Relatively optimal control with characteristic polynomial assignment

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 2, FEBRUARY 2006 183

Relatively Optimal Control With CharacteristicPolynomial Assignment and Output Feedback

Franco Blanchini, Senior Member, IEEE, and Felice Andrea Pellegrino

Abstract—A relatively optimal control is a stabilizing controllersuch that, if initialized at its zero state, produces the optimal (con-strained) behavior for the nominal initial condition of the plant(without feedforwarding and tracking the optimal trajectory). Inthis paper, we prove that a relatively optimal control can be ob-tained under quite general constraints and objective function, inparticular without imposing 0-terminal constraints as previouslydone. The main result is that stability of the closed-loop systemcan be achieved by assigning an arbitrary closed-loop character-istic stable polynomial to the plant. An explicit solution is provided.We also show how to choose the characteristic polynomial in sucha way that the constraints (which are enforced on a finite horizon)can be globally or ultimately satisfied (i.e., satisfied from a certaintime on). We provide conditions to achieve strong stabilization (sta-bilization by means of a stable compensator) precisely, we showhow to assign both compensator and closed-loop poles. We considerthe output feedback problem, and we show that it can be success-fully solved by means of a proper observer initialization (based onoutput measurements only). We discuss several applications of thetechnique and provide experimental results on a cart-pendulumsystem.

Index Terms—Characteristic polynomial assignment, con-strained control, linear control, optimal control.

I. INTRODUCTION

IN THIS paper, we consider the problem of determining

an optimal control for linear discrete-time systems under

general constraints and cost functions. Except for very special

cases, determining an optimal control in a feedback form, under

output or input constraints is a computationally hard task. The

techniques based on dynamic programming [4], [14], which

provide general solutions, are known to be effective only for

systems of small dimensions. There have been several attempts

in the literature to approach optimal constrained control prob-

lems such as those considering the minimum-time problem

[8], [9] or the linear quadratic constrained problem [2], [3].

However, these techniques are based on set-computation in

the state space, and therefore may encounter some difficulties

when applied to high dimensional systems. Optimal control

can be faced by means of the receding-horizon approach (see

the survey work in [7], [11], and [12]) whose main difficulty is

the necessity of solving online an optimization problem.

In [6], we have proposed a different approach named rela-

tively optimal control motivated by the observation that in many

Manuscript received June 10, 2004; revised October 4, 2005. Recommendedby Associate Editor M. Rotea. This work was supported by MURST, Italy.

The authors are with the Dipartimento di Matematica e Informatica,the Università di Udine, 33100 Udine, Italy (e-mail: [email protected];[email protected]).

Digital Object Identifier 10.1109/TAC.2005.863493

problems the request of optimality and constraints satisfaction

is essential only for some special initial conditions. Indeed there

are many examples of plants (such as lifts, bascule bridges,

automatic gates, floodgates, cranes, etc.) which are explicitly

built to perform, under normal circumstances, specific operation

through a specific trajectory with known initial and final states.

Moreover, in many problems the initial state is determined by

perturbations, such as impulses, therefore they have privileged

directions given by the column of the disturbance input matrix.

Optimality from a specific initial condition can be achieved

by a feedforward feedback compensator. However, the feed-

forward solution is not practical since it allows the system to

perform appropriately only for the nominal operation while for

nonnominal initial conditions the behavior is inadequate. In [6]

a pure feedback control is proposed which is stabilizing but is

required to satisfy the constraints and to be optimal only for a

given (or a set of given) initial condition(s). However the as-

sumption of zero terminal constraint was considered in the opti-

mization problem and this restriction led to the achievement of

dead-beat compensators only.

In this paper, we still work with general convex (integral and

pointwise) constraints and costs but without zero terminal state.

Removing the assumption of finite horizon arrival to the origin

reproposes the question of how stability can be assured for the

closed-loop system. Indeed, even for the nominal initial state,

which results in the optimal trajectory, the problem of the global

transient (precisely, the evolution after the optimization horizon)

is open.

As a first result, we show that closed loop stability can be in-

deed assured by the assignment of the closed-loop characteristic

polynomial (and, therefore, of the closed-loop poles). Assigning

the closed-loop poles is a desirable task since it allows to impose

proper modes to the closed-loop plant. Then we further inves-

tigate the problem of the choice of the closed-loop polynomial

and its influence on the global constraint satisfaction. We also

solve the problem of output feedback, left as open in the pre-

vious work. The main contribution of the paper can be summa-

rized in the following points.

