Robust observer-controller compensator design using the loop shaping design procedure and the algebraic method

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(1)INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control 2010; 20:1176–1196 Published online 5 August 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1501. Robust observer–controller compensator design using the loop shaping design procedure and the algebraic method An-Chen Lee∗, † , Yi-Ren Pan and Yuan-Yong Huang Department of Mechanical Engineering, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsinchu City, Taiwan. SUMMARY This paper investigates robust observer-controller compensator design using Vidyasagar’s structure (VS). VS has a unit matrix parameter H similar to the Q parameter for the Youla–Kucera parameterization. VS can be designed based on the left coprimeness of the central controller in the H∞ -loop shaping design procedure (H∞ -LSDP) and therefore can preserve the intrinsic properties of the H∞ -LSDP. This paper introduces algebraic methods to simplify the design of H in the VS controller by solving specific algebraic equations. In particular, the algebraic design of H can achieve two things. First, a dynamic H adjusts the tracking performance and yields the integral action. Second, a dynamic H rejects the input and output sinusoidal disturbances with known frequencies. These attributes are indications of the flexibility of the proposed method since the output-feedback controller design of the H∞ -LSDP cannot easily deal with such conditions. This paper discusses the achieved loop and the closed-loop behavior of the system with VS, and also gives two numerical examples. The first example shows that the proposed method results in a better design in many aspects than the resulting from H∞ -LSDP. The second example shows the application of the proposed method to rejecting input and output step disturbances, and input and output multiple sinusoidal disturbances, for which the H∞ -LSDP can hardly be used. Copyright q 2009 John Wiley & Sons, Ltd. Received 12 August 2008; Revised 12 June 2009; Accepted 15 June 2009 KEY WORDS:. Youla–Kucera parameterization; H∞ -loop shaping design procedure; left coprime factorization; tracking control; sinusoidal disturbance rejection. 1. INTRODUCTION The observer–controller compensator (OBC) has been widely used in control system design because of its great flexibility [1–10]. In 2001, Giua et al. [3] proposed an OBC for a three-degree-of-freedom. ∗ Correspondence. to: An-Chen Lee, Department of Mechanical Engineering, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsinchu City, Taiwan. † E-mail: aclee@mail.nctu.edu.tw Contract/grant sponsor: Republic of China National Science Council; contract/grant number: NSC 94-221-E-009-181 Copyright q. 2009 John Wiley & Sons, Ltd.. overhead crane with a time-varying suspending rope. Pandian et al. [4] proposed design methods for the estimation of chamber pressure variables and a sliding-mode controller to control a cylinder actuator. Noijen et al. [5] proposed a state-feedback controller combined with an observer that estimated the orientation error based on available trajectory information and measurement of the position coordinates for a unicycle mobile robot system. Driessen and Duggirala [6] proposed an OBC for a relatively large class of systems with hysteresis, and Alazard and Apkarian [7] used the Youla–Kucera parameterization (YKP) to give arbitrary-order H∞ or controllers. Gao and Ho [8] suggested that the YKP may be non-proper and.

(2) 1177. ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. proposed a modified parameterization of all proper stabilizing compensators. Gao and So [9] proposed a unified doubly coprime factorization, which is applicable to both descriptor systems and state-space systems. Gao [10] developed a PD observer parameterization for descriptor systems using the coprime factorization technique. Vidyasagar’s structure (VS) is an OBC with an observer observing the partial state and a controller with an H parameter [11, 12]. Instead of the Q parameter over the set of proper and stable real rational functions denoted by RH ∞ in the YKP of [13–15], this H parameter is a unit over the set of proper and stable real rational functions denoted by U(R H∞ ). On the other hand, applications using the H∞ -loop shaping design procedure (H∞ -LSDP) of [16–18] have been the subject of further study, e.g. [19–23]. Panagopoulos and Astrom [19] showed that traditional methods for the design of PID controllers can be related to the H∞ -loop shaping method. Zhu et al. [21] designed a robust power system stabilizer using H∞ -loop shaping approach. Schinstick et al. [22] applied H∞ -loop shaping method to linear motor stages and non-contacting machines control. Patra et al. [23] used the H∞ -loop shaping method to design a robust load frequency controller. Hence, applying the principle of the H∞ -loop shaping method to VS and relating VS to the H∞ -LSDP fulfills the specified robustness requirement for more powerful design. The H∞ -LSDP is an open-loop shaping approach that follows the elementary open-loop shaping principles specifying the closed-loop objectives in terms of requirements on the open-loop singular values, denoted by (•). That is, for a plant G and a controller K , the controller design achieves the desired loop (and controller) gains in the appropriate range: (GK) 1,. (KG) 1,. (K ) 1. (1). in a low frequency range [0, l ] and (GK) ¯ 1,. (KG) ¯ 1,. (K ¯ ). (2). in a high frequency range [h , ∞] where is not too large. (•) ¯ and (•) denote the maximum and minimum singular values, respectively. Hence, such a controller Copyright q. 2009 John Wiley & Sons, Ltd.. design makes ((I ¯ +GK)−1 ), (K ¯ (I +GK)−1 ),. ((I ¯ +KG)−1 ) ((I ¯ +GK)−1 G). (3). small in [0, l ] for good performance, and (GK(I ¯ +GK)−1 ),. (K ¯ (I +GK)−1 G). (4). small in [h , ∞] for good robustness. Section 2 shows that VS itself is a simple scheme and its equivalent output-feedback controller has a subset of solutions to the YKP in terms of feedback properties. In comparison with the equivalent controller and the YKP, the VS observer can be derived from the left coprimeness of the central (output-feedback) controller. The different feedback solutions to VS from YKP are indicated, and the feedback and tracking properties for the H parameter in the VS are also presented. Section 3 reviews the H∞ -LSDP and describes the VS design procedure. The first step of the VS design procedure uses the pre- and post-weighting matrices to shape the nominal plant, as in the H∞ -LSDP. These weighting matrices can be constant or dynamic. Using constant weighting matrices can reduce the dimension of VS and increase the crossover frequency. The dynamic weighting matrices are used for the integral action with the phase-advance term so that the feedback system can reject step disturbances. The VS observer can then be designed from the left coprimeness of the central controller of the H∞ -LSDP for a strictly proper plant. For a specific performance, the final step uses algebraic methods to design the H parameter in the VS controller to obtain the controller. Algebraically designing H provides solutions to four cases: constant H yields the integral action, dynamic H yields the integral action with the phase-advance term, dynamic H adjusts the tracking performance and yields the integral action, and dynamic H rejects the input and output sinusoidal disturbances with known frequencies, according to the algebraic equations. A constant pre-weighting matrix W1 and post-weighting matrix W2 are considered in the first three cases. The first two cases for rejecting step disturbances can also be achieved by W1 or W2 . The last two cases present the flexibility of VS since the output-feedback controller design of the H∞ -LSDP cannot easily cope with the sinusoidal disturbance Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(3) 1178. A.-C. LEE, Y.-R. PAN AND Y.-Y. HUANG. rejection and the tracking control. H can affect the robust stability in addition to performance. Thus, the observer design of VS inherits the properties of the controller in the H∞ -LSDP and preserves certain robust stability. The VS controller design provides a robust stability/performance tradeoff. Section 4 provides two numerical examples. The first considers the case of a VS design that uses a pre-weighting matrix to reject the step disturbance and H to adjust the robust stability and the step tracking response. The case is compared to the output-feedback of the H∞ -LSDP with and without a pre-filter. The second example is designed to reject the input–output sinusoidal and step disturbances using the H parameter and the post-weighting matrix, respectively. Section 5 provides the conclusions of this paper.. 2. DISCUSSION OF VS A nominal plant G(s) is assumed to be a proper real rational matrix. G(s) is said to have a doubly coprime ˜ factorization if (N (s), M(s)) and ( N˜ (s), M(s)) are a right coprime factorization (RCF) and a left coprime factorization (LCF) of G(s), i.e. G = NM −1 = M˜ −1 N˜ , ˜ respectively, and N (s), M(s), N˜ (s), M(s), X r (s), Yr (s), X l (s), and Yl (s) exist over RH ∞ such that X r Yr X r Yr M −Yl M −Yl = N Xl N Xl − N˜ M˜ − N˜ M˜ =I. (5). Viswanadham [11] and Vidyasagar [12] proposed the VS, the OBC of Figure 1(a), where the nominal plant G(s) has an RCF, i.e. G = N M −1 ; the observer composed of X r and Yr observes the ‘internal state’ z to be zˆ , and the controller Cv (s) feeds zˆ back. Moreover, r , di and do denote the command reference, input disturbance and output disturbance, respectively; y is the system output, and er and ed are the internal signals. The system of Figure 1(a) is internally stable [12] if and only if M(s)+Cv (s) = H (s) ∈ U(RH ∞ ) Copyright q. 2009 John Wiley & Sons, Ltd.. (6). er. r. −. di. G. ed. Cv (s). do z. −1. X r (s). y. N (s). M ( s). Yr ( s) zˆ. Vidyasagar's structure (a) di r. −. ( I + Cv X r ) −1. er. do ed. y G. C vYr. (b). Figure 1. Vidyasagar’s structure: (a) observer–controller compensator and (b) equivalent compensator.. The system of Figure 1(a) also can be transformed to the system of Figure 1(b) in terms of input–output equivalence. The notation U(RH ∞ ) denotes a unit over RH ∞ . When a square matrix and its inverse are stable, the matrix belongs to U(RH ∞ ). In terms of the feedback properties, the outputfeedback controller K v (s) = (I +Cv X r )−1 ·(Cv Yr ) in Figure 1(b) can be represented according to Equations (5) and (6) as K v (s) = (I +Cv X r )−1 (Cv Yr ) = [H −1 (I +Cv X r )]−1 [H −1 (Cv Yr )] ˜ = [X r +H −1 Yl N˜ ]−1 [Yr −H −1 Yl M]. (7). The YKP, all stabilizing controllers, can be parameterized as follows: ˜ K YK (s) = (X r + Q N˜ )−1 (Yr − Q M) Q(s) ∈ RH ∞ and det(I + Q N˜ X r−1 )(∞) = 0. (8). When Q(s) is zero, the result is the central controller of K YK (s), X r−1 Yr . In comparison with X r and Yr of Equation (8), X r and Yr of Equation (7) play the similar role as central (output-feedback) controller, although H in Equation (7) cannot be zero. Nevertheless, this shows that the observer design of VS can be derived from the LCF of the central controller. Moreover, Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(4) ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. 1179. has the inherent properties of the H∞ -LSDP. The the term H −1 Yl in Equation (7) corresponds to Q in Equation (8). This means that K v (s) is a subset of H∞ -LSDP is first reviewed below. the YKP since H is a unit over RH ∞ . However, VS 3.1. Review of the H∞ -LSDP is simpler than the YKP structure. Noted that VS in Figure 1(a) is stable itself while K v (s) of Equation (7) The H∞ -LSDP of [17, 18] incorporates the perforand K YK (s) of Equation (8) are not necessarily stable. mance/robustness tradeoff obtained in loop shaping, VS is different from the YKP in terms of input– with the guaranteed stability properties of the robust output relationships. The transfer functions from stabilization H∞ (r, di , do ) to (er , ed , y) in Figure 1(b) are as follows: ⎤⎡ ⎤ ⎡ ⎤ ⎡ MH −1 M(X + H −1 Y N˜ )− I −M(Y − H −1 Y M) r er r l r l ˜ ⎥ ⎢ ⎢ ⎥ ⎢ −1 ˜ ⎥ ⎣ di ⎥ −M(Yr − H −1 Yl M) M(X r + H −1 Yl N˜ ) ⎦ (9) ⎣ ed ⎦ = ⎢ ⎣ MH ⎦ −1 −1 −1 ˜ y do N (X r + H Yl N˜ ) NH I − N (Yr − H Yl M) Replacing K v (s) in Figure 1(b) with the YKP of Equation (8), produces transfer functions from (r, di , do ) to (er , ed , y), ⎡ ⎤ ⎡M er ⎢ ⎥ ⎢ ⎣ ed ⎦ = ⎢ ⎣M y. M(X r + Q N˜ )− I M(X r + Q N˜ ). N N (X r + Q N˜ ) ⎡ ⎤ r ⎢ ⎥ × ⎣ di ⎦. ˜ −M(Yr − Q M). ⎤. ⎥ ˜ ⎥ −M(Yr − Q M) ⎦. ˜ I − N (Yr − Q M). (10). do Even if Q in Equation (10) is replaced with H −1 Yl , Equation (10) is still not equivalent to Equation (9) with respect to r . H exists in nine elements in Equation (9). The first column in Equation (9) corresponds to the tracking performance and the last two columns correspond to the feedback performance. This shows that H has both tracking and feedback properties. Finally, VS is more flexible than the central controller since the controller is included in VS [24].. problem [16]. The main results of [17, 18] are as follows. The H∞ -LSDP uses a pre-weighting matrix W1 and/or a post-weighting matrix W2 to shape the singular values of the nominal plant G to the desired open-loop shape G S = W2 GW 1 . W1 and W2 are selected such that G S has no hidden modes. The two weighting matrices can increase the loop gain and the crossover frequency, and/or introduce the integral action with or without the phase-advance term. Then, the robust stabilization H∞ problem is applied to the normalized RCF of G S to produce a robust controller K ∞ . Several researchers [12, 25–27] advocate an expression of coprime factor uncertainty in terms of additive stable perturbations to coprime factors of the nominal plant. Such a class of perturbations has advantages over additive or multiplicative unstructured uncertainty models. For example, the number of unstable zeros and poles may change as the plant is perturbed. That is, if the shaped plant is G S = N S M S−1 a perturbed plant can be written as G = (N S + N )(M S + M )−1. 3. COMPENSATOR DESIGN This section uses the VS properties to obtain the observer directly from the LCF of the controller in the H∞ -LSDP in [17, 18]. This method ensures that the proposed VS design procedure is convenient and Copyright q. 2009 John Wiley & Sons, Ltd.. (11). (12). where M S and N S are the RCF of G S , and M and N are stable, unknown transfer representing the. functions

(5). N. uncertainty and satisfying M <ε, where ε(>0) ∞ presents the stability margin. The design objective is to find a feedback controller K ∞ that stabilizes all such Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(6) 1180. A.-C. LEE, Y.-R. PAN AND Y.-Y. HUANG. G for a given ε. Such a stabilizing controller K ∞ satisfies M S−1 (I + K ∞ G S )−1 [K ∞ I ]∞ ε −1. (13). Suppose the shaped plant of G S has the minimal realization (A, B, C, D). A state-space construction for the normalized RCF can be obtained in terms of a solution to the generalized control (filter) algebraic Riccati equation as follows. The generalized control algebraic Riccati equation (GCARE) is. where • H denotes the Hankel norm, and εmax is the maximum stability margin. is always selected to be /εmax , where is a constant greater than zero. The final feedback controller K is constructed as W1 K ∞ W2 . 3.2. VS design procedure This section presents the VS design procedure and discusses only the strictly proper rational plant.. (A −BS−1 D T C)T X + X (A −BS−1 D T C) −XBS−1 B T X +C T R −1 C = 0. (14). and generalized filter algebraic Riccati equation (GFARE) is. 3.2.1. Selections of weighting matrices. The first step is to select the constant or dynamic weighting matrices W1 and/or W2 to shape the singular values of G S (= W2 GW1 ) as the LSDP does [17].. (A −BS−1 D T C)Z + Z (A −BS−1 D T C)T where R ≡ I +DDT , and S ≡ I + D T D. Then, the normalized RCF (M S , N S ) is given as ⎤ ⎡ A +BF BS−1/2 MS ⎥ ⎢ −1/2 ⎥ −−− = ⎢ (16) ⎣ - - - -F- - - - - - - -S- - - - ⎦ NS DS−1/2 C +DF. 3.2.2. The design of the observer. The second step is to find an observer in VS. Suppose the shaped plant G S has a minimal realization [A, B, C, D] where D = 0, and the perturbed shaped plant is presented by Equation (12). Then, according to Equation (16), the normalized RCF of G S can be represented by. where F ≡ −S −1 (D T C + B T X ). The normalized RCF of G S means M ST (−j)M S (j)+ N ST (−j)N S ( j) = I for all . (17). Copyright q. 2009 John Wiley & Sons, Ltd.. . A +BF. (20). where F = −B T X . Theorem 1 shows that the observer composed of X r and Yr in VS can be derived from the LCF of the central controller in terms of feedback properties. Hence, the observer can be obtained. McFarlane and Glover [17] showed that a central controller satisfying Equation (13) can be obtained as follows: A +BF +2 (W T )−1 Z C T (C +DF) −2 (W T )−1 ZC T K∞ = BT X DT where W ≡ I +(X Z −2 I ) and are defined as 1/ε. In addition, a maximum value of ε can be obtained by a non-iterative method, and is given by ⎛ ⎞1/2. N S 2. ⎠. εmax = ⎝1−. (19). MS. H. ⎡. ⎤ B ⎢ ⎥ MS I⎥ −−− = ⎢ ⎢ F ⎥ ⎣ ------------ ⎦ NS C 0. . (18). from the LCF of the central controller in Equation (18), which satisfies Equation (13) with stability margin ε as follows: ⎡ ⎤ B −QL A +QLC ⎦ X r Yr = ⎣ (21) −F I 0 -----------. (15). ---. −ZCT R −1 CZ +BS−1 B T = 0. Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(7) 1181. ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. Yl ] = ⎣. [ N˜ S. --. ⎡ M˜ S ] = ⎣. A +BF. B. C +DF. I. A +QLC. B. C. 0. −QL. ⎤ ⎦. QL. =. M˜ S MS. −Yl. NS. Xl. NS . Yr. − N˜ S. M˜ S. ⎦. (23). I. =I. (24). 3.2.3. Design of H (the controller). The third step is to find the controller Cv in VS such that M S−1 (I + K v G S )−1 [K v I ]∞ εv−1 (= v ). (25). where K v = (I +Cv X r )−1 Cv Yr and εv are the stability margin with respect to the shaped plant for the system with VS. The internal stability condition of Equation (8) can be rewritten as M S (s)+Cv (s) = H (s) ∈ U(RH ∞ ). (26). According to Equations (9), (24), and (26), Equation (25) can be written as Equation (27) or (28). [H −1 Cv Yr [Yr − H −1 Yl M˜ S. H −1 (I +Cv X r )]∞ εv−1. (27). X r + H −1 Yl N˜ S ]∞ εv−1. (28). Equation (28) shows that the H parameter in Cv can affect the value of the stability margin εv . Here, H is selected according to the control requirements and then the value of εv can be checked. H may require several redesigns to obtain a satisfactory value of εv . Copyright q. 2009 John Wiley & Sons, Ltd.. (ii ) G. W2. −. observer. Yr. Xr. controller. ⎤. Xl Xr. (i ) W1. Figure 2. VS design procedure.. where N˜ S and M˜ S are the LCF of G S , but not the normalized LCF. Such coprime factorizations in Equations (20)–(23) satisfy Equation (24). Xr Yr M S −Yl − N˜ S . GS. Cv. (22). 0. -----------. [X l. --. ⎡. -----------. where Q = −2 (W T )−1 , F = −B T X and L = −ZC T . Such an observer design preserves all advantages of the H∞ -LSDP in the design of VS. Moreover, the realizations of X l , Yl , N˜ S and M˜ S can be presented as follows:. Remark 3.1 Equation (25) implicitly considers minimizing the H∞ norm of the transfer functions from (dõ , d˜i ) to (y1 , y2 ) in Figure 3(a) as follows: M S−1 (I + K v G S )−1 [K v I ]∞. . I. (I + K v G S )−1 [K v I ]. =. GS. ∞. −1 . W. 1 (I + K G)−1 [KW −1 =. W ]. (29) 1 2. W2 G. ∞

