1030 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 5, MAY 2011

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1 1030 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 L L 2 Low-Gain Feedback: Their Properties, Characterizations Applications in Constrained Control Bin Zhou, Member, IEEE, Zongli Lin, Fellow, IEEE, Guang-Ren Duan, Senior Member, IEEE Abstract Low-gain feedback has found several applications in constrained control, robust control, nonlinear control In this paper, we first generalize the existing low-gain design methods by introducing the notion of -vanishment by providing a full characterization of feedback gains that achieve such a property We observe that low-gain feedback can lead to energy peaking, namely, the control energy required by low-gain feedback increases toward infinity as the low-gain parameter decreases to zero Motivated by this observation, we consider the notion of 2-vanishment establish several of its characterizations, based on which a new design approach referred to as the 2 low-gain feedback approach for linear systems is developed Different from the low-gain feedback, the 2 low-gain feedback is instrumental in the control of systems with control energy constraints As an application of 2 low-gain feedback, the problem of semiglobal stabilization of linear systems with control energy constraints is solved in this paper The notions of 2 -vanishment also allow us to establish a systematic approach to the design of 2 low-gain feedback The advantage of this new design approach is that it results in a family of control laws, including those resulting from the existing design methods Index Terms Constrained control, energy constraints, 2 low-gain feedback, 2 -vanishment, parametric Lyapunov equation, semiglobal stabilization, slow peaking I INTRODUCTION L OW-GAIN feedback was initially proposed in [18] to deal with semiglobal stabilization of linear systems under actuator saturation A key feature of the low-gain feedback, parameterized in a low-gain parameter, is that if the linear system is asymptotically null controllable with bounded controls (ANCBC), then for any given bounded initial condition, Manuscript received March 25, 2009; revised December 03, 2009; April 26, 2010; August 23, 2010; accepted August 23, 2010 Date of publication September 07, 2010; date of current version May 11, 2011 This work was supported in part by the National Natural Science Foundation of China under Grants , , ; the China Postdoctoral Science Foundation under Grant ; the Heilongjiang Postdoctoral Foundation of China under Grant LRB10-194; the Foundation for Innovative Research Group of the National Natural Science Foundation of China under Grant ; the Development Program for Outsting Young Teachers at the Harbin Institute of Technology under Grant HITQNJS ; a Zhiyuan Chair Professorship at Shanghai Jiao Tong University, Shanghai, China Recommended by Associated Editor A Loria B Zhou G-R Duan are with Center for Control Theory Guidance Technology, Harbin Institute of Technology, Harbin , China ( binzhoulee@163com; binzhou@hiteducn; grduan@hiteducn) Z Lin is with the Charles L Brown Department of Electrical Computer Engineering, University of Virginia, Charlottesville, VA USA ( zl5y@virginiaedu) Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TAC the peak magnitude of the control signal goes to zero as the low-gain parameter approaches zero As a result, for any given bounded set of initial conditions, actuator saturation can be avoided by decreasing the value of the low-gain parameter In other words, an ANCBC linear system is semiglobally stabilizable by linear low-gain feedback We recall that [27] a linear system is said to be ANCBC if it is stabilizable in the usual linear systems sense all its open-loop poles are located in the closed left-half -plane The low-gain feedback laws in [18] were constructed by eigenstructure assignment Alternative ways of constructing low-gain feedback laws were later proposed based on the solution of a parameterized algebraic Riccati equation (ARE) [19], ARE [29], parametric Lyapunov matrix equations [35], respectively Low-gain feedback has found applications in control [15], global stabilization with input saturation [6], output regulation with input saturation [12], external internal stabilization with input saturation [24], semiglobal stabilization with input magnitude rate saturations [13], [16], stability region analysis with actuator saturation [30], nonlinear stabilization [1], [14], [15], nonlinear control [15], stabilization of time-delayed systems [17] While low-gain feedback has been widely used in the literature, its characteristics have yet to be fully understood Slow peaking is one of them As the state of an ANCBC linear system can be made to converge arbitrarily slowly by placing the poles of the closed-loop system close to the imaginary axis, its magnitude will climb slowly to an arbitrarily high value during the convergence process Roughly speaking, the state cannot converge to zero very slowly without experiencing expansion in its magnitude Slow peaking phenomenon was recognized in [15] has been utilized in semiglobal stabilization of cascade nonlinear systems [25] More intricate aspects of the slow peaking phenomenon are yet to be revealed We observe that although the slow peaking phenomenon cannot be avoided in the state evolution of the closed-loop system under low-gain feedback, the control signal that results from the multiplication of the state the feedback gain can, however, be kept under an arbitrarily low level by decreasing the value of the low-gain parameter Thus, the peaking in the different components of the state can be viewed as canceling each other when they are summed with feedback gains as weighting factors This is the key property on which the low-gain design was conceived Clearly, in the low-gain feedback, the gain approaches zero as its parameter does However, does this property itself account for the cancellation of the slow peaking? In other words, is every stabilizing feedback gain /$ IEEE