• The problem of determining a relatively optimal con-

trol is generalized by removing the assumption of zero

terminal constraint. The consequence is that there is no

restriction to FIR closed-loop systems.

• In the state feedback case, the relatively optimal con-

troller, if initialized at its zero state, produces the op-

timal open-loop constrained trajectory if the plant ini-

tial state is the nominal one. The compensator is linear

time-invariant dynamic, and its order is equal to the

time horizon minus the order of the plant.

0018-9286/$20.00 © 2006 IEEE

184 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 2, FEBRUARY 2006

• The controller is stabilizing for all initial conditions

and the closed-loop polynomial can be assigned by

means of an additional linear constraint in the opti-

mization problem.

• In the output feedback case, the problem can be solved

by means of an observer which has to be properly ini-

tialized. In this case, the compensator order is equal

to the open-loop optimization horizon. It is shown that

the mentioned initialization (clearly based on the mea-

sured output information only) is necessary to achieve

the same open-loop performances.

• It is shown how to choose the characteristic polynomial

to assure that, as long as the optimal finite horizon tra-

jectory satisfies the constraints, then these will be sat-

isfied at each time.

• A solution to the problem of ultimate constraints sat-

isfaction (the ultimate boundedness in a target) is pro-

posed. Again this reduces to an appropriate choice of

the characteristic polynomial.

• The approach can be applied to several optimization

problems such as optimal arrival to a target, optimiza-

tion of the feedback matrix for multiple-input pole as-

signment and impulse response optimization.

• An application to the time-optimal arrival to a target set

under constraints for a cart-pendulum system is pro-

posed. Experimental results are provided to show the

efficacy of the approach and the behavior for nonnom-

inal initial states.

II. PROBLEM STATEMENT

Consider the discrete-time reachable system

(1)

where is the state, is the control input,

is the performance output, is the mea-

sured output, and , , , , and are matrices of appro-

priate dimensions. We assume that the system is detectable from

the measured output . For this system, we consider the

following locally bounded convex functions:

(we assume them 0-symmetric, i.e., ,

and ) and the constraint

(2)

where is a 0-symmetric convex and closed set (as well as all

the sets we will introduce in the following). Consider also the

final constraint

(3)

Fix and fix any Schur-stable (i.e., having roots of

modulus strictly less than one) polynomial

(4)

Fig. 1. Feedforward + feedback and pure feedback solution.

and consider the next optimization problem with assigned initial

condition (we assume as initial time)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

The formulation of the problem is essentially that in [6], with the

only substantial difference of the last two constraints. The con-

straint (12) is a generalization of the terminal constraint

which can be considered as a special case by assuming

Note that we have imposed a constraint on the final part of the

horizon from to , where . This allows

for solving the problem of ultimate constraints satisfaction for

and suitably chosen, as we will show in Section V.

Note also that the cost includes a weight on the terminal state

. The problem of achieving the optimal trajectory can

be solved by means of a stabilizing feedback feedforward

control as in Fig. 1 (left), according to the control law

(14)

where and denote Z-transformed optimal input se-

quence and the measured output sequence corresponding to the

optimal trajectory, respectively. It is obvious that, if the initial

condition is the corresponding trajectory is the op-

timal one, by definition of . As already discussed in [6], this

scheme presents several disadvantages. In particular, the main

trouble is that the controller basically tracks the optimal trajec-

tory originating in , therefore if is far from (for instance

the opposite state ), then the resulting transient will be com-

pletely unsatisfactory. On the contrary, in this work we consider

a feedback scheme as in Fig. 1 (right). The basic request for this

scheme is that the system has to be stable and the transient has

to be optimal if the initial condition is . As we will see next,

we will achieve many advantages from this kind of scheme. We

BLANCHINI AND PELLEGRINO: RELATIVELY OPTIMAL CONTROL 185

are now in the position of formalizing the problem. We initially

consider state feedback, namely .

Problem 1: Find a state feedback compensator having a pure

feedback structure (Fig. 1, right) such that for both

and the control and state trajectories are optimal

and which is stabilizing (possibly with constraints violations)

for all initial conditions.

