulation. Controllability analysis demonstrated nonlinear stabilizability in the

region of interest. Also, a dynamics factorization chosen on the basis of its strong point-

wise controllability properties guaranteed factored stabilizability where needed. The design

problem showed that both types of controllability are separately important to successful ap-

plication of the SDRE nonlinear regulation algorithm, and demonstrated how the desired

properties may be established.

AcknowledgmentsThis work was supported by the Air Force Office of Scientific Research.

References

[1] Pearson, J. D., “Approximation Methods in Optimal Control,”Journal of Electronics

and Control, Vol. 13, No. 5, 1962, pp. 453–465.

[2] Burghart, J. A., “A Technique for Suboptimal Control of Nonlinear Systems,”IEEE

Transactions on Automatic Control, Vol. 14, No. 5, 1969, pp. 530–533.

[3] Wernli, A. and Cook, G., “Suboptimal Control for the Nonlinear Quadratic Regulator

Problem,”Automatica, Vol. 11, No. 1, 1975, pp. 75–84.

[4] Ehrler, D. and Vadali, S. R., “Examination of the Optimal Nonlinear Regulator Prob-

lem,” In Proceedings of the AIAA Guidance, Navigation, and Control Conference, Min-

neapolis, MN, August 1988.

30

[5] Friedland, B.,Advanced Control System Design, Prentice-Hall, Englewood Cliffs, NJ,

1996, pp. 110–112.

[6] Cloutier, J. R., D’Souza, C. N., and Mracek, C. P., “Nonlinear Regulation and Nonlinear

H1 Control Via the State-Dependent Riccati Equation Technique: Part 1, Theory,” In

Proceedings of the First International Conference on Nonlinear Problems in Aviation

and Aerospace, Daytona Beach, FL, May 1996.

[7] Qu, Z., Cloutier, J. R., and Mracek, C. P., “A New Sub-optimal Nonlinear Control De-

sign Technique-SDARE,” InProceedings of the 13th IFAC World Congress, pp. 365–

370, San Francisco, CA, July 1996.

[8] Vidyasagar, M.,Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, NJ, sec-

ond edition, 1993, pp. 51–52.

[9] Zhou, K., Doyle, J. C., and Glover, K.,Robust and Optimal Control, Prentice-Hall,

Englewood Cliffs, NJ, 1995, Chap. 3.

[10] Kwakernaak, H. and Sivan, R.,Linear Optimal Control Systems, Wiley Interscience,

New York, 1972, Chap. 1.

[11] Isidori, A., Nonlinear Control Systems, Springer-Verlag, Berlin, Third edition, 1995,

Chaps. 1 and 2.

31

[12] Hammett, K. D.,Control of Nonlinear Systems Via State Feedback State-Dependent

Riccati Equation Techniques, PhD thesis, Graduate School of Engineering, Air Force

Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1997.

[13] Hughes, P. C.,Spacecraft Attitude Dynamics, John Wiley and Sons, New York, 1996,

Chaps. 6 and 11.

[14] Hall, C. D. and Rand, R. H., “Spinup Dynamics of Axial Dual-Spin Spacecraft,”Jour-

nal of Guidance, Control, and Dyamics, Vol. 17, No. 1, 1994, pp. 30–37.

[15] Hall, C. D., “Spinup Dynamics of Gyrostats,”Journal of Guidance, Control, and Dy-

namics, Vol. 18, No. 5, 1995, pp. 1177–1183.

[16] Hall, C. D., “Momentum Tranfer in Two-Rotor Gyrostats,”Journal of Guidance, Con-

trol, and Dynamics, Vol. 19, No. 5, 1996, pp. 1157–1161.

[17] D’Azzo, J. and Houpis, C.,Linear Control System Analysis and Design: Conventional

and Modern, McGraw-Hill, New York, third edition, 1988, pp. 185–191.

[18] Hall, C. D., “Equivalence of Two Classes of Dual-Spin Spacecraft Spinup Problems,”

Journal of Guidance, Control, and Dynamics, Vol. 15, No. 4, 1992, pp. 1032–1034.

32

Figure 1. Gyrostat Model of Two-Rotor Spacecraft

Figure 2. Transverse SDRE State Histories for 5-State Gyrostat (H = H1)

Figure 3. Transverse SDRE State Histories for 5-State Gyrostat (H = H2)

Figure 4. Transverse State Histories for 5-State Gyrostat (u = 0:01)

Figure 5. Transverse Control Histories for 5-State Gyrostat

33

Figure 1: This page left blank.

34

0 50 100 150 200 250−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

x2

x3

mu2

nu

Figure 2: Transverse SDRE State Histories for 5-State Gyrostat (H = H1)Hammett,

Hall &

Ridgely

G5427

Fig. 2

35

0 50 100 150 200 250−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

nu

mu2

x2

x3

Figure 3: Transverse SDRE State Histories for 5-State Gyrostat (H = H2)Hammett,

Hall &

Ridgely

G5427

Fig. 3

36

0 50 100 150 200 250−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

nu

mu2

x2

x3

Figure 4: Transverse State Histories for 5-State Gyrostat (u = 0:01)Hammett,

Hall &

Ridgely

G5427

Fig. 4

37

0 50 100 150 200 250−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

us1

uc

us2

Figure 5: Transverse Control Histories for 5-State GyrostatHammett,

Hall &

Ridgely

G5427

Fig. 5

38