IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 54, NO. 2, FEBRUARY 2007
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Controller Synthesis for Positive Linear Systems With Bounded Controls M. Ait Rami and F. Tadeo
Abstract—This brief solves some synthesis problems for a class of linear systems for which the state takes nonnegative values whenever the initial conditions are nonnegative. In particular, the synthesis of state-feedback controllers is solved in terms of Linear Programming problem, including the requirement of positiveness of the controller and its extension to uncertain plants. In addition, the synthesis problem with nonsymmetrical bounds on the stabilizing control is treated. Index Terms—Bounded controls, linear programming, positive systems, stabilization.
I. INTRODUCTION
M
ANY physical systems involve quantities that have intrinsically constant sign. For example, absolute temperatures, level of liquids in tanks and concentration of substances in chemical processes are always positive. These examples belong to the important class of systems which have the property that the state is nonnegative whenever the initial conditions are nonnegative. In the literature, such systems are referred to be positive (see [4], [8], [11], and [14] for general references). Control of this kind of systems must take into account this positivity constraint; otherwise the mathematical model of these systems might move into infeasible regions, impossible to reach by the real system, causing loss of stability or performance when the controller is implemented in the real plant. This brief proposes a new approach for the stabilization of MIMO linear positive systems by means of state feedback. Following this approach one can solve different problems for these positive systems: nominal stability and stabilization, robust stabilization with polytopic uncertainty, stabilization with restricted sign controls and stabilization with bounded controls. For the stabilization of positive systems, some previous works, based on algebraic approaches, can be found in [4], [6]. Note that these works are only concerned with the single-input case. Based on Gersgorin’s theorem, a sufficient condition is provided in [10] and formulated as a quadratic programming problem. Recently, in [12] a necessary and sufficient Linear Matrix Inequality (LMI) condition is proposed for the stabilization of linear positive systems, although it was already published in [3] (see chapter 10, page 144). These results are based on a Lyapunov inequality with a diagonal positive definite solution [1], [2].
Manuscript received November 21, 2005. This work was supported by Morocco-Spain AECI under Research Project 8/03/P and CICYT Project DPI2004-07444-C04-02. The work of M. A. Rami was supported by AECI-BecasMAE.This paper was recommended by Associate Editor B. Jean-Pierre. The authors are with the Departamento de Ingenieria de Sistemas y Automatica, Universidad de Valladolid, 47005 Valladolid, Spain (e-mail: ait.rami.
[email protected];
[email protected]). Digital Object Identifier 10.1109/TCSII.2006.886888
In comparison with these previous works, this brief provides a new treatment for the stabilization of positive linear systems where all the proposed conditions are necessary and sufficient and expressed in terms of Linear Programming (LP). It is worth mentioning that the proposed approach is more simple than the LMI approach and possesses a low computational complexity. Also, this brief solves an important synthesis problem: finding bounded stabilizing state-feedback controls. Although the stabilitation of constrained plants have been studied in the literature (see, for example, [5], [13], [15], and [16]), as far as the authors’ knowledge, the proposed approach is original. The remainder of the brief is structured as follows. Section II gives some preliminary results. Section III studies the stabilization problem. A robust stabilization problem is addressed in Section IV. Then the state-feedback synthesis problem with controls having nonsymmetrical bounds is considered in Section V, and finally, Section VI gives some concluding remarks. denotes the nonnegative orthant of the -diNotation: . The following descriptions of elemensional real space ments of matrices are used throughout the brief: and . denotes the transpose of the real (resp. matrix . For a real matrix (or a vector) ) means that its components are positive: (resp. ). II. PRELIMINARIES This section presents some definitions and preliminary results which will be used throughout the brief. Consider the following autonomous linear system:
(1) Let us state the following definition of positive system (1). Definition 2.1: Given any positive initial condition , the system (1) is said to be positive if the corresponding trajectory for all . Next, we need to find under which condition the system (1) is positive. In fact, this condition is provided by a classical result (see [14]). Lemma 2.2: The system (1) is positive if and only if the offdiagonal elements of are positive (2) The above result permits to determine whether a system is positive or not, by simply looking at the entries of the dynamic matrix of the system. In connection with the above Lemma the following definition will be used.