In the formulation of the problem in [6], we assumed that the

compensator state is not initialized, namely we assumed

. In the new formulation, we remove this assump-

tion. Actually, the condition is maintained in the state

feedback case, while for the measurement feedback we have to

suitably initialize the compensator. Also, the optimality require-

ment for both and (which is redundant in the state feed-

back case being the provided compensator linear) is introduced

to exclude a trivial solution for the output feedback problem.

Finally, we note that the constraints of the problem have to be

considered as a design specification “soft” constraints and that

their violation implies only a performance loss.

III. RELATIVELY OPTIMAL CONTROL WITH CHARACTERISTIC

POLYNOMIAL ASSIGNMENT—THE STATE FEEDBACK CASE

In this section, we show how to solve Problem 1. We intro-

duce the following technical assumption (which can be easily

removed, see [6]).

Assumption 1: The initial state does not belong to any

proper —invariant subspace1 of .

Denote by

(15)

the matrix containing the optimal state trajectory and by

(16)

the matrix containing the optimal input sequence. By

construction, and satisfies

(12). This means that matrices and satisfy the next equation

(17)

where the square matrix is the Frobenius matrix associated

with

(18)

which is, therefore, a stable matrix. In view of Assumption 1,

the next Lemma holds [6].

Lemma 3.1: The matrix has rank , namely has full row

rank.

1A subspace S is said (A;B)—invariant if for all x 2 S there exists u(x)such that Ax + Bu(x) 2 S [1]. It is said proper if S 6= IR .

First consider the case . Let us consider any

matrix of the form

(19)

(note that the first column is zero) such that the next square

matrix is invertible

(20)

Clearly, finding in such a way that the matrix is invertible

is always possible (see [6, App. A]). Now, denote by

(21)

Consider the linear compensator

(22)

(23)

where , , , and are achieved as the unique solution of

the linear equation

(24)

The next theorem states that such a compensator is stabilizing

and, denoting by , , the generic solution of the

closed-loop system, if we take and , then

the resulting trajectory is the optimal one, namely ,

, and .

Theorem 3.1: The compensator given by (22)–(24) is a solu-

tion of Problem 1.

Proof: By combining (17) and (21), we derive the fol-

lowing equation:

(25)

The closed-loop system with compensator (22) and (23) has

state matrix

(26)

By (24) and (25), we have that

(27)

Since is invertible, the matrix , so that it is

similar to , hence is stable. If we consider the expression of

the th column in (27) and the expression of in (20), denoting

by the th column of , we see that if we take the initial

condition


then the solution

(28)

is such that

and

and, denoting by the th column of

(29)

and, therefore, the state sequence is the optimal one. The corre-

sponding control sequence is achieved by considering (22) and

(23) and, in particular

so that, from (24), the control sequence is ,

, the optimal one.

In the special case , is square and no augmentation

is necessary. In this case we get the linear static compensator

with .

Remark 3.1: We point out that the role of constraint (12) is

essential. Precisely if we formulate the problem without (12)

and we seek for a compensator which is both relatively optimal

and produces the closed-loop characteristic polynomial ,

then such a compensator must be constructed on a trajectory

which satisfies constraints (12) (as any other closed-loop un-

forced trajectory).

A. Compensator Eigenvalue Assignment

In this section we present a synthesis procedure which allows

for the simultaneous assignment of both the closed-loop and the

compensator eigenvalues.

Procedure: Simultaneous placement of closed-loop and

compensator poles.

• Fix a stable (whose eigenvalues are those of the com-

pensator) and a stable

(whose roots are the closed-loop poles).

• Consider the linear equation

(30)

where is the th row of and

columns

• Each of the infinite solutions of (30) provides a pair

, . If the

resulting matrix is invertible, compute

and and from (24).

It is not difficult to see that the previous equation corresponds to

, and, according

to (27)

where the last equality holds since .

IV. OUTPUT FEEDBACK RELATIVELY OPTIMAL CONTROL WITH

OBSERVER INITIALIZATION

The relatively optimal control proposed in this paper is

thought for state feedback although we can use it for output

feedback provided that the problem allows for the initialization

of an observer. As mentioned in the introduction, our compen-

sator is suitable for special problems such as point-to-point

operations in which the initialization is reasonable. Basically, in

this way we achieve a compensator which is not time-invariant.

Let us reconsider the output equation

with . We introduce the following technical as-

sumption.