(8) S where the inner function M N S is pre-multiplied to go to the four-block problem. Corollary 1 of [18] shows that Equation (29) also equals Equation (30) by interchanging K v and G S : M S−1 (I + K v G S )−1 [K v I ]∞. . I. −1 (I + G S K v ) [G S I ]. =. Kv. ∞. . . W2. −1. −1 (I + G K ) =. [GW W ] (30) 1 2. W −1 K. 1 ∞ Equation (30) presents the transfer functions from (d˜i , dõ ) to (y2 , y1 ) in Figure 3(b). Equations (29) and (30) show how all the closed-loop objectives of Equations (3) and (4) are incorporated. The following will show how to design H algebraically to achieve the four cases: constant H yields the integral action, dynamic H yields the integral action with the phase-advance term, dynamic H adjusts the Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(9) 1182. A.-C. LEE, Y.-R. PAN AND Y.-Y. HUANG. ~ di. y1 −1. W1. y = W1 N S H −1 (I +Cv X r )W1−1 ·di. −1. W2. W1. (a). Kv. W2. ~ di. y1 −1 W1. ~ do −1 W2. W1 − W1. y2. y = W1 N S (X r + H −1 Yl N˜ S )W1−1 ·di. W2. (X r + H −1 Yl N˜ S )|s=0 = 0. W2. Figure 3. Two cases of the transfer functions from (d˜i , dõ ) to (y1 , y2 ).. tracking performance and yields the integral action in the feedback system, and dynamic H rejects the input and output sinusoidal disturbances with known frequencies according to the algebraic equations. The first case uses constant H to provide the integral action in the feedback system for the input and output disturbance rejections. If the disturbance is a step from the input and/or output of the plant, rejecting the disturbance requires that the Smith–Macmillan form of W1 K v W2 G [point (i) of Figure 2] and GW 1 K v W2 [point (ii) of Figure 2] have a pole at 0 for every channel. If three conditions hold (W1 and W2 are constant, at least one channel of GY r and Yr G does not have any poles at 0, and K v can be represented as K v = (Cv−1 + X r )−1 Yr ), then the requirements equal (Cv−1 + X r )|s=0 = 0. (31). The third condition implies that the DC gain of Cv must not be a zero matrix so that Equation (31) equals (I +Cv X r )|s=0 = 0. (32). Equation (32) presents the Smith–Macmillan in the form where the term I +Cv X r has a zero at 0 for every channel. The right-hand sides of Equations (31) and (32) are zero matrices with compatible dimensions. Copyright q. 2009 John Wiley & Sons, Ltd.. (34). Equations (33) and (34) show that the requirement of Equation (32) for the input step disturbance rejection also can be written as. G. Kv. (33). When each element in the vector di is a unit step, each element in y indeed has a zero steady state according to Equation (32) and the final value theorem. Because of H −1 (I +Cv X r ) = X r + H −1 Yl N˜ S , Equation (33) can be rewritten as. G. −. (b). W2. W1. The proof of Equation (32) is as follows. The transfer functions from the input disturbance to the output are. ~ do. y2. (35). since the DC gain of H must not be a zero matrix. Because of X l N˜ S = N S X r in Equation (24), Equation (35) can be rewritten as (X l + N S H −1 Yl )|s=0 = 0. (36). Moreover, the transfer functions from the output disturbance to the output are y = W2−1 (X l + N S H −1 Yl ) M˜ S W2 ·do. (37). Hence, when each element in the vector do is a unit step, each element in y indeed has a zero steady state, according to Equation (36) and the final value theorem. According to Equations (26) and (31), H in Cv should satisfy H |s=0 = (M S − X r−1 )|s=0. (38). Equation (38) shows that the simplest way to select H with integral action is H = (M S − X r−1 )|s=0. (39). There is only one requirement for the constant H in Equation (39) that makes the system’s input and output unit step disturbances zero at the steady state. The stability margin may be small, although Equation (39) leads to integral action in the feedback system and good step disturbance rejection at low frequencies. Hence, the second case will give the feedback system integral action with the phase-advance term using dynamic H to increase the stability margin. Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(10) 1183. ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. G K W1. Kv. W2. output-feedback controller. Figure 4. Positive output-feedback controller derived from VS.. Each channel in the open-loop shaping with a zero at 2 (>0) can be achieved by satisfying (Cv−1 + X r )−1 |s=2 = 0. (41). Thus, the integral action with the phase-advance term requires that H satisfy Equations (38) and (41). In that case, the element Hij of the ith row and jth column in H can be selected as Hij (s) = h ij. s +kij s+ p. (42). where the real number p(>0) is given, and h ij and kij (>0) are selected according to Equations (38) and (41). This results in Cv . The final compensator can be in the form of the observer–controller configuration of Figure 2 or the (positive) output-feedback controller K = −W1 K v W2 of Figure 4. W1 and W2 are assumed to be I in Figure 4. Rejecting the step input and output disturbance in VS requires that only parameter H satisfy Equations (38) and (41). This property of VS is better than that of the controller in the H∞ -LSDP, which may require W1 and W2 to improve the input and output step disturbances, respectively [17]. Copyright q. 2009 John Wiley & Sons, Ltd.. t→∞. s→0. 