2 ZHOU et al: AND LOW-GAIN FEEDBACK 1031 that approaches zero as the low-gain parameter does able to cancel the slow peaking in the control signal? As will be seen later, the answer is that the cancellation cannot be achieved by such arbitrary stabilizing feedback gains Another peaking phenomenon, we referred to as energy peaking, is that the energy required of the control input resulting from low-gain feedback may increase to infinity as the low-gain parameter does This means that though the magnitude of the control signal can be made arbitrarily small by decreasing the low-gain parameter in the low-gain feedback, the required control energy may increase unboundedly Motivated by the slow peaking phenomenon that exists in closed-loop systems under low-gain feedback, in this paper, we will take a closer look at this category of feedback gains, with the peaking measured both in the norm in the norm More specifically, we consider more general problems we call the norm vanishment norm vanishment Roughly speaking, we say a matrix/vector valued time signal that depends on a scalar is -vanishing ( -vanishing) if its norm approaches zero as does These notions of norm vanishment allow us to give a strict definition of low-gain feedback, namely, it is reasonable to say that a stabilizing feedback gain parameterized in the scalar is an low gain if the resulting control signal is -vanishing ( -vanishing) By characterizing the necessary sufficient conditions for -vanishment ( -vanishment), we are able to establish conditions to test whether a feedback gain is an low-gain feedback It is shown that low-gain feedback plays a similar critical role in the design of control systems with energy constraints in the control input as the low-gain feedback plays in the design of control systems with magnitude constraints on the control input As an illustration of the use of low-gain feedback in energy-constrained control, we show that semiglobal stabilization of some linear systems with energy constraint is solvable via low-gain feedback Another aspect of the results in this paper is that we propose a systematic approach to the design of low-gain feedback based on the necessary sufficient conditions we are to characterize for the -vanishment -vanishment The primary advantage of this approach is that it can provide a family of solutions to the low-gain feedback design As a result, the low-gain feedback gains resulting from the eigenstructure assignment approach [15] the parametric Lyapunov matrix equation approach [35] can be regarded as special cases of the proposed solutions A possible application of this design advantage is that the free parameters in the solutions can be exploited for the full utilization of the actuator capacities, which in turn can improve the performances of the resulting closed-loop system The remainder of this paper is organized as follows An introduction to low-gain feedback the slow peaking phenomena that result from it are recalled in Section II In Section III, we give a formal definition of the notions of -vanishment -vanishment develop a series of necessary sufficient conditions to test these two properties Our results obtained in Section III allow us to reconsider the well-developed low-gain feedback for linear systems under actuator saturation the new low-gain feedback for linear systems with control energy constraint in Section IV Based on the necessary sufficient conditions for - -vanishment, a systematic approach to the design of low-gain feedback is proposed in Section V, an illustrative numerical example is also included Finally, Section VI concludes this paper Notation: The notation used in this paper is fairly stard We use,,,, to denote the transpose, the eigenvalue set, the trace, the determinant, the rank of matrix, respectively The symbol, are two integers with, denotes the set, For two matrices, denotes their Kronecker product For two sets, If, then The notation refers to any norm of matrix if the subscript is omitted, refers to the Frobenius norm of, denotes the combination number We use to denote the (block) Jordan matrix whose diagonal elements are For example Furthermore, we use to denote a nilpotent matrix Let,, be a series of matrices of appropriate dimensions We denote a (block) diagonal matrix whose diagonal elements are, For a function, the norm norm of, if they exist, are respectively defined as Finally, we define II SLOW PEAKING UNDER LOW-GAIN FEEDBACK A Peaking Phenomenon Consider a linear system subject to actuator saturation (1) are, respectively, the state input vectors, the function is a vector-valued stard saturation function, ie,, for each, Here, we have slightly abused the notation by using to denote both the scalar valued vector valued function We have also assumed, without loss of generality, the unity saturation level Nonunity saturation level can be absorbed by the matrix the feedback gain It is well known (see, for example,

3 1032 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 [11], [18], [27]) that such a system as (1) can be globally or semiglobally stabilized if only if it is ANCBC For system (1), the second author of this paper his coauthors have developed a low-gain feedback design approach to solve a host of control problems in a semiglobal framework An underlying feature of this design framework is that both the resulting stabilizing feedback gains, parameterized in a scalar low-gain parameter, the norm of the control signal under any given bounded initial condition approach zero as the parameter does [15] Therefore, for any given bounded set of initial conditions, we can decrease the low-gain parameter to make the peak values of the control signals as small as needed to avoid actuator saturation A natural question that arises is this: Does every stabilizing feedback gain that is parameterized in a scalar goes to zero as decreases to zero achieve arbitrarily small peak value in the control signals for a given bounded set of initial conditions? The answer is no, as the following simple example indicates Example 1: Consider a linear system in the form of (1) with The following matrix: is a stabilizing feedback gain such that Since holds true, can be naturally referred to as a low gain However, the peak value of the control signal can be arbitrarily large when approaches zero To see this, we note which implies that there exists a bounded initial condition such that the peak value of approaches infinity as approaches zero The above example shows that not every parametric stabilizing feedback gain that approaches zero as the parameter does is a meaningful low-gain feedback The reason is the so-called slow peaking phenomenon that results from the low-gain feedback [15], [25] That is, under such a low-gain feedback, the value of the state of the closed-loop system increases toward infinity as the time goes to infinity the value of the low-gain parameter decreases to zero Slow peaking phenomenon has been recognized in [15] Roughly speaking, it is found that for a single input system that is ANCBC, the unique feedback gain such that can always cancel the peak effect in the control signal However, this is not the case for general multiple-input systems as illustrated in Example 1 Indeed, special care must be taken to arrive at a low-gain feedback that would cancel the slow peaking effect in the control signal [15] This motivates us to formalize the notion of low-gain feedback develop conditions on to guarantee that the peak value in the control signal can be eliminated To this end, we (2) (3) will consider solve a more general problem associated with the notion of -vanishment, of which the existing low-gain feedback is a special case B Peaking Phenomenon We begin our discussion with an example Example 2: Consider a system in the form of (1) with given in (2) Since is ANCBC, the system can be semiglobally stabilized by low-gain feedback Indeed, the following feedback gain: which assigns all the eigenvalues of at, is a low-gain feedback because (see Section V-B for an explanation) Therefore, the peak value in the control signal (in the absence of input saturation) can be reduced to an arbitrarily low level by decreasing the value of That is, the actuator saturation can be avoided by decreasing the value of for any bounded set of initial conditions Now, let Then, it is easy to compute that (see, for example, [3]) It follows that, which, in view of (5), implies that there exists a bounded initial condition such that This indicates that if we decrease the value of toward zero so as to avoid input saturation, the control energy that is used to steer the initial state to the origin approaches infinity The above example shows that although the norm of the resulting control signal can be made arbitrarily small by decreasing the value of, there exists an energy peaking phenomenon, namely, the norm of the control signal approaches infinity as approaches zero The following question then arises: For a linear system, can we design a stabilizing feedback gain such that the norm of the resulting control signal also approaches zero as does, namely In other words, can we steer bounded initial state to the origin with a control input that has arbitrarily small energy? The answer to this problem is quite positive, as will be seen later In fact, it is natural important to consider control problems with energy constraints, as practical systems can only be powered with finite energy However, these problems have not received as much attention as control systems with magnitude saturation, which have been widely studied in the past several decades (see, for instance, [5], [7], [9], [11], [28], [31], [32], (4) (5)