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Definition 2.3: A real matrix is called a Metzler matrix if its off-diagonal elements are positive: . The analysis of the asymptotic stability of the system (1) is presented in the following result which can be seen as a particular case of a more general result concerning the stability of definite positive matrix linear systems (see the Appendix of [7]). Theorem 2.4: Assume that system (1) is positive (or equivalently that the dynamic matrix is Metzler); then the following statements are equivalent. (i) is a Hurwitz matrix: the real part of the eigenvalues of A is strictly negative. (ii) System (1) is asymptotically stable for every initial con. dition (iii) System (1) is asymptotically stable for any arbitrary initial in the interior of . condition (iv) There exists such that
and Find necessary and sufficient conditions on such that there exists a matrix satisfying the following. is a Metzler matrix. • • is a Hurwitz matrix. Now, let us state the main theorem of this section. Theorem 3.1: The following statements are equivalent. such that the (i) There exists a state-feedback law closed-loop system is positive and asymptotically stable. such that is (ii) There exists a matrix both a Metzler and a Hurwitz matrix. vectors and (iii) There exist such that
for
(3) Proof: (i) and (ii) are equivalent since the system is linear and any initial condition can be expressed as the difference of two positive vectors. The proof will be complete if we show the equivalence between (iii) and (iv), and the implication (iv) (i). Assume that (iii) holds for some initial condition in the interior of . Then by integrating System (1) we get
knowing that the trajectory is continuous and is in the interior of we must have . This proofs that (iii) implies (iv). Now, let us show that (iv) also implies (iii): is positive and stable if Knowing that the dual system and only if System (1) is positive and stable, it suffices to prove that the dual system is stable if condition (iv) is satisfied. This fact is also straightforward since it is easy to check that the linear is a Lyapunov function of the dual system. function Finally, the implication (iv) (i) results also from the fact that is a Lyapunov function of the dual system and then the proof is complete. Remark 2.5: The condition and other properties of Metzler matrices can be found in [9]. III. STABILITY SYNTHESIS This section studies the stabilization problem of the following governed linear system:
(4) where . First, we consider the case where the control law is a state-feedback, not sign restricted. This control law must be designed in such way that the resulting governed system is positive and asymptotically stable. In other words, with regard to the previous preliminary results, the problem reduces to look for a state-feedback law , where the matrix has to be determined to satisfy the following problem.
with and Moreover, the gain matrix be calculated as follows:
(5)
. in conditions (i) and (ii) can
Proof: Regarding to the considerations discussed previously the equivalence between (i) and (ii) is straightforward. The proof will be completed by showing that (ii) and (iii) are equivalent. Assume that condition (iii) holds and define the matrix with for . Then, by calculation we got and using condition (iii) . Since then by using Thethis leads to is a Hurwitz matrix if orem 2.4 we can conclude that it is proved that it is a Metzler matrix. Now it is easy to see that is a Metzler matrix since condition (iii) implies for that
The implication and the rest of the proof follows the same line of argument. The importance of the above result is relevant, because it provides not only a checkable necessary and sufficient conditions but also a simple approach to address numerically the computation of the problem. Indeed, one can see that the conditions in statement (iii) of Theorem 3.1 are linear. Thus, these conditions can be solved as a standard Linear Programming problem. In addition, based on the same formulation we can take into account the positiveness of the state-feedback control law by just adding an additional constraint on the variables . This is shown in the following result, that is straightforward from Theorem 3.1. Theorem 3.2: The following statements are equivalent. (i) There exists a positive state-feedback law such that the closed-loop system is positive and asymptotically stable. (ii) There exists a matrix such that and is both a Metzler and a Hurwitz matrix.
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(iii) The following LP problem, in the variables and , is feasible:
for for (with gain matrix
(6)
and ). Moreover, the in conditions (i) and (ii) can be chosen as
where and are given by any feasible solution of the above LP problem. At this stage, some remarks are in order. Remark 3.3: Note that if a negative state-feedback control instead of law is to be considered it suffices to impose in the previous LP formulation. Remark 3.4: We stress out that the LP condition in Theorem 3.1 do not impose any restriction on the dynamics of the governed system. For instance, is not necessarily a Metzler matrix. In this case, the free system is not positive, so that the synthesis problem can be interpreted as enforcing the system to be positive. Remark 3.5: Notice that in the very special case when is a Metzler matrix but not a Hurwitz matrix and is positive, then it impossible to stabilize the system by any positive state-feedback control law. This is a consequence of the previous result since the existence of such control necessarily implies that . Then, from the result of Theorem 2.4 we must have a Hurwitz matrix. Remark 3.6: Observe that selecting in Theorem 3.1 reproduces the main result (sufficient condition) given in [10].
