AN OVERVIEW OF THE DEVELOPMENT OF LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK

Transcription

1 Jrl Syst Sci & Complexity (2009) 22: AN OVERVIEW OF THE DEVELOPMENT OF LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK Zongli LIN Received: 3 August 2009 c 2009 Springer Science + Business Media, LLC Abstract Low gain feedback refers to certain families of stabilizing state feedback gains that are parameterized in a scalar and go to zero as the scalar decreases to zero. Low gain feedback was initially proposed to achieve semi-global stabilization of linear systems subject to input saturation. It was then combined with high gain feedback in different ways for solving various control problems. The resulting feedback laws are referred to as low-and-high gain feedback. Since the introduction of low gain feedback in the context of semi-global stabilization of linear systems subject to input saturation, there has been effort to develop alternative methods for low gain design, to characterize key features of low gain feedback, and to explore new applications of the low gain and low-and-high gain feedback. This paper reviews the developments in low gain and low-and-high gain feedback designs. Key words Actuator saturation, composite nonlinear feedback, input saturation, low-and-high gain feedback, low gain feedback, nonlinear control. 1 Introduction The concept underlying low gain feedback is that of asymptoticity and, roughly speaking, by low gain feedback we mean certain families of feedback laws in which a parameterized gain matrix, say F(ε), approaches zero as the parameter ε approaches zero. The parameter ε is referred to as the low gain parameter. Low gain feedback was initially proposed in [1] to establish semi-global stabilizability of linear systems subject to actuator saturation (or input saturation). A key feature of the family of low gain feedback laws proposed in [1] is that, if the linear system is asymptotically null controllable by bounded controls (ANCBC), then for any given bounded initial condition 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 subject to actuator saturation is semi-globally stabilizable by linear low gain feedback. We recall that a linear system is said to be ANCBC if it is stabilizable in the usual linear systems sense and all its open-loop poles are located in the closed left-half s-plane [2]. Since the introduction of low gain feedback in [1], much effort has been reported in the literature that aims to develop alternative design methods and to explore new applications. Zongli LIN Department of Electrical and Computer Engineering, University of Virginia, P.O. Box , Charlottesville, VA , USA. zl5y@virginia.edu.

2 698 ZONGLI LIN The feedback gains constructed in [1] (and in [3] for discrete-time systems) are based on an eigenstructure assignment algorithm. Alternative approaches to the low gain feedback design were later developed based on the solution of a parametric H 2 algebraic Riccati equation (ARE) and a parametric H ARE in [4] and [5], respectively. More recently, a new low gain design was proposed that is based on the solution of a parametric Lyapunov equation in [6]. Besides semi-global stabilization of linear systems subject to input saturation, low gain feedback has found applications in solving several other control problems. These problems include semi-global stabilization of linear systems subject to simultaneous input magnitude and rate saturation (e.g., [7 8]), robust semi-global stabilization of linear systems with imperfect actuator input output characteristics (e.g., [9]), finite gain L p stabilization and robust control of linear systems in the presence of actuator saturation and input additive disturbances/uncertainties (e.g., [10 16]), semi-global stabilization of minimum-phase input-output linearizable nonlinear systems (e.g., [17 18]), nonlinear H -control (e.g., [19 20]), H 2 and H suboptimal control (e.g., [21 24]), stabilization of linear time delayed systems (e.g., [25 27]), semi-global stabilization and output regulation of singular linear systems subject to input saturation (e.g., [28 29]), improvement of the transience response (e.g., [30 32]), and stabilization of an inverted pendulum on a carriage with restricted travel (e.g., [33]). In solving the above problems, there have been situations where both low gain and high gain feedback are needed. In such situations, the design procedure is typically sequential. A family of low gain feedback laws, parameterized in ε, is first constructed. Based on the low gain design, a family of high gain feedback laws is then constructed. The high gain feedback laws could be parameterized in a new parameter, say ρ, or in the same parameter ε. The two families of feedback laws are then combined in a certain way to arrive at the desired final feedback laws, called low-and-high gain feedback laws. Also, in achieving certain control objectives, it is sometimes necessary to schedule the low gain parameter ε and the high gain parameter ρ as functions of the state of the system. The resulting feedback laws are consequently referred to as scheduled low gain, scheduled high gain, and scheduled low-and-high gain feedback, respectively. Several earlier results on the developments of the low gain and low-and-high gain feedback have been collected in [34]. Since the publication of [34], there has been new effort to develop alternative methods for low gain design, to characterize key properties of low gain feedback, and to explore new applications of the low gain and low-and-high gain feedback. This paper aims to give a brief up-to-date review of the developments in low gain and low-and-high gain feedback designs. We will first review the low gain feedback results in Section 2 and then proceed to review the results on low-and-high gain design in Section 3. Various design methods are described, with their key features and applications mentioned. A brief conclusion is drawn in Section 4. We will use standard notation. In particular, R := the set of real numbers, N := the set of natural numbers, C := the entire complex plane, C := the open left-half complex plane, C 0 := the imaginary axis in the complex plane, x := the Euclidean norm, or 2-norm, of x R n, x := max x i for x R n, i L n p := the set of all measurable functions x : [0, ) R n such that 0 x(t) p dt <,