4 ZHOU et al: AND LOW-GAIN FEEDBACK 1033 the references therein) Only very recently has the null controllability with vanishing energy problem been considered in [8] [23] Roughly speaking, a system is null controllable with vanishing energy (NCVE) if any bounded initial state of it can be steered to the origin with arbitrarily small energy cost According to the results obtained in [8] [23], it is interesting to note that the condition for null controllability with vanishing energy happens to be the same as conditions for asymptotically null controllability with bounded controls (see, for example, [15], [26], [27]) Motivated by the peaking phenomenon, in this paper we also formalize the notion of low-gain feedback establish conditions on to guarantee that the peaking can be eliminated To this end, we consider solve a more general problem associated with the notion of -vanishment, of which the low-gain feedback problem is a special case At the end of this section, we should point out that considering peaking phenomena of control systems is not new in the literature For example, [4] considered the bounded peaking in the optimal linear regulator with cheap control However, the feedback gain considered in [4] is of a high-gain type, while the feedback gain considered in this paper is of a low-gain type To the best of our knowledge, this problem has not been fully studied in the literature, though some specific designs have been proposed in [15] Moreover, we would also like to point out that the method we are to use the results we obtained are totally different from those in [4] Finally, [4] was concerned with the peaking phenomenon in the state trajectories, while we are interested in eliminating the peaking phenomenon from the control signals (since peaking phenomenon cannot be avoided in the state trajectories of the system considered in this paper) III -VANISHMENT AND ITS CHARACTERIZATIONS We begin this section by introducing a condition Condition 1: The matrix is a continuous matrix function of is such that The above condition means that should be asymptotically stable for all should not be exponentially unstable Moreover, it is clear that is bounded for all We then define the following notions of -vanishment -vanishment Definition 1: Given a continuous function Let satisfy Condition 1 The matrix pair is said to be -vanishing ( or )if exists equals zero To proceed, we introduce the following lemma, whose proof is very simple is omitted Lemma 1: Let, Assume that for all all, are nonsingular bounded Then, for or, if only if The following proposition is then a consequence of Definition 1 Proposition 1: Given satisfying Condition 1 Then, we have the following 1) The matrix pair is -vanishing ( or ) only if 2) If is -vanishing ( or ), then for an arbitrary integer, is also -vanishing ( or ) Proof: For brevity, we only prove Item 1 with We show this by contradiction First, we recall that the Lyapunov equation, is asymptotically stable, has a unique positive semidefinite solution Now assume that be the unique positive semidefinite solu-,wehave Then, by letting tion to (6) (7) Consider two cases Case 1) Since satisfies Condition 1, may contain unbounded terms as approaches 0 Moreover, as satisfies (7), we conclude that the limit of approaches infinity as approaches 0 Case 2) If the limit of exists as approaches 0, because of (7), we must have In both cases, in view of (8), we know that A contradiction In fact, Item 1 of Proposition 1 can be strengthened with the help of Lemma 1 Proposition 2: Assume that satisfies Condition 1 1) If, further, is marginally stable, namely, its eigenvalues on the imaginary axis are simple, then is -vanishing if only if (6) is satisfied 2) If is asymptotically stable, then is -vanishing if only if (6) is satisfied Proof: 1) We only need to show that if is marginally stable (6) is satisfied, then is -vanishing In fact, if is marginally stable, we know that is bounded for all Then, by continuity Condition 1, is also bounded for all for sufficiently small The result then follows from Lemma 1 2) We only show that if (6) is satisfied, then is -vanishing Note that if is asymptotically stable, then the Lyapunov matrix equation has a unique nonnegative definite solution Therefore, by continuity, we know that the solution to (22) satisfies The result then follows from (8) (8)