Assumption 2: The vector and the matrix are such that

We will comment on this assumption later. Consider any ob-

server with initialization2

(31)

(32)

where is Schur-stable. Then the compensator can be

computed exactly as shown in the previous sections with the

2The trivial initialization x̂(1) = �x is not feasible since we require optimalityfrom both �x and ��x.


difference that now the feedback of the estimated state is con-

sidered instead of (22) and (23)

(33)

(34)

We formalize the result in the next theorem.

Theorem 4.1: Consider any matrix such that

is a Schur matrix. Then, the output feedback compensator (33)

and (34) with the observer (32) is stabilizing and, if initialized

as in (31), then it is optimal for the initial state . If the

cost and the constraints are symmetrical, then it is optimal also

for and satisfies the constraints for all

for . If the problem is unconstrained and the functions

and are homogeneous then it is optimal for all the initial

conditions , .

Proof: The fact that the compensator is stabilizing is an

immediate consequence of the fact that if we take into account

the error we have that

, so that we can write the closed-loop system as

which is stable as long as is Schur. Furthermore, if

the observer is initialized as (31), as long as , we have

so that . Then, for .

As far as the observer initialization is concerned, we can

easily see that it cannot be avoided in a relative optimality con-

text for the following reason. If we take as a fictitious

input matrix and we consider the augmented system

then the closed-loop output impulse response with our control

turns out to be . The corresponding resulting

closed-loop transfer function is

Unfortunately, such a closed-loop transfer function can

be the result of unstable zero-pole cancellations as it can be seen

by very simple examples. Indeed, the admissible closed-loop

impulse responses are subject to well-known restrictions [10],

[16], precisely, that they must preserve the unstable open-loop

zeros to assure internal stability and these constraints are not

included in the original optimization problem to compute

and . As we have seen, this problem disappears if we allow

for an observer initialization since this basically means that we

are considering a time-varying compensator (although of a very

special type and very simple to implement).

Finally we would like to discuss about the Assumption 2.

If we introduce the fictitious matrix , this is a “rela-

tive degree” assumption. If it fails, then solving the relatively

optimal control problem is hopeless. Indeed, if we

have also therefore the controller cannot discrim-

inate the two situations or and therefore

cannot (unless for trivial cases) be optimal for both as required

in the problem statement. To summarize, we make the following

points.

• As long as we have state feedback, any open-loop (op-

timal) trajectory from can be achieved by a dynamic

pure feedback compensator.

• In the output feedback case the trajectory can be

achieved by means of a proper observer initialization

(under Assumption 2).

• If the observer initialization is not possible, achieving

relative optimality might be not compatible with in-

ternal stability.

V. GLOBAL AND ULTIMATE CONSTRAINTS SATISFACTION WITH

APPLICATIONS

The assumption of final zero state considered in [6] is a re-

striction and may have undesirable consequences. The provided

scheme removes such a restriction but leaves open the problem

of what happens after the first steps. We have seen that the

closed-loop modes are fixed by the characteristic polynomial

. In brief, we have to face two problems.

• How can we assure that the point-wise constraints im-

posed on the finite horizon are globally satisfied?

• How can we conveniently fix ?

Interestingly, we can simultaneously face these problems, as

shown in the sequel.

A. Constraint Satisfaction Via Polynomial Assignment

Let us consider the constraint

To assure that this constraint is satisfied even after the horizon

, one can take in such a way that

(35)

Indeed we can state the following result.

Theorem 5.1: Assume that is chosen according to (35)

and that the optimization problem is feasible. Then the closed-

loop system is stable and, if , satisfies the constraints

(9) for all .

Proof: The fact that is a Schur-stable polynomial, if

(35) holds, is immediately derived by (35). The constraint (9)

holds by construction for . We show that

also for . Being the closed-loop matrix, after

reaching the end of the optimal trajectory, the system behaves

according to the following:


By construction and

, hence the previous equations can

be rewritten as

which are equivalent to the following:

By setting , we obtain

(36)

The previous expressions for and can be substituted

in (23), leading to

(37)

Since , from (36) and (37), we get:

therefore, after steps, the output is a weighted sum of

the preceding outputs. Since the weights satisfy (35), the

following hold for :

Hence, being convex and 0-symmetric and

by construction, it follows that

, .

We can solve also a generalized version of the problem

namely the ultimate constraints satisfaction. Let us replace the

constraints (13) and let us choose in such a way that

and (38)

We have the following.