1/s ⎛ ⎡ ⎤⎞ 1 ⎜ −1 −1 ⎢ ⎥⎟ = ⎝W2 N S H ⎣ 1⎦⎠ 1. (40). Equation (40) shows that the Smith–Macmillan form of the term Cv−1 + X r has a pole at 2 for each channel if the pole at 2 is not a hidden mode. Moreover, we assume that the Smith–Macmillan form of the term X r does not have a pole at 2 . Hence, the Smith– Macmillan form of the term Cv−1 must have a pole at 2 for each channel. That is, the term Cv has a zero at 2 for each channel so that the following equation is satisfied: H |s=2 = M S |s=2. In the third case, H is used to adjust the tracking performance and produce the integral action in the feedback system. Suppose that the nominal plant has three inputs and three outputs, W1 is an identity matrix, and W2 is a constant matrix. To achieve unit step demand responses, the DC gain of the H parameter satisfies the final value theorem as follows: ⎡ ⎤ 1/s ⎥ ⎢ lim y(t) = lim sW2−1 N S H −1 ⎣ 1/s ⎦ ⎡ ⎤ 1 ⎢ ⎥ = ⎣ 1⎦ (43). s=0. 1. where y is the nominal plant output; N S is obtained from the normalized RCF of G S (=N S M S−1);W2−1N S H −1 is the transfer function from step demands to y, and the step demands are a 3×1 vector with each element 1/s. It is impossible for the DC gain of the H parameter to satisfy Equations (38) and (43). An extra constant matrix J is required to regulate the output responses and achieve the two objectives as follows: ⎡. 1/s. ⎤. ⎢ ⎥ lim y(t) = lim sW2−1 N S H −1 J ⎣ 1/s ⎦. t→∞. s→0. 1/s ⎛ ⎡ ⎤⎞ ⎡ ⎤ 1 1 ⎜ ⎢ ⎥⎟ ⎢ ⎥ = ⎝W2−1 N S H −1 J ⎣ 1⎦⎠ = ⎣ 1⎦ 1 1 s=0. (44). where the DC gain of H satisfies Equation (38). Introducing the constant J yields the final compensator in Figure 5, where r is the demand, y is the plant output, and di and do are the input and output disturbances, respectively. The extra constant matrix J does not affect the openloop shape, as shown in below J (I +J −1 X r Cv J )−1 · J −1 Yr =(I +X r Cv )−1 Yr = K v (45) Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(11) 1184. A.-C. LEE, Y.-R. PAN AND Y.-Y. HUANG. di r. J. do y. G. W1. −. W2. J −1 Yr. Xr Cv. Figure 5. Control scheme of VS for tracking and step disturbance rejections.. If H is selected to be constant, as shown in Equation (39), the dynamic responses of the outputs depend only on N S due to constants W2 and J . However, N S is often not sufficiently good to obtain satisfactory tracking performance. Hence, the dynamic H matrix is required and can be designed using ⎤ ⎡ h1 h2 h3 ⎥ det(N S ) ⎢ H −1 = ⎣ h 4 h 5 h 6 ⎦ · (46) (s) h7 h8 h9 where det(N S ) is the determinant of N S , the roots of the polynomial (s) with the same orders as det(N S ) are designed in the left-half plane, and h 1 , . . . , h 9 can be obtained relying on the fact that the DC gains of H in Equations (38) and (46) are identical. The transfer function from r to y is then given by ⎡ ⎤ h1 h2 h3 Num(N S ) ⎢ ⎥ y = W2−1 · · ⎣ h 4 h 5 h 6 ⎦ · J ·r (47) (s) h7 h8 h9 where Num(N S ) is the numerator matrix of N S . The term det(N S ) in Equation (46) can be chosen freely and has no effect on the tracking performance since it cancels. Equation (47) shows that (s) can be designed to control the tracking performance. If the absolute values for the roots of (s) are larger, the tracking response is faster but the system robustness is reduced. The robustness can be checked using Equation (B1). Thus, if the dynamic H matrix is designed as shown in Equation (36), the objectives of good step disturbance rejections and tracking performance can be achieved. Copyright q. 2009 John Wiley & Sons, Ltd.. H is designed in a similar way for other dimensions of the square or for wide plants. Suppose that di and do are sinusoidal with frequencies at . If H is used to reject di and do , the following two equivalent equations must be satisfied according to Equations (34) and (37), respectively: (X r + H −1 Yl N˜ S )|s=j = 0. (48). (X l + N S H −1 Yl )|s=j = 0. (49). Equations (48) and (49) can be rewritten as Equations (50) and (51), respectively: H |s=j = −Yl N˜ S X r−1 |s= j . (50). H |s=j = −Yl X l−1 N S |s= j . (51). Because of X l N˜ S = N S X r in Equation (24), Equation (50) is equivalent to Equation (51). This means that the input and output sinusoidal rejections with a known frequency require only one of either Equation (50) or (51), which shows good VS properties. The following discussion uses Equation (50). Since each element for H at is a complex number, solving Equation (50) requires two unknown coefficients in each element of H . Hence, the ith row and jth column element of H can be given algebraically as Hij (s) = h i j1 +. h i j2 s + ph. (52). where ph (>0) is given, and h i j1 and h i j2 can be solved according to Equation (50). If the result of H is not unimodular, another value of ph is given and Equation (50) is solved again. In a similar manner, if the input and output disturbances have two known sinusoidal frequencies, 1 and 2 , then each element of H needs four coefficients to be solved as Hij (s) = h i j1 +. h i j2 h i j3 h i j4 + + s + ph (s + ph )2 (s + ph )3. (53). The four coefficients for each element of H in Equation (53) are obtained according to Equation (50) with respect to at 1 and 2 . That is, the number of the coefficients in Hij (s) to be solved is twice the number of the different frequencies. Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(12) ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. The H in VS provides flexibility to the compensator. When designing H , it is straightforward to obtain solutions to the four cases described earlier. More complex H design methods are not presented here. For example, H adjusts the tracking performance and rejects the sinusoidal input disturbance simultaneously. Since H , W1 , or W2 can achieve integral action with or without the phase-advance term, it is possible to implement the integral action using W1 or W2 , and to solve other problems such as tracking control or sinusoidal disturbance with H . 