5 1034 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 We note that the condition in Item 2 of Proposition 2 is more restrictive than the condition in Item 1 of Proposition 2 A Algebraic Characterizations for -Vanishment Clearly, using the definition to test whether a matrix pair is -vanishing ( or ) is difficult in general Here, we give an algebraic characterization of the -vanishment ( or ) property Let exists Clearly, implies that To derive the results, we give the following lemma, whose proof is provided in Appendix A for clarity Lemma 2: Let,,be some continuous matrix functions, is some given integer, Denote (9) Theorem 1: Let, be related with (12) (16) Then, is -vanishing if only if is -vanishing, moreover, is -vanishing if only if for all,, (17) (18) for all,, is defined in (11) Proof: We only prove the case that since can be proven similarly By using (13) (16), we have (19) Then, if only if (10) Since is bounded for, it follows from Lemma 1 (19) that is -vanishing if only if is -vanishing Moreover, with the partition of in (16) relation (19), the -vanishment of is equivalent to if if (11) In the remainder of this subsection, we will suppress the independent variable for simplicity For example, will be denoted as Let be a matrix such that is transformed into its real Jordan canonical from with (12) (13) are respectively given by (14) By identifying the special structure of,wehave (20) (21) with Note that are commutable Therefore, it follows from (21) that Similarly, we have (22) with,, (15) Note that, for all (23) Clearly, we have Consequently, we let with (16) namely, is nonsingular bounded for arbitrary By applying Lemma 1 again, it follows from (22) (23) that the relations in (20) are true if only if in which,,, Then, there exists a sufficiently small scalar such that is bounded nonsingular for all We then have the following result for testing the -vanishment ( or ) property of a matrix pair (24)

6 ZHOU et al: AND LOW-GAIN FEEDBACK 1035 Since,, are nilpotent matrices, we have Then, by using (28) (26), we have which, according to Lemma 2, is equivalent to (17) Similarly, the relation in (24) is also equivalent to (18) The proof is completed Note that to apply Theorem 1, we need to first transform to its Jordan form, which is not very convenient in general In a special case, such transformation is not required This is stated as follows Theorem 2: Assume that all the eigenvalues of are with Moreover, let be the maximal algebraic multiplicity of such eigenvalue Then, is -vanishing if only if, for all for any from which it follows that Consequently, we conclude that (29) (25) is defined in (11) Proof: Again, we only prove the case Since all the eigenvalues of are, we conclude that all the eigenvalues of are zero, namely is a nilpotent matrix Moreover, the nilpotent index of is, that is, Therefore, we have The result then follows directly from Lemma 2 Theorems 1 2 clearly imply that if is -vanishing, then is -vanishing This is not difficult to underst as this situation also exists in peaking peaking for cheap control problems studied in [4] Indeed, the -vanishment requires a rather weaker condition than -vanishment The distinction can also be observed in Proposition 2 B Lyapunov Inequality Characterizations of -Vanishment Here, we present Lyapunov inequality characterizations of the -vanishment property Again, throughout this subsection, we will also suppress the independent variable for simplicity Theorem 3: Let satisfy Condition 1 Then, is -vanishing if only if there exists a scalar such that the following Lyapunov inequality: (26) has a solution that is continuous in, positive definite for all, bounded as approaches zero, satisfies (27) Proof: The proof for the Only If part of this theorem is rather involved is provided in Appendix B for clarity We only provide here a proof for the If part For any, let (28) On the other h, since is continuous for is bounded as approaches zero, we know that is bounded for all Then, taking the limit on both sides of (29) with respect to using (27) leads to Therefore, we conclude that, which ends the proof We next establish a Lyapunov inequality characterization of the -vanishment property Theorem 4: Let satisfy Condition 1 Then, the matrix pair is -vanishing if only if there exists a scalar such that the following Lyapunov inequality: (30) has a solution that is a continuous matrix function of, positive definite for all, (31) Proof: Since is Hurwitz for all, similar to the proof of Proposition 1, we have (32) is the positive semidefinite solution to the following Lyapunov matrix equation: If : Comparing (30) to (33), we conclude that, Consequently, it follows from (32) (31) that (33) That is, is -vanishing Only If : Let denote, respectively, the minimal the maximal eigenvalues of matrix Let

7 1036 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 solution to be the unique positive definite (34) with, namely Then, we can compute Therefore, we have, namely satisfies all the conditions in Theorem 3, thus is -vanishing IV APPLICATION: LOW-GAIN FEEDBACK FOR CONSTRAINED CONTROL A Low-Gain Design With Magnitude Constraint on Input Here, we revisit the well-developed low-gain design method [15] for constrained linear systems by using the theory developed in Section III Consider a linear system with input saturation (36) Therefore, we have (35) Let, which is strictly positive definite for all Then, it follows from (33) (34) that satisfies the inequality in (30) On the other h, from (32) we have Assume that is ANCBC Without loss of generality, it is further assumed that is controllable Let be a continuous matrix function of such that is Hurwitz for all Then, matrix satisfies Condition 1 For an arbitrary initial condition, the control signal of the closed-loop system consisting of (36) in the absence of saturation the feedback is Therefore, if the limit which, together with (35), imply that such also satisfies (31) The proof is completed Similar to the algebraic characterizations for -vanishment given in Theorems 1 2, we can also obtain the following statement from the Lyapunov inequality characterizations of -vanishment given in Theorems 3 4 Corollary 1: If there exists an a such that all the conditions in Theorem 4 are satisfied, then there exists an a such that all the conditions in Theorem 3 are also satisfied, namely, if is -vanishing, then is -vanishing Proof: Let be such that all the conditions in Theorem 4 are satisfied Then, are continuous functions of, is a unitary matrix for all Moreover, let Then, is also a continuous function of is such that Let (37) is well defined the initial condition is bounded, the peak value of can be made arbitrarily small by decreasing the value of [15] As a result, the actuator saturation can be avoided, stability of the closed-loop system can be guaranteed Motivated by this observation, we give the following definition Definition 2: ( Low-Gain Feedback) Assume that is ANCBC is stated above Then, is called an low-gain feedback for if (37) is satisfied By the definition of -vanishment low-gain feedback, we can see that is an low-gain feedback for in the sense of Definition 2 if only if is -vanishing Then, by applying the theory developed in Section III, we can test whether a given stabilizing feedback gain is an low-gain feedback for by using either Theorem 1, 2, or 4 In the literature on low-gain feedback design for ANCBC linear systems (see [15] for example), it is well recognized that a feedback gain that satisfies (38) from which it follows that Moreover, the inequality in (30) can be written as it follows that makes Hurwitz for all in general does not lead to the cancellation of the effect on the control signal of the slow peaking in the state, namely relation (37) For example, different ways to construct special low-gain feedback that can cancel such effect have been attempted [15], [18] An explicit illustrative example for this fact is Example 1 Indeed, according to Proposition 1, relation (38) is only a sufficient condition for (37) in general For completeness, we present a corollary of Propositions 1 2 as follows