3 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 699 for any p [1, ), L n := the set of all measurable functions x: [0, ) Rn such that esssup x(t) <, t [0, ) x Lp := ( 0 x(t) 2 dt ) 1 2, the L p -norm of any x L n p, p [1, ), x L := x(t), the L -norm of any x L n, esssup t [0, ) I := an identity matrix, X := 2-norm of matrix X, X F := the Frobenious norm of matrix X, X T := the transpose of X, λ(x) := the set of eigenvalues of X. 2 Low Gain Feedback Low gain feedback as originally proposed in [1] was designed based on an eigenstructure assignment algorithm. Alternative design methods were later proposed based on the solution of either a parametric H 2 ARE [4] or a parametric H ARE [5]. The ARE based approach has the appealing feature of directly resulting in a Lyapunov function for the closed-loop system along with the feedback gain. The eigenstructure assignment based approach however leads to feedback gains in the form of a matrix polynomial in the low gain parameter, while the ARE approach requires the solution of an ARE for each value of the parameter. Another alternative approach to low gain feedback design was recently proposed based on the solution of a parametric Lyapunov equation. Such an approach possesses the advantages of both the eigenstructure assignment based approach and the ARE based approach. It directly results in a Lyapunov function for the closed-loop system along with feedback gains in the form of a rational matrix in the low gain parameter. More recently, the existing low gain design methods were generalized by introducing the notion of L -vanishment and by providing a full characterization of low gain feedback that achieves such an L -vanishment property [35]. Parallel to the notion of L -vanishment, the notion of L 2 -vanishment was also developed and low again feedback that achieves the L 2 - vanishment characterized in [36]. In this section, we will review these low gain design methods and their applications. Throughout the session, we will consider the linear system ẋ = Ax + Bu, x R n, u R m, (1) where x is the state and u is the input. We make the following assumption. Assumption 1 The pair (A, B) is asymptotically null controllable with bounded controls (ANCBC), i.e., 1. (A, B) is stabilizable; 2. All eigenvalues of A are in the closed left-half s-plane. 2.1 Eigenstructure Assignment Based Method The eigenstructure assignment based low gain state feedback design is carried out in three steps [34]. This algorithm simplifies the algorithm originally proposed in [1].

4 700 ZONGLI LIN Eigenstructure assignment based low gain design algorithm Step 1 Find nonsingular transformation matrices T S and T I such that the pair (A, B) is transformed into the following block diagonal control canonical form: A B 1 B 12 B 1l 0 A B 2 B 2l T 1 S AT S = A l A 0, T 1 S BT I = B l B 01 B 02 B 0l where A 0 contains all the open left-half plane eigenvalues of A, for each i = 1 to l, all eigenvalues of A i are on the jω axis and hence (A i, B i ) is controllable as given by A i = , B i =. 0, a i,ni a i,ni 1 a i,1 1 and finally, s represent submatrices of less interest. The existence of the above canonical form was established in [37]. Its software realization can be found in [38]. Step 2 For each (A i, B i ), let F i (ε) R 1 ni be the state feedback gain such that λ(a i + B i F i (ε)) = ε + λ(a i ) C, ε (0, 1]. Note that F i (ε) is unique. Step 3 Construct a family of low gain state feedback laws for system (1) as, u = F(ε)x, (2) where the low gain matrix F(ε) is given by F 1 (ε) F 2 (ε) 0 0 F(ε) = T I F l (ε) T 1 S. It is clear from the construction of F(ε) that F(ε) is a polynomial matrix in ε and lim F(ε) = ε 0 0. For this reason the family of state feedback laws (2) are referred to as low gain feedback, and ε the low gain parameter. We however also note that, as the following theorem shows, the low gain feedback laws (2) possess more intricate properties than simply having small feedback gains. Theorem 1 Consider a single input pair (A, B) in the control canonical form: A= a n a n 1 a 1, B =. 0 1.

5 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 701 Assume that all the eigenvalues are in the closed left-half s-plane. Let F(ε) R 1 n be the state feedback gain such that λ(a + BF(ε)) = ε + λ(a). Then, there exists an ε (0, 1] such that, for all ε (0, ε ], F(ε) αε, (3) e (A+BF(ε))t β ε r 1e εt/2, t 0, (4) F(ε)(A+BF(ε)) l e (A+BF(ε))t γl εe εt/2, t 0, l N, (5) where r is the largest algebraic multiplicity of the eigenvalues of A, and α, β and γ l s are some positive scalars independent of ε. Property (3) indicates the asymptotic nature of the feedback laws (2), i.e., lim F(ε) = 0. ε 0 In view of the fact that e (A+BF(ε)t is the transition matrix of the closed-loop system under the low gain feedback, property (4) reveals that the closed-loop system under low gain feedback will peak slowly to a magnitude of order O(1/ε r 1 ), with r being the largest algebraic multiplicity of the eigenvalues of A. Property (5) on the other hand implies that, for any given bounded set of initial conditions, the control input and all its derivatives can be made arbitrarily small by decreasing the value of the low gain parameter ε. This can also be seen in the simulation of the following example. Example 1 Consider the system (1) with A = , B = It is straightforward to verify that (A, B) is controllable with four controllable modes at { j, j, j, j}, where j = 1. Following the eigenstructure assignment based low gain feedback design algorithm, we construct the following family of linear state feedback laws: u = [ ε 4 +2ε 2 4ε 3 +4ε 6ε 2 4ε ] x, ε (0, 1]. (6) Some simulation of the resulting closed-loop system is given in Figures. 1 and 2. It can be easily seen from these figures that, the state peaks slowly and, for the same initial conditions, as the value of ε decreases, the peak value of the state trajectory increases while that of the control input decreases. It is property (5) that enables semi-global stabilization of linear systems subject to input saturation [1] or simultaneous input magnitude and rate saturation [7]. Property (5) also enables the stabilization of linear systems subject to time delay in the input. In particular, it was recently shown in [25], by using the eigenstructure assignment based low gain feedback design, that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open loop system is not exponentially unstable (i.e., all the open loop poles are on the closed left-half s-plane). It was further shown that such systems, when subject to actuator saturation, are semi-globally asymptotically stabilizable by linear state or output feedback. Other applications of the eigenstructure assignment based low gain feedback design include semi-global stabilization of robust semi-global stabilization of minimum-phase input-output