Theorem 5.2: Assume that is chosen according to (38)

and that the optimization problem is feasible. Then, if ,

the closed-loop system satisfies the constraints (13) for all

.

Proof: To prove the theorem, set . Then, it is

enough to notice that in view of (38), expression (36) becomes

(39)

Then, the proof proceeds exactly as in the previous theorem by

replacing by .

B. Ultimate Arrival to a Target

A possible alternative way to impose the pointwise con-

straints over the infinite horizon is to impose the arrival of the

final state to a target set as follows:

(40)

where must be suitably taken.

A first possibility is that (or a subset) is controlled in-

variant, that is there exists a local control that renders posi-

tively invariant and such that the constraints are satisfied for all

initial conditions inside the set [5]. In this case, one can switch

to the local controller as soon as the condition is

satisfied. This idea is well-known in the receding horizon frame-

work (see [15]).

A second possibility is the following. Consider the convex

hull of the vectors of the optimal sequence and their opposite

and assume that is small enough to be included in

(41)

Then the inclusion (40) implies the condition

(42)

which means that (27) is satisfied with characteristic polynomial

. Therefore, instead of

arbitrarily fixing one can proceed as follows.

• Remove the constraint (12) and take a sufficiently

small final set in such a way that (41) is satisfied

for the optimal sequence.

• Compute the vector such that and

.

• Determine the compensator as previously shown.

The first of the previous steps might require an iteration, because

if the condition (41) is not satisfied one should reduce , (pos-

sibly increase ) and repeat the computation.

Remark 5.1: In the “ultimate confinement” problem to a

target one can consider free and solve a minimum-time

problem of reaching a target. This is what we actually do in the

experiments of Section VI.

C. Optimal Pole Assignment

A known problem in pole assignment via static state feedback

for multiple input systems is to exploit the degree of freedom

given by the multiple choices of the feedback matrix [13]. This

problem can be faced in the present context as a special case for

. We can fix the first column of as , and solve

the optimization problem to determine the remaining columns

of and the columns of . Matrix turns out to be square


Fig. 2. Cart-pole system.

invertible as long as Assumption 1 holds. From the equation

, well-known in the eigenvalue assignment

context, we have the static linear control

Just as an example, consider the problem of finding the con-

troller which places the poles in the desired positions and that

minimizes the maximum control effort, in terms of a proper

norm , in the transient starting from . The natural problem

to be solved is the following:

(43)

(44)

where carries the desired polynomial as previously shown.

This problem is a convex programming one and it can be

reduced to a linear programming problem if one chooses the

-norm for . Clearly the solution does not assure that the the

condition is satisfied after the transient even

for the initial condition . However, the condition can

be assured for any if we consider a characteristic polynomial

subject to the constraints (35).

VI. EXPERIMENTAL RESULTS

In this section, we report some experimental results obtained

by applying a relatively optimal controller to the cart-pole

system shown in Fig. 2. The system has one input (the current

given to the electrical motor, proportional to the force applied

to the cart) and two outputs (the position of the cart and the

angle of the pole , measured by means of two encoders). The

system has the following state vector: . A scheme

of the system is reported in Fig. 3. The continuous-time model,

linearized around a stable equilibrium point is

where is the acceleration of gravity, the length of the pole,

and the friction coefficients for the pole and the cart, re-

spectively, and a coefficient that takes into account the whole

Fig. 3. Scheme of the cart-pole system.

Fig. 4. Open-loop optimal trajectory.

mass of the cart and the pole and the proportionality between the

input and the force applied to the cart (note that the dynamics of

the pole does not influence that of the cart). By substituting the

proper values ( , and having been determined experimen-

tally) and performing a zero-order-hold sampling of the system

(sampling time: 0.125 s) we get the state and input matrices:

Now, we solve the following open-loop minimum-time con-

strained problem: Find the shortest input sequence ,

that drives the system from to a

neighborhood of the origin under the constraints

and . We stress that these constraints are not hard:

they can be violated without particular consequences. They have

to be considered as design specification, limiting the accelera-

tion of the cart and the angle of the pole of the optimal trajectory.

As a final constraint, we impose

For , , , , ,

we get the optimal

open-loop trajectory reported, for the cart position and the pole

angle, in Fig. 4. A phase-plane representation of such trajectory

(which is steps long) is shown in Fig. 5.