8 ZHOU et al: AND LOW-GAIN FEEDBACK 1037 Corollary 2: Assume that is ANCBC is stated before 1) is an low-gain feedback for in the sense of Definition 2 only if (38) is satisfied 2) Assume that is marginally stable, namely, its eigenvalues on the imaginary axis are simple Then, is an low-gain feedback for if only if (38) is satisfied As mentioned earlier, there are several existing approaches to the design of a low-gain feedback, including the AREbased approach [19], the eigenstructure assignment-based approach [15], the parametric Lyapunov equation-based approach [35] In what follows, for illustration, we will show that these approaches can provide low gains in the sense of Definition 2 We first recall the ARE-based low-gain design [15] Assume that is ANCBC Let be positive definite for all, Then, for any, the ARE (39) has a unique positive definite solution such that Moreover, the low gain is given by (40) Let It follows from (39) (40) that (41) Moreover, we conclude from that is bounded for all Therefore, all the conditions in Theorem 3 are satisfied for We thus conclude that the ARE-based low-gain design can lead to an low gain in the sense of Definition 2 Remark 1: Since the parametric Lyapunov equation-based low-gain design is based on the solution to the following parametric matrix equation [35]: which is also in the form of (39), this low-gain design approach can also lead to an low gain in the sense of Definition 2 In the literature, an important low-gain design approach for a matrix pair in the form of (42) is frequently used Such a system represents a chain of integrators It has many applications has received much attention in the past several decades (see [20], [21], [28], the references therein) This type of low-gain design method is recalled as follows (see, for example, [15]) Let be in the form of (42), let be a vector such that the polynomial is Hurwitz Then, the pole assignment low-gain feedback is given by We call this method pole assignment because, the roots of can be arbitrarily assigned by the choice of Corollary 3: The pole assignment low-gain design approach is an low-gain design approach in the sense of Definition 2 Proof: Let be the unique solution to the Lyapunov matrix equation (43) is given Let, is defined in (67) in Appendix B With the special structures of, we have,, Multiplying both sides of (43) by in view of the above relations, (43) is equivalent to Then, we have Therefore, all the conditions in Theorem 3 are satisfied for, the proof is ended At the end of this subsection, we use the developed theory to verify the result in Example 1 Example 3: Let be in the form of (2) be given by (3) Since has a single eigenvalue it follows from Theorem 2 that is not -vanishing, thus is not an low-gain feedback for, which validates the conclusion in Example 1 B Low-Gain Design With Energy Constraint on the Input Consider a linear system in the following form (44), are, respectively, the state input vectors Let denote the solutions of (44) with initial condition input We recall the following definition of null controllability with vanishing energy for system (44) Denote Definition 3: [8] The system (44) [or the matrix pair ] is said to be NCVE if for each initial, there exists a sequence of pairs,, such that Roughly speaking, a system is NCVE if, for any bounded initial condition, there exists a control signal with arbitrarily small energy that steers the state of the system to the origin This class of systems the related control problems have many applications in practice Regarding the criterion for null controllability with vanishing energy, it is shown in [23] that the linear system (44) is NCVE