6 702 ZONGLI LIN linearizable systems via partial state and output feedback [18], almost disturbance decoupling with internal stability for nonlinear systems [19 20], global stabilization of nonlinear systems [39], perfect regulation [23 24], H control problems [21 22], and semi-global stabilization of singular linear systems subject to actuator saturation [28 29]. States x(t) Time Control u(t) Time Figure 1 Evolutions of the states and control input: x(0) = (1, 2, 2, 1) and ε = 0.1 States x(t) Time Control u(t) Time Figure 2 Evolutions of the states and control input: x(0) = (1, 2, 2,1) and ε = 0.01

7 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK Algebraic Riccati Equation Based Method Two different ARE based low gain design algorithms have been proposed, one involving the H 2 ARE [4] and the other H ARE [5]. We recall the H 2 ARE based low gain design algorithm from [34] in the following two step algorithm. H 2 ARE based low gain design algorithm Step 1 Solve the following algebraic Riccati equation for the unique positive definite solution P(ε), A T P + PA PBB T P + Q(ε) = 0, ε (0, 1], (7) where Q(ε) : (0, 1] R n n is any matrix that is positive definite for all ε (0, 1] and satisfies lim Q(ε) = 0. ε 0 The existence of such a solution is established in [40]. Step 2 Construct a family of low gain state feedback laws as u = F(ε)x, where F(ε) = B T P(ε). The following theorem establishes some basic properties of this family of ARE based low gain feedback laws. Theorem 2 Let Assumption hold. Then, for each ε (0, 1], there exists a unique matrix P(ε) > 0 that solves the ARE (7). Moreover, such a P(ε) satisfies, 1) lim P(ε) = 0; ε 0 2) There exists a constant α > 0, independent of ε, such that P 1 2 (ε)a l P 1 2 (ε) α l, ε (0, 1], l N. (8) Like Theorem 1, Theorem 2 reveals that, for any given bounded set of initial conditions, the control input and all its derivatives can be made arbitrarily small by decreasing the value of the low gain parameter ε. As such, semi-global stabilization of the linear system (1) subject to input saturation can also be achieved by this family of ARE based low gain feedback laws. As seen in [5] and [34], the ARE based low gain feedback design has the advantage of directly resulting in a quadratic Lyapunov function V (x) = x T P(ε)x, which is used to establish semiglobal stabilizability of the system. In addition, with the H ARE low gain design algorithm, additional robustness with respect to input nonlinearities can be achieved [5]. The quadratic Lyapunov functions resulting from the ARE based low gain designs enable the scheduling of the low gain parameter ε as a function of the state x to enhance the semi-global results to global results [11,41]. 2.3 Lyapunov Equation Based Method The Lyapunov equation based low gain design for the system (1) was recently developed in [6]. Without loss of generality (see [6], assume that all eigenvalues of A are on the jω axis. Then, the Lyapunov equation based low gain design is carried out in the following 3 step algorithm. Lyapunov equation based low gain design algorithm Step 1 Solve the following Lyapunov equation for the unique positive definite solution W(ε), W (A + ε ) T 2 I + (A + ε ) 2 I W = BB T, ε (0, 1]. (9)

8 704 ZONGLI LIN The existence of such a solution is guaranteed by its analytic expression, W(ε) = Step 2 Compute the matrix P(ε) as, 0 e (A+ ε 2 I)t BB T e (A+ ε 2 I)Tt dt. P(ε) = W 1 (ε). Step 3 Construct a family of low gain state feedback laws as u = F(ε)x, (10) where F(ε) = B T P(ε). Some key properties are summarized in the following theorem. Further properties can be found in [27]. Theorem 3 1) The matrix A BB T P(ε) is asymptotically stable and satisfies A BB T P(ε) = P 1 (ε) ( A T εi ) P(ε), that is, the eigenvalues of the matrix A BB T P(ε) and those of A are symmetric with respect to the line s = ε 2 on the s-plane. 2) The system (1) under the control law (10) converges to the origin no slower than e ε 2 t. 3) lim P(ε) = 0. ε 0 + 4) The matrix P(ε) is differentiable and monotonically increasing with respect to ε, i.e., dp (ε) dε > 0, ε (0, 1]. 5) If m = 1, then P(ε) is a polynomial matrix, and, if m > 1, then P(ε) is a rational matrix but is generally not a polynomial matrix. These properties indicate that the Lyapunov equation based low gain design method possesses several nice features. The resulting feedback gain is parameterized explicitly in terms of the low gain parameter ε (see the example below). It directly results in a quadratic Lyapunov function V (x) = x T P(ε)x. The convergence rate of the resulting closed-loop system is also explicitly specified. Example 2 Consider the linear system in Example 1. To construct the Lyapunov equation based low gain feedback laws, we solve the Lyapunov equation (9) and obtain W(ε) 1 = ε 3 (ε ε ε ) 4(5ε 2 +4) 2(5ε 2 +4) 4(ε 4 ε 2 4) (ε 4 10ε 2 24) 2(5ε 2 +4) 2(3ε 4 +6ε 2 +8) (3ε 4 +6ε 2 + 8) ε 6 2ε 4 12ε (ε 4 ε 2 4) (3ε 4 +6ε 2 +8) 2(ε 6 +4ε 4 +10ε 2 +8) (ε 6 +4ε 4 +10ε 2 +8). (ε 4 10ε 2 24) ε 6 2ε 4 12ε 2 16 (ε 6 +4ε 4 +10ε 2 +8) ε 8 +8ε 6 +30ε ε Then we have P(ε) = W 1 (ε) ( ε 6 + 4ε 4 + 6ε ) ε ( 3ε 4 + 8ε ) ε ( 2 3ε 4 + 6ε ) ε ( ε ) ( ε 2 = 3ε ε ) ε 2 2 ( 5ε 4 + 8ε ) ε ( 11ε ) ε 2 4 ( ε ) ( ε 3ε ε ) ( ε 11ε ) ε 2 2 ( 7ε ) ( ε 6ε 2 ε ) ε 2 4 ( ε ). ε 6ε 2 4ε