By choosing a that ensures invertibility of , we get a

relatively optimal state-feedback dynamic controller of order

(note that being , the


Fig. 5. Phase-plane representation of the open-loop optimal trajectory.

Fig. 6. Actual trajectory (squares) and the optimal one (points).

Fig. 7. Trajectory of the controlled system from �x = [0 0� 0:3 0] .

characteristic polynomial assigned to the closed-loop system is

). Fig. 6 shows a comparison

between the optimal trajectory (dot-marked line) and that ob-

tained by means of the proposed compensator (square-marked

line) starting from the nominal initial condition. Notwith-

standing the extremely simple model (that neglects the strong

static friction and some dependencies of the parameters from

the position of the cart), the controlled system behaves well

and the actual trajectory is very close to the optimal. As

expected, the proposed feedback scheme produces a scaled

version of the optimal trajectory when the (nonnominal) initial

condition is aligned with the nominal one: Fig. 7 shows the

trajectory of the system from the nonnominal initial condi-

tion . In Fig. 8, we report the behavior

of the controlled system for the nonnominal initial condition

. The relatively optimal controller employed

Fig. 8. Trajectory of the controlled system from �x = [�0:51 0 0 0] .

Fig. 9. Comparison between a state-feedback trajectory (continuous line)and an output-feedback trajectory (dashed line) obtained by means of areduced-order observer.

for the previous experiments is in practice a state-feedback

controller, because the cart and pole velocities are computed by

means of second-order filters based on the position encoders,

read at each 1 ms. As said in Section IV, whenever it is possible

to initialize an observer, an output-feedback relatively optimal

controller can be implemented. As an example, Fig. 9 shows the

state-feedback trajectory (continuous line) and the output-feed-

back trajectory (dashed line) from the same initial condition

(aligned with the nominal). We employed a second-order dis-

crete-time (reduced) observer having the same sampling time

of the discretized system, i.e., 0.125 s according to the scheme

in [1]). In this case, this reduced observer has to be initialized at

its zero state when activating the controller. The behavior of the

system under the output-feedback relatively optimal controller

is close (although slightly worst) to that obtained when using

the state-feedback. Movies of the experimental results reported

here, are available at http://www.dimi.uniud.it/pellegri.

VII. CONCLUSION

In this paper, we have shown that a relatively optimal con-

trol (i.e., a state feedback controller which produces the optimal

transient from a “nominal initial condition” and that is stabi-

lizing, possibly with constraint violation, from any other ini-

tial state) can be achieved without considering a zero terminal

constraint. As a result, we obtained a closed-loop system which

is not a finite-impulse response (FIR) but whose characteristic


polynomial can be arbitrarily assigned. We have shown that we

can skip the assignment but achieve stability if we impose some

terminal constraints to the open-loop optimization problem. The

problem of strong stabilization has been considered as well, pre-

cisely we faced the problem of simultaneous assignment of the

closed-loop and compensator eigenvalues. In the output feed-

back case, a relatively optimal control can be achieved by an

observer that has to be properly initialized. The experimental

results provided show that the presented technique works sig-

nificantly well on a real problem.

The proposed subject is suitable to further investigation.

Among the possible significant extensions we mention the goal

of determining a (nonlinear) static relatively optimal control,

instead of a linear dynamic one. Furthermore, the constraints

considered in this paper are “soft” performance constraints.

If hard constraints are included in the problem, such as input

saturations, then, the stabilization may be clearly not achiev-

able from any initial condition. Combining the proposed design

methodology with a suitable (possibly maximal) domain of

attraction is a future subject of research.

REFERENCES

[1] G. Basile and G. G. Marro, Controlled and Conditioned Invariants in

Linear system Theory. Englewood Cliffs, NJ: Prentice-Hall, 1992.[2] A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, “The explicit

linear quadratic regulator for constrained systems,” Automatica, vol. 38,no. 1, pp. 3–3, Jan. 2002.

[3] A. Bemporad, F. Borrelli, and M. Morari, “Model predictive controlbased on linear programming—The explicit solution,” IEEE Trans.

Autom. Control, vol. 47, no. 12, pp. 1974–1985, Dec. 2002.[4] D. P. Bertsekas, Dynamic Programming and Optimal Control. Bel-

mont, MA: Athena Scientific, 2000.[5] F. Blanchini, “Set invariance in control—A survey,” Automatica, vol. 35,

no. 11, pp. 1747–1767, 1999.[6] F. Blanchini and F. A. Pellegrino, “Relatively optimal control and its

linear implementation,” IEEE Trans. Autom. Control, vol. 48, no. 12,pp. 2151–2162, Dec. 2003.