9 1038 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 if only if is controllable in the ordinary sense all the eigenvalues of are located in the closed left-half -plane (in that paper, the results are developed for infinite dimensional linear systems) It follows that a linear system is NCVE if only if it is ANCBC [27] Parallel to the low-gain feedback, we can state the low-gain feedback as follows Definition 4: ( Low-Gain Feedback) Assume that is NCVE A stabilizing feedback gain is said to be an low-gain feedback if is -vanishing, namely Based on the above definition the results developed in Section III, we can use Theorems 1 4 to test whether a given feedback gain is an low gain Take the ARE-based low-gain design for example In view of (41), since we can see that all the conditions in Theorem 4 are satisfied for as Therefore, the ARE-based low gain given in Section IV-A is an low gain To illustrate the application of the low-gain feedback, we consider the following semiglobal stabilization problem Let,, namely Finally, the asymptotic stability of (47) follows from the fact that is asymptotically stable for all To conclude, we use the theory we have developed to verify the result obtained in Example 2 Example 4: Let be in the form of (2) be given by (4) According to (25), we have Therefore, it follows from Theorem 2 that is not -vanishing, thus is not an low-gain feedback for, which coincides with the conclusion obtained in Example 2 V SYSTEMATIC APPROACH TO LOW-GAIN FEEDBACK DESIGN In this section, we establish a systematic approach to the design of ( ) low-gain feedback based on the results obtained in Section III provide an illustrative example to validate its effectiveness A Design Approach To simplify the idea, we consider a special class of feedback gains defined as Problem 1: ( Semiglobal Stabilization) Consider the linear system (44), which is NCVE For any given set that is arbitrarily large bounded, find a control such that the closed-loop system is locally asymptotically stable with contained in the domain of attraction Theorem 5: Assume that is NCVN If is an low-gain feedback for the matrix pair, then there exists an such that the following control law solves Problem 1: (45) Proof: Since is an low-gain feedback for matrix pair, by definition, we have (46) Note that the closed-loop system consisting of (44) (45) is from which it follows that (47) is ANCBC (see Remark 4 for an explanation) We are interested in the problem of designing a feedback gain such that is an low gain for the matrix pair in the sense of Definitions 2 4 Note that holds if only if there exists a nonsingular matrix such that (48) It follows from Corollary 2 Proposition 2 that is an low gain only if Then, by taking the limit on both sides of (48) using the fact that is nonsingular, we have, which means that is similar to Denote, or equivalently Then, the equation in (48) can be equivalently rewritten as We recall that a pair of polynomial matrices is right coprime if only if (49) (50) Therefore, as is bounded in view of (46), we conclude that there exists an such that, Then, regarding the solutions to the linear matrix equation (50), we have the following result Theorem 6: Let be some right coprime polynomial matrices such that (51)

10 ZHOU et al: AND LOW-GAIN FEEDBACK 1039 Assume that are given by Consequently, we let be partitioned as (52) with being some integer Clearly,,, are polynomial matrices in Then, for an arbitrary, the complete solutions to the linear matrix equation (50) can be characterized as (53) is an arbitrary parametric matrix is continuous in, representing the degree of the freedom in the solution Proof: Since satisfies (51), we conclude that the matrix pair given in (52) satisfies On the other h, since is right coprime, we know from (52) that is also right coprime for an arbitrary With this, the result then follows from [33, Theorem 2] The proof is completed Remark 2: The polynomial matrix pair satisfying (51) is guaranteed to exist provided is controllable Explicit construction of has been proposed in [2], [22], [34] To proceed, we need the following technical lemma Lemma 3: Let be defined by is a positive scalar is given by (53) Then, there exists an such that is nonempty Proof: Let Note that the controllability of implies that there exists a matrix such that for an arbitrary real number, or equivalently, there exist two real matrices such that (50) is satisfied Since is the complete solution to the (50) according to Theorem 6, by choosing in (50), we conclude that there exists a matrix such that is nonsingular, that is to say, Since is continuous in, we conclude that there exists a sufficiently small such that This completes the proof Let be given by,,, are in the form of (14) with in the form of (15), is replaced by with, in which,,, Let for some Then, by using the relations (48) (49), we have Since, we know that both are bounded over Therefore, is an low gain for if only if is -vanishing Then, according to Theorems 1 2, we can state the following results on low-gain design The proof is straightforward omitted here Theorem 7: Let be parameterized as (49) with given in (53) Then, is an low gain for in the sense of Definitions 2 4 if only if there exists a matrix for some such that for all,, for all,, is defined in (11) Theorem 8: Assume that matrix has a single eigenvalue the maximal algebraic multiplicity of the eigenvalue is Let be parameterized as (49) with given in (53), Then, is an low gain for in the sense of Definitions 2 4 if only if there exists a matrix for some such that, for all (54) is defined in (11) We conclude with two remarks Remark 3: Theorems 7 8 give complete solutions to the low-gain feedback gains that belong to the set The remaining degree of freedom offered by the parameter matrix can be further utilized to achieve some other requirements, for example, fuller utilization of the actuator capacities (see the example in Section V-B) However, further study is required along this line Remark 4: In Theorems 7 8, we have shifted all the poles of the open-loop system to their left by The imaginary parts of the closed-loop system remain unchanged The results in these two theorems can be easily extended to deal with more general situations such as when the change of the imaginary parts of the eigenvalues is desired In other words, the Jordan form of the closed-loop matrix can be chosen arbitrarily such as given in (13) (14) B Illustrative Example Consider a linear system in the form of (1) with given in (2) For brevity, we only consider low-gain design By

11 1040 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 using the method given in [34], a pair of polynomial matrices satisfying (51) can be obtained as Denote According to Theorem 8, an low-gain feedback for the matrix pair exists if only if the condition in (54) is satisfied, namely (55) in which Denote Then, it follows from Theorem 8 that all the low-gain feedbacks for the matrix pair within the set can be characterized by Fig 1 Control signals state evolutions of system (1) (2) under the lowgain feedback laws K (") K ("), respectively Top plot: the 2-norm of the states Middle plot: control u Bottom plot: control u Case 3: If we choose with is given by Again, it is easy to verify that The resulting low gain is given by (4) However, as shown in Example 4, such low gain is not an low gain Case 4: If we choose with in which In what follows, we discuss several special cases Case 1: If we choose with is an arbitrary scalar such that (namely, ), we obtain the following feedback gain: (57) (to guarantee that ), we get the following low gain: (56) which is exactly what is obtained by using the eigenstructure assignment based method in [15] Since, we know that is an low gain Case 2: If we choose with we obtain the feedback gain matrix Indeed, we have given in (3) Note that Thus, the gain in (3) is not an low gain ( thus is not an low gain either) This validates the conclusion arrived at in Example 1 Choosing the free parameter in (57) such that the Frobenius norm of is minimized, we obtain the optimal solution which can be verified to be an low gain for matrix pair as well Finally, we note that under the feedback gain in (56), the output of the second actuator is always zero, which indicates that does not fully utilize the capability of the actuator On the contrary, the low gain uses both actuators capacities, which may lead to better transient performances of the closed-loop system For comparison purposes, by choosing an initial condition, the control signals state evolutions of the closed-loop system are shown in Fig 1 Here the parameter is chosen as the maximal value such that the actuators are not saturated It seems that can indeed lead to a better transient performance