Fig. 1. Trajectories from different initial values.
evolution, from random initial positive conditions, can be seen in Fig. 1. IV. SYNTHESIS WITH UNCERTAIN PLANT An important extension of the proposed approach is the possibility of handling the case when the dynamics of the system are not exactly known, as it is now presented. Consider the following uncertain system:
(7) are supposed to be not exactly is assumed to belong to the following
Matrices determined. But convex set:
(8)
A. Example: Stability Synthesis Suppose we have the following nonpositive system:
are known given matrices. where The proposed robust synthesis problem consists in finding a fixed matrix such that the following closed-loop system is positive and asymptotically stable for every
Using state feedback we want to stabilize the system and enforce the state to be positive. Based on Theorem 3.1, the following inequalities must be fulfilled:
(9) Theorem 4.1: There exists a robust state-feedback law such that the resulting closed-loop system (9) is positive and asymptotically stable for every , if the following LP problem in the variables and , is feasible for
since the gain of a stabilizing control is given by , one feasible solution of the above LP problem provides . The state
for (with
and
(10) ).
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V. SYNTHESIS WITH BOUNDED CONTROLS This section studies the problem of bounded controls for different kinds: bounded positive control and control not restricted in sign, but with nonsymmetric bounds. The following key role lemma will be used for our purpose. Lemma 5.1: Consider the trajectory of the autonomous system ; then, for a given we have for any initial condition satisfying if and only if is Metzler and . Proof: Sufficiency: notice that it suffices to prove the inequality . Because, since is Metzler, then so that if we have . Now, it is shown that , or equivalently that Fig. 2. Trajectories for extreme plants.
Moreover, the gain matrix computed as
of the robust controller can be
where correspond to any feasible solution of the above LP problem. Proof: By a simple convexity argument the proof is straightforward.
One can easily see that . Since and , then we have . Necessity: the system is necessarily positive and then must be Metzler. Now, let , then all the components of the derivative of the state at zero are negative because the state at any time, so that we necessarily have must satisfy . A. Positive Bounded Control This part considers the following constrained system:
A. Example: Uncertain Plant Suppose that the nominal system in the previous example is subject to an uncertain parameter , as follows:
Using state feedback the objective is to determine the set of controllers that stabilize all the possible plants, keeping the state positive for every plant in the given set. From Theorem 4.1, the following conditions must be fulfilled:
(11)
and
That is, the trajectory of the system is positive and the input is constrained to be positive and bounded by a given value . The aim here is to address the following problem. Given find corresponding to the set of initial conditions for which we can determine a positive bounded state-feedback control law such that the resulting closed-loop system is positive and asymptotically stable. Now, we state the main result of this section. Theorem 5.2: Consider the following LP problem in the variables and
for
for Since the gain of any stabilizing control is given by , one feasible solution of the above LP problem provides . With this controller the feedback system is positive: the state evolution for the two most extreme plants ( and ) can be seen in Fig. 2, starting from random initial positive conditions.
with
and
(12)
. Define the matrix
and consider the closed-loop system state-feedback control . Then
under the for any
RAMI AND TADEO: CONTROLLER SYNTHESIS FOR POSITIVE LINEAR SYSTEMS
initial state satisfying . Moreover, the closed-loop system is positive and asymptotically stable. Proof: Take any and that ; then since for solve (12) and define
we have that the matrix is Metzler. The inequality is equivalent to , then by Lemma 5.1 the trajectory of the system is such that for any initial condition satisfying . Using this fact and recalling the inequalities for , or equivalently and , it is easy to see that the state-feedback control is such that for any initial state satisfying . Finally, as we have shown that the closed loop system is posiare positive, because the off-diagonal components of is nothing else than tive, and since we can apply the result of Theorem 2.4 to conclude that the closed-loop system is asymptotically stable, so the proof is complete.
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and consider the closed-loop system under the state-feedback control . Then for any . Moreover, the closed-loop initial state satisfying system is positive and asymptotically stable. Proof: The proof follows the same line of the Proof of can be Theorem 5.2 by using the fact that any gain matrix expressed as the difference of two positive matrices. VI. CONCLUSION This brief has proposed a novel approach to solve some synthesis problems for positive linear systems. The stabilization problem has been considered, and necessary and sufficient conditions for its solvability have been proposed, also for the uncertain case. Moreover, the synthesis problem with bounded controls have been addressed. It has been shown that all the proposed conditions are solvable in terms of simple linear programming problems. ACKNOWLEDGMENT The authors would like to thank Prof. A. Benzaouia for giving them the opportunity to work together.
B. Nonpositive Control With Nonsymmetrical Bounds Consider the following constrained system:
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(13)
and
we will address the following problem with nonsymmetrical bounds on the control. Given find corresponding to the set of initial conditions for which we can determine a bounded state-feedback control law such that the resulting closed-loop system is positive and asymptotically stable. Theorem 5.3: Consider the following LP problem in the and variables
for
for
for with
and
. Define the matrix
(14)
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