9 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 705 Consequently, the low gain feedback laws are given by u = B T P(ε)x = [ ε 4 + 2ε 2 4ε 3 + 4ε 6ε 2 4ε ] x, ε (0, 1]. 2.4 Characterization of Low Gain Feedback Laws In Example 1, the open-loop system has all its poles on the jω axis. The low gain feedback law (6) shifts these poles to their left by ε. Indeed, Theorem 1 indicates that, for a single input system with all its open loop-poles on the jω axis, the unique feedback gain that shifts the open loop poles to their left by ε possesses the key properties that enables the successful applications of low gain feedback. For a multiple input system, the feedback law that shifts the open-loop poles to their left by ε is non-unique. A natural question then is whether or not every stabilizing feedback gain that is parameterized in a scalar ε and goes to zero as ε decreases to zero achieves arbitrarily small peak value in the control signals for a given bounded set of initial conditions. The answer to this question is negative as the following example indicates [35]. Example 3 Consider the linear system (1) with A = , B = The following feedback gain F (ε) = [ ] ε 3 + 3ε 2 3ε 2 + ε 3ε, ε > 0 ε 0 0 is a stabilizing feedback gain such that λ(a + BF (ε)) = { ε, ε, ε}. Note that lim F (ε) = 0 ε 0 + holds true. However, the peak value of the control signal u (t) = F (ε)e (A+BF(ε))t x 0 can be arbitrarily large when ε approaches zero. To see this, we note that { F } (ε)e (A+BF(ε))t 2 F (ε)e (A+BF(ε)) 1 ε F sup t 0 = 1 4 ε ε ε ε ε 2, which implies that there exists a bounded initial condition x 0 such that the peak value of u (t) approaches infinity as ε approaches zero. Motivated by this example, a complete characterization of conditions under which a stabilizing feedback law that shifts the open-loop poles to their left by ε would possess the key property of low gain feedback as described in Theorem 1 was recently made in [35]. To do so, we introduce the notion of L 2 -vanishment. Definition 1 Given S (ε) : (0, 1] R m n and A(ε) : [0, 1] R n n. Let A(ε) be such that λ(a(ε)) C, ε (0, 1]. Then, (S (ε),a(ε)) is said to be L -vanishing if S lim (ε)e A(ε)t = 0. ε 0 + L 2 F

10 706 ZONGLI LIN Let F = { f(ε) : (0, 1] R } f(ε) > 0 and lim f(ε) exists. ε 0 + We then have the following characterization of the L -vanishing low gain feedback. Theorem 4 Assume that all the eigenvalues of A(ε) R n n are f (ε) with f (ε) F. Moreover, let r be the maximal algebraic multiplicity of such eigenvalue. Then, (S (ε), A(ε)) is L -vanishing if and only if where 1 f i (ε) S (ε)(a(ε)+f (ε)i n) i O m n (ε), i = 0, 1,, r 1, (11) O m n (ε) = { } L(ε): (0, 1] R m n : lim L(ε) = 0. ε 0 + We now consider Example 3. In that example, r = 3, f(ε) = ε, S(ε) = F(ε), and A(ε) = ε 0 1. ε 3 3ε 2 3ε 2 ε 3ε It is now straightforward to verify that (11) does not hold true for i = 1. The above characterization is restricted to the situation where all eigenvalues of A(ε) are at a same location f(ε). More details on L -vanishing low gain feedback and its more general characterization can be found in [35]. 2.5 L 2 Low Gain Feedback In the low gain feedback discussed so far, the emphasis has been on making the peak value of the control input small, in the sense of L -vanishment discussed in Subsection 2.4. The next example shows that an L -vanishing low gain feedback does not guarantees that the L 2 -norm of the control input to be small. Example 4 Consider the system in Example 3. For this system, the following L -vanishing low gain feedback law was constructed in [35], F(ε) = ε ε 3 ε ε 9 4 ε ε ε 2 ε ε 5 4 ε ε ε ε ε 1 4 ε This feedback law places the eigenvalues of A + BF(ε) at { ε, ε, ε}. By the L -vanishing property, F lim (ε)e (A+BF(ε))t = 0. ε 0 + L Therefore, the peak value in the control input u ε (t) = F(ε)e (A+BF(ε))t x 0, x 0 < can be reduced to an arbitrarily low level by decreasing the value of ε. Now, let U ε (t) = F(ε)e (A+BF(ε))t.

11 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 707 Then, it is easy to compute that (see, e.g., [42]), where It follows that 0 F(ε)e (A BF(ε))t 2 dt = g( 4 ε) F 16 ε( 4 ε + 1) 2, g(s) = 4s 24 6s s s 16 20s s s s s lim ε 0 + U ε(t) L2 =, which implies that there exists a bounded initial condition x 0 such that lim ε 0 + u ε (t) L2 = lim ε 0 + U ε (t)x 0 L2 =. This indicates that, if we decrease ε so as to avoid input saturation, the control energy that is used to steer the initial state to the origin approaches infinity. Motivated by this example, the notion of L 2 -vanishment was recently introduced in [36] and it characterization made. Definition 2 Given S (ε) : (0, 1] R m n and A(ε) : [0, 1] R n n. Let A(ε) be such that λ(a(ε)) C. Then (S (ε),a(ε)) is said to be L 2 -vanishing if ( ) 1 S(ε)e A(ε)t lim = lim S(ε)e A(ε)t 2 2 dt = 0. ε 0 + L2 ε 0 + Theorem 5 Assume that all the eigenvalues of A(ε) R n n are f(ε) with f(ε) F. Let r be the maximal algebraic multiplicity of f(ε) as the eigenvalue of A(ε). Then (S(ε), A(ε)) is L 2 -vanishing if and only if 1 f 1 2 +i (ε) S(ε)(A(ε) + f(ε)i)i O m n (ε), i = 0, 1,, r 1, where O m n (ε) is as defined in (11). 3 Low-and High-Gain Feedback Several applications of low gain feedback that were mentioned above actually involve high gain feedback. In other words, it is combinations of low gain and high gain feedback that solve these control problems. There are different ways of constructing these low-and-high gain feedback laws. In this section, we review some of these low-and-high gain design methods. 3.1 Additive Low-and-High Gain Design Low-and-high gain design was initiated in [12] for a chain of integrator system, based on the eigenstructure assignment based low gain feedback. The design for general systems was developed in [16] based on the ARE based low gain feedback design. The development of the eigenstructure assignment based low-and-high gain design was later completed in [13] for general systems. The low-and-gain feedback design algorithm again applies to system (1), where (A, B) is asymptotically null controllable with bounded controls. 0