12 ZHOU et al: AND LOW-GAIN FEEDBACK 1041 VI CONCLUSION In this paper, we have introduced the notion of -vanishment ( 2) established a series of characterizations of this new notion Our first characterization is expressed in an algebraic form, which is easy to use in practice, while the second characterization is expressed via a Lyapunov inequality, which is theoretically more appealing With the help of this notion of -vanishment, the mechanism of low-gain feedback developed in the past decade or so for the control of linear systems subject to actuator saturation was fully reexamined In particular, the slow peaking phenomenon that results from the low-gain feedback is fully analyzed Just as the low-gain feedback does for control systems with constraints on their inputs, the low-gain feedback plays an important role in the control of systems with constraints on the energy of their inputs As applications of this new design methodology, the problems of semiglobal stabilization of linear systems with actuator energy constraints was solved Our study of the -vanishment property has also enabled us to establish a systematic approach to the design of low-gain feedback on the basis of a class of linear matrix equations whose solutions can be explicitly expressed It turned out that the eigenstructure assignment approach the parametric Lyapunov matrix equation-based approach developed by the authors can be rendered two special cases of this proposed systematic approach The free parameters in the design can be further utilized to achieve other system performances It is hoped that the notion of -vanishment would also facilitate the solution of some nonlinear control problems such as those considered in [25], which are currently under study APPENDIX A PROOF OF LEMMA 2 Proof of Case: We first show that By definition, condition (10) is equivalent to which, together with the fact that We next show that that, for any, we can write, imply that Note (61) is a scalar to be specified Consider two cases Case 1) There exist some integers such that the limit of as approaches zero is not well defined Case 2) There exist some integers such that Then, in both cases, there clearly exists a scalar such that either the limit of as approaches zero is not well defined or Then, it follows from (61) that either the limit of as approaches zero is not well defined or A contradiction Proof of Case: We first show that Similar to the proof of case, the condition in (10) is equivalent to the existence of,, such that (62) which clearly implies that there exists such that (58) Let be as defined in (59) Then, we deduce from (62) (59) that Define a scalar function as follows: (59) Therefore, by using (58) (59), we have (60) Then, by using (9) the above inequality, we get the inequality Invoking (9) (60), we get

13 1042 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 has a unique positive definite solution Moreover,, Lemma 5: Let be the unique positive definite solution to the parametric ARE 1) The following identity holds: (64) (63) (65) As, it follows from (63) (59) that is as defined in (14) (66) 2) For an arbitrary positive scalar, let (67) We next show that that, for any Note Then, the following identity is valid: (68) is the unique positive definite solution to the Lyapunov matrix equation (69) Moreover, the following relation holds: (70) is any positive scalar By a change of integral variable, the above inequality can be continued as Proof: Proof of Item 1 Let Then, it is clear that is a companion matrix Moreover, it follows from Lemma 4 that Notice that the above inequality is very similar to (61) Then, similarly to the proof for the case, we can show that if (10) is not satisfied, we conclude that either the limit of is not well defined or Both contradict with the assumption The proof is completed Therefore, the Jordan canonical form of is It is well known [10] that the matrix APPENDIX B PROOF OF THE NECESSITY PART OF THEOREM 3 Here, we will also suppress the independent variable for simplicity We first introduce some technical lemmas The first lemma is the parametric Lyapunov equation-based low-gain feedback developed in [35] Lemma 4: Assume that all the eigenvalues of are on the imaginary axis is controllable, Then, the ARE transforms into its Jordan canonical form, namely Rewrite the ARE (64) as follows: (71)

14 ZHOU et al: AND LOW-GAIN FEEDBACK 1043 Then, by inserting (71) into the above equation, we get after simplification Hence, by using (66), we get is given by (72) which is (68) Finally, it follows from (79) the above relation that (73) Note that (72) is already in the form of (65) Therefore, to complete the proof, we need only to show (66) Multiplying both sides of the ARE (64) to the left by to the right by, we obtain (74) (75) With the expressions of, direct calculation shows that, from which from (75) we deduce Therefore, by using the ARE (64) again, we can compute which is (70) The proof is completed Lemma 6: Let be in the form of (14), If the matrix pair is -vanishing, then there exists an matrix such that all the conditions in Theorem 3 are satisfied Proof: Let (80) Then, it follows from Theorem 1 that is -vanishing if only if (18) is satisfied, which, in view of the special structure of, is equivalent to,, which is further equivalent to the existence of vector functions,, such that, Consequently, in view of (80), we can write (81) is defined in (67) (76) Since is also controllable all the eigenvalues of are zeros, it follows from Lemma 4 that the ARE (76) has a unique positive definite solution Therefore, by comparing (74) to (76) noting (75), we have Inserting this relation into (73) leads to (66) Proof of Item 2: Multiplying both sides of (64) to the left by to the right by, we get (82) with Now let be the unique positive definite solution to the following ARE: Then, it follows from Lemma 5 that (83) (84) Direct calculation shows that (77) satisfies the following inequality: (85) Now, let, from which the inequality in (85) can be written as Therefore, by denoting the equation in (77) to, we simplify (78) which is exactly (69) has a unique positive definite solution In fact, the unique positive definite solution to (78) is given by The above inequality is exactly in the form of (26) associated with the positive definite matrix the matrix pair We next show that is bounded as approaches zero We consider two cases Case 1) Then, it follows from (70) that (79) Case 2) Then, the boundedness of clearly implies the boundedness of In both cases, we conclude that is bounded as approaches zero