[7] C. E. Garcia, D. M. Prett, and M. Morari, “Model predictive controltheory and practice—A survey,” Automatica, vol. 25, no. 3, pp. 335–348,1989.

[8] P. O. Gutman and M. Cwikel, “An algorithm to find maximal state con-straint sets for discrete-time linear dynamical systems with bounded con-trol and states,” IEEE Trans. Autom. Control, vol. AC-32, no. 3, pp.251–254, Mar. 1987.

[9] S. S. Keerthi and E. G. Gilbert, “Computation of minimum-timefeedback control laws for discrete-time systems with state-control con-straints,” IEEE Trans. Autom. Control, vol. AC-32, no. 5, pp. 432–435,May 1987.

[10] V. Kucera, Analysis and Design of Discrete Linear Control Systems,ser. Prentice-Hall International Series in Systems and Control Engi-neering. Englewood Cliffs, NJ: Prentice-Hall, 1991.

[11] D. Q. Mayne, “Control of constrained dynamic systems,” Eur. J. Control,vol. 7, pp. 87–99, 2001.

[12] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Con-strained model predictive control: Stability and optimality,” Automatica,vol. 36, pp. 789–814, 2000.

[13] P. Petkov, N. D. Christov, and M. KonstantinovM, Computational

Methods for Linear Control Systems. Englewood Cliffs, NJ: Pren-tice-Hall, 1991.

[14] A. S. Sage and C. C. White, Optimum system control. Upper SaddleRiver, NJ: Prantice-Hall, 1997.

[15] M. Sznaier and M. Damborg, “Suboptimal control of linear systems withstate and control inequality constraints,” in Proc. 26th Conf. Decision

and Control, Los Angeles, CA, Dec. 1987, pp. 761–762.[16] D. C. Youla, H. A. Jabr, and J. J. Bongiorno, “Modern Wiener-Hopf

design of optimal controllers, II. The multivariable case,” IEEE Trans.

Autom. Control, vol. AC-21, no. 3, pp. 319–338, Mar. 1976.

Franco Blanchini (M’92–SM’05) was born in Leg-nano (MI), Italy, on December 29, 1959. He receivedthe Laurea degree in electrical engineering from theUniversity of Trieste, Trieste, Italy, in 1984.

He is a Full Professor with the Engineering Fac-ulty of the University of Udine, Udine, Italy, where heteaches dynamic system theory and automatic con-trol. He is a member of the Department of Mathe-matics and Computer Science of the same university,and is the Director of the System Dynamics Labora-tory.

Dr. Blanchini has been Associate Editor of Automatica since 1996. In 1997,he was a member of the Program Committee of the 36th IEEE Conference onDecision and Control, San Diego, CA. In 1999, he was a member of the ProgramCommittee of the 38th IEEE Conference on Decision and Control, Phoenix, AZ.In 2001, he was member of the Program Committee of the 40th IEEE Confer-ence on Decision and Control, Orlando, FL. In 2003, he was a member of theProgram Committee of the 42nd IEEE Conference on Decision and Control,Maui, HI. He was Chairman of the 2002 IFAC workshop on Robust Control,Cascais, Portugal. He has been Program Vice-Chairman for the conference JointConference on Decision and Control-ECC, Seville, Spain, December 2005. Heis the recipient of 2001 ASME Oil and Gas Application Committee Best PaperAward as a coauthor of the article “Experimental evaluation of a high—gaincontrol for compressor surge instability.” He is the recipient of the 2002 IFACprize survey paper award as author or the article “Set Invariance in Control—Asurvey,” Automatica, November, 1999.

Felice Andrea Pellegrino was born in Conegliano,Italy, in 1974. He received the Laurea degree (magna

cum laude) in engineering from the University ofUdine, Udine, Italy, in 2000, and the Ph.D. degreefrom the same university, in 2005, discussing a thesison constrained and optimal control.

From 2001 to 2003, he was a Research Fellow atInternational School for Advanced Studies, Trieste,Italy. Currently, he is Research Fellow with theDepartment of Mathematics and Computer Science,University of Udine. His research interests include

control theory, computer vision, and pattern recognition.

Relatively optimal control with characteristic polynomial assignment

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