12 708 ZONGLI LIN Low-and-high gain design algorithm: Step 1 Low gain design Construct a low gain feedback law Let P(ε) > 0 be such that u L = F L (ε)x, ε (0, 1]. (A + BF L (ε)) T P(ε) + P(ε)(A + BF L (ε)) Q(ε) (12) for some Q(ε) > 0. If an ARE based low gain design is adopted, then such a P(ε) is the solution of the ARE. If the eigenstructure assignment based low gain feedback design is adopted, a procedure for constructing such a P(ε) can be found in [34, Subsection 4.5]. Step 2 High gain design Form the high gain state feedback control law as, u H = F H (ε, ρ)x, ε (0, 1], ρ 0, where F H (ε, ρ) = ρb T P(ε), ρ 0. The nonnegative parameter ρ is referred to as the high gain parameter. Step 3 Low-and-high gain design The family of low-and-high gain feedback laws is simply formed by adding together the low gain feedback and the high gain feedback as designed in the previous two steps. Namely, where u = u L + u H = F LH (ε, ρ)x, F LH (ε, ρ) = F L (ε) + F H (ε, ρ) = ( (1 + ρ)b T P(ε), for ARE based design). We refer to the above low-and-high gain design as the additive low-and-high gain design due to the fact that the final feedback law is the addition of low gain feedback and high gain feedback, in contrast with the low-and-gain feedback laws where the low gain component is only embedded in the high gain design (see Subsection 3.3). For the low gain feedback F L (ε) and the resulting matrix P(ε) in Step 1 of the above algorithm, let us define an ellipsoid Ω(P(ε)) = { x R n : x T P(ε)x 1 } and the region where control input does not saturate L(F L (ε)) = {x R n : F L (ε)x 1}. Here we have assumed, without loss of generality, that the level of saturation is unity. Clearly, if ε can be chosen such that Ω(P(ε)) L(F L (ε)), (13) then the closed-loop system remains linear within Ω(P(ε)), and by (12), Ω(P(ε)) is a contractively invariant set of the closed-loop system under input saturation. That is, the closed-loop system is asymptotically stable at the origin with Ω(P(ε)) enclosed in the domain of attraction. It can be shown (see, e.g., [5 6,34]) that, for all the low gain feedback designs, Ω(P(ε)) that

13 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 709 satisfies (13) can be made large enough to enclose any given bounded set as a subset by choosing the value of ε sufficiently small. The key feature of the low-and-high gain feedback is that, with the addition of the high gain component, the ellipsoid Ω(P(ε)) remains contractively invariant. Thus, the tuning of the low gain and high gain parameters can be carried out sequentially. The low gain parameter is first tuned small enough to obtain a large enough stability region. Once the low gain parameter is fixed, the high gain parameter ρ is then tuned high enough to obtain closed-loop performances beyond the large stability region. We note that the low-high-gain algorithm recalled above also apply to systems have poles in the open right half plane. In this situation, any stabilizing feedback gain can be design in Step 1, instead of low gain feedback in Step 1. This stabilizing feedback gain will result in a contractively invariant ellipsoid, which, unlike with a low gain feedback, is fixed and cannot be made arbitrarily large. Such a fixed ellipsoid however will still remain contractively invariant when a corresponding high gain feedback is added to the original stabilizing feedback law that makes it contractively invariant. Indeed, as shown in [16], the low-and-high gain feedback laws achieve robust semi-global practical stabilization of the system (1) in the presence of actuator saturation, input-additive uncertainties and input additive bounded disturbances. It is further shown in [9] that robust semi-global practical stabilization can still be achieved by the low-and-high gain feedback even the system is also to subject to deadzone nonlinearity (see the example below). Example 5 Consider the following system in the presence of actuator satuation/deadzone nonlinearities and input additive uncertainties and disturbances, ẋ = x σ(u + g(x) + d(t)), where the actuator nonlinearity σ : R R is as shown in Figure 3. σ(v) v Figure 3 Actuator nonlinearity σ We design the following low-and-high gain feedback law, u = (ε ε 2 ρ)x 1 (3ε ερ)x 2 (3ε ρ)x 3, which as shown in [9] achieves robust semi-global practical stabilization of the system. Some simulation of the resulting closed-loop system is shown in Figure 4. These robust semi-global practical stabilization results can be made global by scheduling both the low gain parameter ε and the high gain parameter ρ as functions of the state x [11].