15 1044 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 56, NO 5, MAY 2011 At last, we show that the associated relation (27) is true In view of (81) (82), we have Again, we consider two cases Case 1) it follows from Item 2 of Lemma 5 (86) that (86) Then, The above inequality is exactly in the form of (26) associated with the positive definite matrix the matrix pair Similarly to the proof of Lemma 6, we can also show that there exists an such that is bounded for all Finally, in view of (88) (89), we have (92) Case 2) In this case, since is bounded, we clearly get from (86) that Similarly to the proof of Lemma 6, we can show that if is bounded for sufficiently small if Therefore, it follows from (92) that In both cases, we can see that the relation (27) associated with is true The proof is completed Lemma 7: Let be in the form of (14) If the matrix pair,, is -vanishing, then there exist an a matrix such that all the conditions in Theorem 3 are satisfied Proof: Denote It follows from Theorem 1 that is -vanishing if only if (17) is satisfied, which, in view of the special structure of, can be written as (87) Similar to (81), relation (87) is equivalent to the existence of such that Now let is as defined in (83) (84) Then, we have (88) (89) (90) In view of (14), we can write Then, we conclude from (90) that Therefore, if we let in (91) becomes (91), then the inequality Hence, the relation (27) associated with is true With the above lemmas, we are ready to prove the necessity part of Theorem 3 Let be the nonsingular bounded matrix such that the relations in (12) (16) are satisfied Then, it follows from Theorem 1 that is -vanishing if only if is In view of the special structure of,we conclude that all the matrix pairs,,,, are -vanishing According to Lemma 7, for each matrix pair,, there exists a scalar a matrix,, such that all the conditions in Theorem 3 are met Similarly, it follows from Lemma 6 that, for each matrix pair,, there also exist a scalar a matrix such that all of the conditions in Theorem 3 are satisfied Consequently, if we let, then is the matrix such that all the conditions in Theorem 3 are satisfied for the matrix pair The proof is completed Remark 5: It follows from the above proof that a possible positive definite matrix, that satisfies all the conditions in Theorem 3 has been explicitly constructed REFERENCES [1] S Battilotti, A unifying framework for the semiglobal stabilization of nonlinear uncertain systems via measurement feedback, IEEE Trans Autom Control, vol 46, no 1, pp 3 16, Jan 2001 [2] T G J Beelen G W Veltkamp, Numerical computation of a coprime factorization of a transfer-function matrix, Syst Control Lett, vol 9, no 4, pp , 1987 [3] T Chen B A Francis, Optimal Sampled-Data Control Systems New York: Springer, 1995 [4] B A Francis K Glover, Bounded peaking in the optimal linear regulator with cheap control, IEEE Trans Autom Control, vol AC-23, no 4, pp , Aug 1978 [5] G Grimm, A R Teel, L Zaccarian, The l anti-windup problem for discrete-time linear systems: Definition solutions, Syst Control Lett, vol 57, no 4, pp , 2008

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Hong Kong, from December 2007 to March 2008, a Visiting Fellow with the School of Computing Mathematics, University of Western Sydney, Sydney, Australia, from May to August 2009 Since January 2010, he has been a reviewer for Mathematical Reviews He has published over 50 journal papers within his research interests, which currently include constrained control systems, time-delay systems, nonlinear control systems Zongli Lin (S 89 M 90 SM 96 F 07) received the BS degree in mathematics computer science from Xiamen University, Xiamen, China, in 1983, the Master of Engineering degree in automatic control from the Chinese Academy of Space Technology, Beijing, China, in 1989, the PhD degree in electrical computer engineering from Washington State University, Pullman, in 1994 He is a Professor of electrical computer engineering with the University of Virginia, Charlottesville His current research interests include nonlinear control, robust control, control applications Prof Lin is an elected Member of the Board of Governors of the IEEE Control Systems Society He was an Associate Editor of IEEE TRANSACTIONS ON AUTOMATIC CONTROL He currently serves on the Editorial Boards of several journals book series, including Automatica, Systems & Control Letters, the IEEE/ASME TRANSACTIONS ON MECHATRONICS, the IEEE Control Systems Magazine Guang-Ren Duan (M 91 SM 95) received the BSc degree in applied mathematics the MSc PhD degrees in control systems theory from the Harbin Institute of Technology, Harbin, China From 1989 to 1991, he was a Post-Doctoral Researcher with the Harbin Institute of Technology, Harbin, China, he became a Professor of control systems theory in 1991 Since August 2000, he has been elected Specially Employed Professor with Harbin Institute of Technology sponsored by the Cheung Kong Scholars Program of the Chinese Government He is currently the Director of the Centre for Control Systems Guidance Technology, Harbin Institute of Technology His main research interests include robust control, eigenstructure assignment, descriptor systems, missile autopilot control, magnetic bearing control Dr Duan is a Chartered Engineer in the UK a Fellow of the Institute of Electrical Engineers (IEE)