14 710 ZONGLI LIN State Actuator Output Figure 4 g(x) = x 1x x 3, d(t) = 2sin t, ε = 0.1, ρ = The low-and-high gain feedback law can also make the restricted L 2 -gain from the input additive disturbance to the state arbitrarily small when the magnitude of the disturbances are bounded by an (arbitrarily large) given value [15]. The boundedness requirement on the disturbance can be removed by allowing the high gain parameter to be dependent on the state x [10]. In an another application of the low-and-high gain feedback, the high gain parameter is scheduled in such a way that it has the effect of increasing the damping ratio of the closed-loop system as its output approaches the set value it is to reach, thus achieving a fast transience without a large overshoot. Such a special class of low-and-high gain was referred to as the composite nonlinear feedback in [30]. We will review the composite nonlinear feedback in Subsection 3.2 below. 3.2 Composite Nonlinear Feedback Design Composite nonlinear feedback (CNF) control was conceived in [30] for the tracking control of linear systems subject to actuator saturation. This nonlinear feedback design method takes advantages of the quick responses of systems with small damping ratios and small overshoots of systems with large damping ratios. The idea is to start with a nominal linear feedback law that will result in a small damping ratio for the system, thus insuring a fast response. A design parameter is then tuned as a nonlinear function of the state of the system so that the damping ratio increases as the response approaches the reference input, thus avoiding the overshoot. Since its conception in [30] in the context of second order systems, which facilitate the physical interpretation of damping ratios, the CNF design methods have been extended to higher dimensional and multiple input systems by several authors (see, e.g., [31 32]). Successful implementations of the CNF control have also been reported (see, e.g., [43 44]).

15 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 711 Consider a linear system with actuator saturation: {ẋ = Ax + Bsat(u), x(0) = x0, y = Cx, (14) where x R n is the state, u R m is the control input, A, B and C are matrices of appropriate dimensions with B being of full column rank and C of full row rank, and sat : R m R m is the standard saturation function defined as sat(u) = [ sat(u 1 ) sat(u 2 ) sat(u m ) ] T, with sat(u i ) = signmin {1, u i }. Here, we have slightly abused the notation by using sat to denote both a scalar valued and vector valued saturation functions. Also assumed is that (A, B) is controllable. The objective is to design a feedback law that causes the system output to track a constant reference r quickly but without experiencing large overshoot. The first step in the CNF design is to construct a nominal (low gain) linear feedback law of the form u L = Fx + Gr, where F and G are chosen such that A + BF is Hurwitz, C(sI A BF) 1 B has a small damping ratio, and C(A + BF) 1 BG = I. The small damping ratio ensures a fast response of the system, assuming actuator saturation does not occur. The last condition is to guarantee tracking of a constant input with a zero steady state error, again in the absence of actuator saturation. Let P > 0 be the solution to the Lyapunov equation (A + BF) T P + P(A + BF) = Q, where Q is some positive definite matrix. Then, for the nominal control u L not to saturate the actuators, the state of the system and the reference input must satisfy (x, r) {(x, r) : Fx + Gr 1}. (15) Under this feedback law, the dynamics of the tracking error x = x + (A + BF) 1 BGr can be written as x = (A + BF) x and condition (15) is implied by x { x R n : x T Px c }, r, (16) for some positive scalars c and. We then construct a nonlinear (scheduled high gain) feedback of the form u N = ρ(x)b T P x, where ρ( x) = diag {ρ 1 ( x), ρ 2 ( x),, ρ m ( x)}, with each ρ i ( x) being a nonnegative function, locally Lipschitz in x. A typical form of ρ i ( x) was suggested in [30] as ρ i ( x) = k i e δi x, for some positive constants k i and δ i. The CNF law is then constructed by adding the two components u L and u N, u = u L + u N = F x ρ( x)b T P x ( F(A + BF) 1 B I ) Gr.

16 712 ZONGLI LIN As shown in [30], the nonlinear feedback component increases the damping ratio as the output approaches the reference input and thus helps to avoid overshoot. In the mean time, with the addition of the nonlinear component, for any nonnegative function ρ( x), tracking will continue to occur for all initial conditions and the reference input r that satisfy condition (16). The following example, taken from [30], illustrates the effectiveness of the CNF design. Example 6 Consider an F-16 aircraft derivative. At the flight condition corresponding to an altitude of 20,000 feet and a Mach number of 0.9, the short period dynamics of this aircraft model are given by (14) (see [45]) with [ A = C = [ ], ] [ 0, B =, 1] where the output is the pitch rate and the control input is the elevator deflection scaled such that the maximum actuator capacity is unity. The pitch rate and the commanded pitch rate r are scaled accordingly. Following the CNF design algorithm, we choose and F = [ ] G = , resulting in a closed-loop system damping ratio of 0.7. Letting Q = I, we yield the following CNF law, u = ( ρ(x, r))x ρ(x, r)x 2 ( ρ(x, r)) r, (17) with ρ(x, r) = 5e y r. Some simulation results of the resulting flight control system are shown in Figure 5. We see from the simulation results that the overshoot in the response due to the nominal linear feedback is avoided and hence better tracking performance is obtained. Given the ability of the proposed nonlinear feedback to suppress overshoot of the response due to the nominal linear feedback, one could intentionally design a nominal linear feedback that results in smaller damping ratio to increase the response speed and use the nonlinear feedback to suppress the resulting larger overshoot. To this end, we redesign the linear feedback law with higher gain of F = [ = 0] and G = This linear feedback law results in a damping ratio of 0.4. Letting Q = I and again following the CNF design algorithm, we obtain the following CNF law u = ( ρ(x, r))x x 2 ( ρ(x, r)) r (18) with ρ(x, r) = 30e 10 y r. Some simulation results are shown in Figure 6. Comparing Figures 5 and 6, we see that the closed-loop system performance is further improved.

17 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 713 Pitch Rate y Time Figure 5 Tracking performance under the CNF (17). The dotted line is the command input, the solid line represents the response due to the nonlinear feedback, and the dashed line the nominal linear feedback (ρ(x,r) = 0) 3.3 Embedded Low-and-High Gain Design In the additive low-and-high gain feedback, both low gain feedback and high gain feedback are designed for the entire system. Even though the design is sequential in that the high gain feedback is designed based on the closed-loop system resulting from the low gain feedback, the low gain feedback is added back to the high gain feedback that results from it. In the embedded low-and-high gain feedback design, the low gain is designed for a part of the system and the high gain feedback is designed for the whole system based on the low gain feedback. The resulting high gain feedback is the final low-and-high gain feedback law. The embedded low-and-high gain design has been used to solve a number of control problems, including semiglobal stabilization of nonlinear systems (e.g., [17 18,46]), and H 2 and H control of linear and nonlinear systems (e.g., [10,21,23]). In what follows, we illustrate the embedded low-and-high gain design procedure with its application in the semi-global stabilization of a class systems in a special normal form [46]. Consider a nonlinear system in a special normal form: η = f(η, ξ j ), j {1, 2,, r + 1}, ξ 2 = ξ 3,. ξ r = u, where ξ r+1 = u. We assume that the zero dynamic is globally asymptotically stable at η = 0. η = f(η, 0) (19)

18 714 ZONGLI LIN Pitch Rate y Time Figure 6 Tracking performance under the CNF (18). The dotted line is the command input, the solid line represents the response due to the nonlinear feedback, and the dashed the line nominal linear feedback (ρ(x,r) = 0) The system (19) is known to be not semi-globally stabilizable by linear high gain feedback of the states ξ i s if j > 1 (see, e.g., [47 48]). Its semi-global asymptotic stabilizbility was first established in [49] by using a feedback law of nested saturation type. The low-and-high gain feedback laws for this system are constructed as follows. First, design the low gain feedback laws that stabilize the states ξ 1, ξ 2,, ξ j 1, u L = ε j 1 c j 1 ξ 1 ε j 2 c j 2 ξ 2 εc 1 ξ j 1, where the scalars c 1, c 2,, c j 1 are such that the polynomial s j 1 + c 1 s j c j 2 s + c j 1 is Hurwitz. Under these low gain feedback laws, the state equations for ξ i s can be rewritten as follows: ξ k = ξ k+1, k = 1, 2,, j 2, ξ j 1 = ε j 1 c j 1 ξ 1 ε j 2 c j 2 ξ 2 εc 1 ξ j 1 + ξ j, ξ k = ξ k+1, k = j, j + 1,, r 1, ξ r = ε j 1 c j 1 ξ r j+2 + ε j 2 c j 2 ξ r j εc 1 ξ r + u, where j 1 ξ k = ξ k + ε j l c j l ξ k j+l, k = j, j + 1,, r. l=1

19 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 715 We can now design a family low-and-high gain feedback laws, also parameterized in ε, to stabilize the dynamic of the states ξ k, k = j, j + 1,, r, as follows: u LH = ε j 1 c j 1 ξ r j+2 ε j 2 c j 2 ξ r j+3 εc 1 ξ r d r j+1 ε r j+1 x j d r j ε r j ξ j+1 d 1 ε ξ r, (20) where d 1, d 2,, d r j+1 are chosen such that the polynomial s r j+1 + d 1 s r j + + d r j s + d r j+1 is Hurwitz. The family of low-and-high gain feedback laws (20) was shown in [46] to achieve semi-global asymptotic stabilization of the nonlinear system (19). To illustrate the above stabilization scheme, we consider the following three systems. Example 7 System 1 (Example 8.2 of [47]): η = (1 ηξ 2 )η, ξ 1 = ξ 2, ξ 2 = u. System 2 (Example 1.1 of [48]): η = 0.5(1 + ξ 2 )η 3, ξ 1 = ξ 2, ξ 2 = u. System 3 (Example 4.1 of [49]): η = η + η 2 u, ξ 1 = ξ 2, ξ 2 = u. Obviously all the three systems are in the special normal form we consider in this paper. The first two systems were shown in [47] and [48] respectively to be not semi-globally stabilizable by linear high gain feedback. They essentially the same structure and are however semi-globally stabilizable by the same family of low-and-high gain feedback control laws, u = ξ 1 ε2 + 1 ξ 2, ε where the the poles of the fast and slow subsystems of the closed-loop linear system have been chosen to be 1 ε and ε respectively. For the third example, j 1 = r Thus, no high-gain feedback is involved in the feedback laws and hence no fast subsystem is present in the closed-loop linear system. Choosing the poles of the linear slow subsystem to be ε and 2ε, we obtain the following family of low gain feedback control laws which semi-globally stabilizes the system, u = 2ε 2 ξ 1 3εξ 2.

20 716 ZONGLI LIN Lθ s s mg θ L u M Figure 7 Inverted pendulum on a cart 3.4 Stabilization of an Inverted Pendulum on a Cart with Restricted Travel In this section, we illustrated how the idea of low-and-high gain feedback can be applied to systems that have open right-half plane poles or zeros. We use a system that is comprised of an inverted pendulum mounted on a cart with restricted travel (Figure 1). We denote the displacement of the carriage at time t by s(t), while the angular rotation of the pendulum at time t is denoted by θ(t). The pendulum consists of a weightless rod of length L with a mass m attached to its tip. The moment of inertia with respect to the center of gravity (the tip) is J. The cart has a mass M. The friction coefficient between the cart and the floor is F. The horizontal force exerted on the cart at time t is u(t). We also assume that m is small with respect to M and J is small with respect to ml 2. Under these assumptions, a nonlinear model for this inverted pendulum on a cart system was derived in [50] as, M s = u Fṡ, φ = g L sin φ L s 1 cosφ. (21) Selecting the state variables x 1 = s, x 2 = ṡ, x 3 = s + Lθ and x 4 = ṡ + L θ, a state space representation of (21) is given as ẋ 1 = x 2, ẋ 2 = F M x M u, ẋ 3 = x 4, ẋ 4 = g sin ( x3 x ) 1 F ( L M 1 cos x 3 x 1 L ) x ( M 1 cos x 3 x 1 L ) u. (22)

21 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 717 Linearizing (22) at the origin of the state space yields ẋ 1 = x 2, ẋ 2 = F M x M u, ẋ 3 = x 4, ẋ 4 = g L x 3 g L x 1. (23) Assuming that the pendulum and the cart are not in motion before the driving force is exerted, the initial conditions for the system (23) are then given by x 1 (0) = s(0), x 2 (0) = 0, x 3 (0) = s(0) + Lθ(0) and x 4 (0) = 0. The design objective is to stabilize the system by means of linear state feedback under the constraints that the cart remains within a certain maximum allowable distance from the origin (s = 0). More specifically, our design objectives can be precisely stated as follows: For any a priori given (arbitrarily small) numbers η 1 and η 2, find a linear state feedback law that stabilizes the system subject to the restriction that { 0 x1 (t) (1 + η 1 )(s(0) + Lθ(0)) + η 2, if θ(0) 0, s(0) + Lθ(0) 0, (24) 0 x 1 (t) (1 + η 1 )(s(0) + Lθ(0)) η 2, if θ(0) 0, s(0) + Lθ(0) 0. We note that s(0) + Lθ(0) is a linearized approximation of the projection of the pendulum tip on the floor at time t = 0 and x 1 (t) is the displacement of the cart at time t. Hence, equation (24) in the design problem sets the maximum allowable travel for the cart, which becomes the initial projection of the pendulum tip on the floor as both η 1 and η 2 approach zero. The same control objective (24) was achieved in [51] by using a nonlinear feedback law. This control objective was achieved in [33] by a linear feedback law that was motivated by the low-and-high gain feedback design approach. We summarize such a low-and-high gain design as follows. Step 1 Taking y = x 1 as the output, the system has two invariant zeros at { g L, } g L and has the following zero dynamics, ẋ 3 = x 4, By choosing and renaming the output as ẋ 4 = g L x 3. ( g g ) ( u 0 = L + ε g ) x 3 L L + ε x 4 ỹ = x 1 = x 1 + L g u 0, we place the poles of the zero dynamics at { ε, g L}, where ε is a positive scalar satisfying { } 1 ε min 2, η 1 η 2, L πl ( 2 g L + 1). g

22 718 ZONGLI LIN Step 2 With x 1 as the new output, we rewrite the system (23) as x 1 = 1 ε x 1 + x 2, x 2 = 1 ε x 2 + L εg u 0 + L g ü0 F M x M u, ẋ 3 = x 4, ( ) g g ẋ 4 = ε L x 3 L + ε x 4 L x g 1, (25) where x 2 = x ε x 1 + L εg u 0 + L g u 0, and where ( u 0 = g )( g ) L + ε L x 3 + x 4, ( g )( g u 0 = L + ε L x 4 + g L x 3 g ) L x 1, ( g )( g ( g ü 0 = L + ε L L x 3 g 1) L x + g L x 4 g ) L x 2. Step 3 Choose the linear state feedback law as u = M µ x 2, (26) where µ is a large positive value. The following result from [33] establishes that the linear feedback law as given by (26) indeed achieves our design objective. Theorem 4 Consider the closed-loop system consisting of the system (23) and the linear state feedback law (26). Then, there exists a µ > 0 such that for each µ (0, µ ], the closed-loop system is stable with (24) satisfied. To demonstrate the design algorithm, we take the numerical values for the system parameters as follows, F M = 1 1s 1, M = g 1kg 1, L = 16s 1, L = 0.613m. Let η 1 = 0.1 and η 2 = 0.1m, we choose ε = With these numerical values, the linear feedback law (26) is given by u = 1 µ (37.36x 1 + x x x 4 ). Some simulation results with the above control law on the nonlinear model (22) is shown in Figure 8, where the dashed line in Plot (a) represents the restriction on the travel of the cart. 4 Conclusions and Acknowledgements This paper reviewed the developments in the low gain feedback and low-and-high gain feedback design methods and illustrated their applications. Many of these design methods

23 LOW GAIN FEEDBACK AND LOW-AND-HIGH GAIN FEEDBACK 719 and their applications were the results of the collaborations of the author with his graduate advisor, students and other colleagues. Through these collaborations, the author has learned from these colleagues various aspects of control theory that have deepened his understanding of the subject. a c b d Figure 8 Inverted pendulum on a cart under the linear feedback law (26) with µ = 0.1: (a) x 1(t), (b) x 2(t), (c) x 3(t), (d) x 4(t) References [1] Z. Lin and A. Saberi, Semi-global exponential stabilization of linear systems subject to input saturation via linear feedbacks, System & Control letters, 1993, 21(3): [2] H. J. Sussmann, E. D. Sontag, and Y. Yang, A general result on the stabilization of linear systems using bounded controls, IEEE Transactions on Automatic Control, 1994, 39(12): [3] Z. Lin and A. Saberi, Semi-global exponential stabilization of linear discrete-time systems subject to input saturation via linear feedback, Systems & Control Letters, 1995, 24(2): [4] Z. Lin, A. A. Stoorvogel, and A. Saberi, Output regulation for linear systems subject to input saturation, Automatica, 1996, 32(1): [5] A. R. Teel, Semi-global stabilization of linear controllable systems with input nonlinearities, IEEE Transactions on Automatic Control, 1995, 40(1): [6] B. Zhou, G. Duan, and Z. Lin, A parametric Lyapunov equation approach to the design of low gain feedback, IEEE Transactions on Automatic Control, 2008, 53(6): [7] Z. Lin, Semi-global stabilization of linear systems with position and rate limited actuators, Systems & Control Letters, 1997, 30: [8] Z. Lin, Semi-global stabilization of discrete-time linear systems with position and rate limited actuators, Systems & Control Letters, 1998, 34(5):

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