3.3. Achieved loop shape As described above, the desired loop shaped was specified as W2 GW 1 , but the final shape of the loop achieved is in fact given by W1 K v W2 G at the plant input [point (i) of Figure 2] and GW 1 K v W2 [point (ii) of Figure 2]. McFarlane and Glover [17] have suggested that if (≡ ε −1 ) is small enough, (G S ), is large enough at low frequencies, and (G ¯ S ) is small enough at high frequencies, then the deteriorations of the loop shapes W1 K ∞ W2 G and GW 1 K ∞ W2 at low and high frequencies are limited for the H∞ -LSDP, according to Theorems A.1 and A.2, respectively. With the help of these results [17], this section shows that the deteriorations of the loop shapes W1 K v W2 G and GW 1 K v W2 at low and high frequencies are also limited, according to Theorems 1 and 2, respectively. The following equations show that (K v ) requires a bound on the deterioration of the loop shapes at low frequencies: (GK) = (GW 1 K v W2 ) (W2 GW 1 )(K v )/c(W2 ). (54). (KG) = (W1 K v W2 G) (W2 GW 1 )(K v )/c(W1 ). Copyright q. 2009 John Wiley & Sons, Ltd.. Theorem 1 At frequencies of high loop gain, the smallest singular value of the controller Cv should increase since (K v ). (X r−1 Yr ) 1 1+ (Cv )(X r ). (56). Proof Equation (56) follows from simple manipulation of singular value inequalities. (X r−1 Yr ) in Equation (56) is the smallest singular value of the central controller K ∞ in Equation (18), and is large enough if is small enough and (G S ) is large enough at low frequencies, according to Theorem A.1. Hence, for such K ∞ , K v is large enough if (Cv ) is large enough. Then, (GK) or (KG) are also large enough at low frequencies according to Equations (54) and (55). The following equations show that (K ¯ v ) requires a bound on the deterioration of the loop shapes at high frequencies: (GK) ¯ = (GW ¯ 1 K v W2 ) ¯ v )c(W2 ) (W ¯ 2 GW 1 )(K. (57). (KG) ¯ = (W ¯ 1 K v W2 G) ¯ v )c(W1 ) (W ¯ 2 GW 1 )(K. (58). In a similar manner, the following result shows that (K ¯ v ) is bounded by (X ¯ r−1 Yr ), (C ¯ v ), and (X ¯ r ), and hence by Equations (57) and (58), while the low gain of (C ¯ v ) limits the deterioration of K v at high frequencies. Theorem 2 At frequencies of low loop gain, the largest singular value of the controller Cv should decrease because. (55). where the designer can select the condition numbers c(W1 ) and c(W2 ). The following result shows that (K v ) is bounded by (X r−1 Yr ), (Cv ), and (X r ), and hence by Equations (54) and (55), and the high gain of (Cv ) limits the deterioration of K v at low frequencies.. 1185. (K ¯ v ). (X ¯ r−1 Yr ) 1 −1 (C ¯ v )(X ¯ r). (59). Proof Equation (59) follows from simple manipulation of singular value inequalities. Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(13) 1186. A.-C. LEE, Y.-R. PAN AND Y.-Y. HUANG. (X ¯ r−1 Yr ) in Equation (59) is the largest singular value of the central controller K ∞ in Equation (18), and is small enough if is small enough and (G ¯ S ) is small enough at high frequencies according to Theorem A.2. Hence, for such K ∞ , K v is small enough if (C ¯ v) is small enough. Then, either (GK) ¯ or (KG) ¯ is also small enough at high frequencies according to Equations (57) and (58). This section shows that the selections of K ∞ and Cv can affect the singular values of K v . Moreover, W1 , W2 , and Cv can be used to set the performance/robustness tradeoff so that Equation (25) is satisfied. 3.4. Closed-loop behavior The H∞ -LSDP in [17] ensures that a number of standard closed-loop design objectives are bounded. According to Equations (30) and (31), Theorem 4.2 in [17] can be rewritten as Theorem B.1. Theorem B.1 shows that the bounds on the behavior of all closedloop transfer functions of Equations (3) and (4) depend on the normalized RCF or normalized LCF of G S = W2 GW 1 , v , W1 , and W2 . Hence, the closed-loop objectives of the VS are also ‘well behaved’, as they are referred to by McFarlane and Glover [17]. The trade-off between stability and performance can be seen in Equations (B1)–(B7). When fewer task requirements are required of H , more-complex W1 and/or W2 is/are needed and c(W1 ) and/or c(W2 ) are larger (bigger than one). Then, to satisfy Equations (B1)–(B7), v will be smaller since c(W1 ) and/or c(W2 ) will be larger (greater than one). Hence, when fewer task requirements are placed on H , the stability margin is larger, and vice versa.. outputs, and five states. The inputs are spoiler angle (u 1 measured in tenths of a degree), forward acceleration (u 2 in m/s2 ), and elevator (u 3 in degrees). The states are altitude relative to some datum (x1 in m), forward speed (x2 in m/s), pitch angle (x3 in degrees), pitch rate (x4 in degree/s), and vertical speed (x5 in m/s). The three outputs (y1 , y2 , y3 ) are just the first three states (x1 , x2 , x3 ), which are to be controlled. The continuous-time state-space matrices of the nominal plant are listed in Appendix C. The plant has no transmission zeros and has poles located at −0.78±1.03j, −0.0176±0.1826j and 0. The design requirements are to achieve a crossover frequency of about 10 rad/s, with reasonably damped responses and zero steady-state error in the face of step demands or disturbances. VS Design Procedure: Step 1. The dynamic pre-weighting matrix W1 is selected as 24(s +0.4) 12(s +0.4) 24(s +0.4) W1 = diag , , (60) s s s to increase the crossover frequency and reject the input and output disturbances. W2 is an identity matrix. Step 2. An eighth-order controller K ∞ of Equation (18) satisfying Equation (13) (ε = 0.361) is obtained. X r and Yr , the LCF of K ∞ , are derived from Equation (21). The observer composed of X r and Yr is obtained in Appendices D.1 and D.2. Step 3. The controller (Cv ) is designed. The normalized RCF of the shaped plant G S (s)(= G(s)W1 (s)) is (N S (s), M S (s)). N S (s) has poles at −12.2268, −0.4001, −7.6683±7.6322j and −2.3696±2.3435j. Then, the transfer function from r to y in Figure 5 is as follows:. 4. TWO DESIGN EXAMPLES Example 1 This example applies to the proposed VS design procedure for designing a robust compensator for the aircraft model AIRC [17]. Algebraic methods are proposed for the input and output step disturbance rejection and the tracking performance. The robust VS compensator is compared with the H∞ -LSDP controller. The model used in a linearized model of the verticalplane dynamics of an aircraft with three inputs, three Copyright q. 2009 John Wiley & Sons, Ltd.. y = W2−1 N S H −1 J ·r. (61). where W2 is an identity matrix. If H −1 is selected to be H. −1. . 0.1(s +12.2268) s +5 0.1(s +12.2268) 0.1(s +12.2268) , s +5 s +5. (s) = diag. (62) Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(14) 1187. ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. then the stable zero of H −1 (s) at −12.2268 cancels the stable pole of N S (s), which can reduce the control effort on r . Moreover, the value of the coefficient, 0.1, in each element of H −1 can increase the stability margin according to Equation (28). The system stability margin εv is 0.326. When the coefficient is near zero, εv will be close to ε. If the value of the coefficient is larger, the value of εv will be smaller. For example, the stability margin is 0.165 when the value of the coefficient is 1. Therefore, H (s) requires several designs to obtain a satisfactory value of εv . This is the disadvantage of the VS design procedure. Moreover, a constant matrix J for regulating the output final values is given by ⎡. −2.4661 −0.3402. ⎢ J = ⎣ −0.2276. 4.0751. 3.2443. 1010. 100. ⎥ 0.2544 ⎦. (63). u 3 <10◦. Comparison with the H∞ -LSDP McFarlane and Glover [17] designed this example with W2 as follows: 24(s+0.4) 12(s+0.4) 24(s+0.4) , , (64) W2 =diag s s s Copyright q. 2009 John Wiley & Sons, Ltd.. 100. 101. 102. 100. 101. 102. rad/sec. 1010. 100. ⎤. The M S of Appendix D.3 and designed H and Cv are obtained according to Equation (26). Then, the final (negative) feedback controller K = W1 K v is obtained. Figure 6(a) and (b) show the singular values of the open loops, GK and KG, respectively; their crossover frequencies are about 10 rad/s. Figure 7(a) and (b) shows the output and input sensitivity functions with the largest peak values of 1.57 and 1.58, respectively. Figure 8(a) and (d) shows the output responses and control inputs, respectively, for demands with (1/s, 0, 0). Figure 8(b) and (e) shows the output responses and control inputs, respectively, for demands with (0, 1/s, 0). Figure 8(c) and (f) shows the output responses and control inputs, respectively, for demands with (0, 0, 1/s). Figure 8 demonstrates that the outputs have almost no overshoots, the interaction between outputs is less than 20%, and the control inputs are limited as follows: u 2 <5 m/s,. 10-1. (a). 10-2. 10-3. (b). −3.2542 −0.0272 −2.4764. u 1 <10◦ ,. 10-2. 10-3. 10-1 rad/sec. Figure 6. Singular values of (a) GK and (b) KG.. 100. 10-10 10-3. 10-2. 10-1. (a). 100. 101. 102. 100. 101. 102. rad/sec 100. 10. (b). -10. 10-3. 10-2. 10-1 rad/sec. Figure 7. Singular values of (a) output sensitivity function and (b) input sensitivity function.. Here, the final controller K = K ∞ W2 of the H∞ -LSDP is obtained with no model reduction. The stability margin ε of the H∞ -LSDP is 0.361. Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(15) 1188. A.-C. LEE, Y.-R. PAN AND Y.-Y. HUANG. 5. 2 y1. 1 y2. 0. u2. 0. u1. -5 u3. y3 -1. -10 0. 1. (a). 2. 0. 3. 1. (d). sec. 2. 3. 2. 3. 2. 3. sec. 5. 2 y2. u2. u3. 1 0 0 u1. y1 y3 -5. -1 0. 1. (b). 2. 0. 3. sec. 1. (e). sec. 10. 2. u1. y3. 1. 0. y2. u2. 0 y1. u3. -1. -10 0. 1. (c). 2. 3. 0. (f). sec. 1 sec. Figure 8. (a)–(c) Unit step responses for outputs and (d)–(f) control inputs.. The controller of the H∞ -LSDP is compared with the case of VS. Figure 9(a) and (b) show that the singular values of GK and KG for the H∞ -LSDP are almost the same as those for VS. Hence, the input and output sensitivity functions, which represent the ability to reject input and output disturbances, respectively, are almost the same for the two approaches. The largest peak values in Figure 10(a) and (b) are 1.59 and 1.57, respectively. These peak values also are almost the same as those of the second case. Figure 12 shows the unit step responses for outputs and control inputs when the control scheme shown in Figure 11 is used for tracking. The overshoot of the outputs is less than 40%, the interaction between outputs is less than 30%, and the control inputs are limited as follows: ◦. u 1 <40 , Copyright q. u 2 <10 m/s,. u 3 <30. 2009 John Wiley & Sons, Ltd.. ◦. Hence, the VS case is the best for overshoots, interaction, and range of control input. Moreover, the VS case has a settling time about of 1 s, which is shorter than that of the H∞ -LSDP (≈ 2 s). Because of the implementation difference with regard to the H∞ -LSDP and the VS, a pre-filter F p (s) is added for the H∞ -LSDP and the transfer function from r to y becomes y = PK(I +PK)−1 F p ·r = W2−1 N S Yr W2 F p ·r. (65). The pre-filter is designed according to F p (s) = diag{ p f, p f, p f }. (66). where p f = 8.454×10−2 ·(s +59.142)/(s +5). The DC gain of pf is 1, and pf will cancel the farthest Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(16) 1189. ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. 1010. r −. 100. 10-3. K. G. y. Figure 11. Control scheme for tracking. 10-2. 10-1. 100. 101. 102. rad/ sec. (a). Equation (62). Figure 13 shows the unit step responses for outputs and control inputs, and illustrates that the overshoot of the outputs is less than 30%, the interaction between outputs is less than 20%, and the control inputs are limited as follows:. 1010. 100. 10-3. 10-2. 10-1. 100. 101. 102. rad/ sec. (b). Figure 9. Singular values of (a) GK and (b) KG (LSDP).. 100. 10-10 10-3. 10-2. 10-1. 100. 101. 102. rad/ sec. (a). 100. 10-10 10-3 (b). 10-2. 10-1. 100. 101. 102. rad/ sec. Figure 10. Singular values of (a) output sensitivity function and (b) input sensitivity function (LSDP).. pole of Yr at −59.142 to improve the control effort. Hence, the function of F p (s) is similar to H −1 (s) of Copyright q. 2009 John Wiley & Sons, Ltd.. u 1 <3◦ ,. u 2 <10 m/s,. u 3 <20◦. F p (s) must be higher order to decrease the overshoot. Moreover, F p (s) is a phase-lag pre-filter. If Yr (or N S ) has poles much farther than −59.142, the phase lag for the unit step response will be significant with respect to the pre-filter design of F p (s). However, the similar function with respect to H (s) in VS does not have the phase-lag problem since H (s) is in the feedback loop. The stability margin of the H∞ -LSDP with or without a pre-filter is a little better than the VS case. Adding a pre-filter in the H∞ -LSDP can improve the control effort, interaction, and overshoot, but cannot improve the settling time due to the phase-lag pre-filter. The VS case seems the best design in many aspects, as summarized in Table I. Example 2 This example presents the design of VS for a 2×2 nominal multiple-input multiple-output plant. W2 is used to reject two channels of input and output unit step disturbances, and H is used to reject two channels of input and output sinusoidal disturbances at frequencies of 1 and 5 rad/s. In this case, the design can hardly be achieved using the H∞ -LSDP. The nominal plant is given by 2 −1 1 G(s) = 2 s +9s +20 0.5 3. (67). Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(17) 1190. A.-C. LEE, Y.-R. PAN AND Y.-Y. HUANG. 20. 1.4. u1. 1. 0. y1. 0.5 0 0.3. u2. y2. y3. -20 u3. 0. 1. (a). 2. 3. -40. 0. 1. (d). sec. 1. 2. 3. 2. 3. 2. 3. sec. 6. y2. u2. 4 0.5. u1 y1. 2. y3. u3. 0. 0 0. 1. (b). 2. 3. 0. 1. (e). sec. sec. 30 20. 1.4 1. u1. y3. 0. 0.5 y1. 0 0.3. (c). u2 y2. 0. 1. 2. 3. -20 -30. (f). sec. u3. 0. 1 sec. Figure 12. (a)–(c) Unit step responses for output responses and (d)–(f) control input responses (LSDP).. with the output vector (y1 , y2 ). W1 is an identity matrix and W2 is given as . 45(s +0.1) 45(s +0.1) , W2 = diag s s. (68). Equation (68) yields two channels of input and output unit step disturbances with zero steady sate. Figure 14(a) and (b) show the singular values of G and the shaped plant W2 G, respectively. The crossover frequency of W2 G in Figure 14(b) is about 10 rad/s. A controller K ∞ of Equation (18) satisfying Equation (13) (ε = 0.549) is obtained resulting in the observer composed of X r and Yr is obtained. When the input and output have multiple sinusoidal disturbances with frequencies at 1 and 5 rad/s, then according to Equations (48) Copyright q. 2009 John Wiley & Sons, Ltd.. and (53), the unimodular matrix H can be designed according to −0.1894 0.0142 H (s) = 0.0142 −0.3528 −1.6398 −0.0319 1 + s +1 −0.0319 −1.2732 2.6995 0.0442 1 + (s +1)2 0.0442 2.1917 −2.4099 −0.0323 1 + (69) (s +1)3 −0.0323 −2.0381 The output and input sensitivity functions are shown in Figure 15, where the peak value is 1.91. Moreover, Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(18) 1191. ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. 20 1. y1. 0.5. u1. 0. y2. u2. y3. 0. u3. 0. 1. 2. (a). 3. -20. 0. 1. (d). sec. 1. 2. 3. 2. 3. 2. 3. sec. 4 y2. u1. 2. 0.5. u3. 0. y3. y1. u2. 0 -2 0. 1. 2. (b). 3. 0. 1. (e). sec. sec. 10 1. u2. y3. 0.5. 0 u1. y1 y2. 0. u3. -10 0. 1. (c). 2. 3. 0. 1. (f). sec. sec. Figure 13. Unit step responses for output responses (a)–(c) and control input responses (d)–(f) (LSDP with pre-filter).. Table I. Comparison of VS and LSDP with/without pre-filter where ¯ is ε or εv , Po and Pi are the peak values of the output and input sensitivity functions, respectively, and ts is the settling time.. VS LSDP LSDP with pre-filter. ε¯. Po. Pi. u 1 (degree). u 2 (m/s). u 3 (degree). Overshoot. Interaction. ts (s). 0.326 0.361 0.361. 1.57 1.59 1.59. 1.58 1.57 1.57. <8 <40 <3. <5 <10 <10. <10 <30 <20. No <40% <30%. <20% <30% <20%. 1 2 2. the stability margin εv of the system is 0.190. Figure 16(a) shows the output responses for the disturbance vector (1/s, 1/s) at the input and output of the plant. Figure 16(b) shows the output for the sinusoidal input disturbance vector (sin t +sin 5t, sin t +sin 5t) at the input and output of the plant. Figure 16(c) shows the output responses for the total disturbances of Figure 16(a) and (b). Copyright q. 2009 John Wiley & Sons, Ltd.. 5. CONCLUSIONS The YKP-like property of VS means that the VS design procedure can improve on the H∞ -LSDP while maintaining its inherent properties. Convenient numerical computation is not a feature of the VS design procedure because of the H parameter. However, H does give the VS design procedure flexibility in Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(19) 1192. A.-C. LEE, Y.-R. PAN AND Y.-Y. HUANG. 100. 1 y1 0. y2. 10-2 -1 0. 5. 10. (a) 10-4 10-3. 10-2. 10-1. (a). 100. 101. 102. 15 sec. 20. 25. 30. 0.4. rad/ sec 0.2. y1. 0. y2. 105. -0.2 0. 5. 10. 15 sec. 20. 25. 30. 5. 10. 15 sec. 20. 25. 30. (b). 100. 1.5 1 y1. 0.5. 10-5 10-3. 10-2. 10-1. (b). 100. 101. 102. rad/ sec. 0. y2. -0.5. (c). Figure 14. Singular values of (a) G and (b) W2 G.. 0. Figure 16. Output responses with respect to (a) the unit step disturbances (1/s, 1/s) at the input and output and (b) the sinusoidal input disturbance (sin t +sin 5t, sin t +sin 5t) at the input and output. (c) The total disturbances of (a) and (b).. 101. the objectives of controlling disturbances and tracking simultaneously. Demonstration files for Examples 1 and 2 are provided online at http://s-web.nctu.edu.tw/users/ u9414829/WWW/.. 100. Input Sensitivity. 10-1. APPENDIX A. 10-2. A.1. Lower bound of the robust central controller. Output Sensitivity. 10-3 10-3. 10-2. 10-1. 100. 101. 102. rad/ sec. Figure 15. Output and input sensitivity functions of Example 2.. tracking control, or for the input-output step and known multiple sinusoidal disturbances. The tracking control in particular does not suffer from the phase-lag problem. The design of H requires more study to meet Copyright q. 2009 John Wiley & Sons, Ltd.. Theorem A.1 (McFarlane and Glover [17], Theorem 3.1). A controller K ∞ satisfying Equation (13), where G S is assumed to be square, also satisfies (G S (j))−(2 −1)1/2 (K ∞ (j)) 2 −1(G S (j))+1. (A1). for all such that. (G S ( j))> 2 −1. where = ε −1 . Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(20) 1193. ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. A.2. Upper bound of the robust central controller Theorem A.2 (McFarlane and Glover [17], Theorem 3.3). A controller K ∞ satisfying Equation (13), where G S is assumed to be square, also satisfies (K ¯ ∞ (j)). (2 −1)1/2 + (G ¯ S (j)) 2 1− −1(G ¯ S (j)). where v = εv−1 ; ( N˜ S , M˜ S ) and (N S , M S ) are a normalized LCF and RCF of G S (= W2 GW 1 ), respectively; c(•) denotes (•)/(•), ¯ the condition number, ¯ S) = ( ¯ N˜ S ) = (N. 1/2 (B8). 1+ ¯ 2 (W2 GW 1 ). (A2). for all such that. ¯ 2 (W2 GW 1 ). and. (G ¯ S (j))<1/ 2 −1. ( ¯ M˜ S ) = (M ¯ S) =. where = ε −1 .. . 1 1+2 (W2 GW1 ). 1/2 (B9). Equations (B8) and (B9) are according to Lemma 4.1 in [18].. APPENDIX B B.1. Bounds of standard closed-loop objectives The following theorem is adapted from Theorem 4.2 in [17]. Theorem B.1 Suppose G is the nominal plant, and K = W1 K v W2 is the final (negative) output-feedback controller from the VS (see Figure 3). If M S−1 (I + K v G S )−1 [K v. I ]∞ v. (B1). (K ¯ (I +GK)−1 )v ( ¯ M˜ S )(W ¯ 1 )(W ¯ 2). (B2). then ((I ¯ +GK). −1. ). ¯ S )c(W2 ), v ( ¯ M˜ S )c(W2 )} min{1+v (N. ((I ¯ +GK)−1 G). v ( ¯ N˜ S ) (W1 )(W2 ). (B4) (B5). ((I ¯ +KG)−1 ) (G(I ¯ +KG). (B6). K). ¯ S )c(W2 ), 1+v ( ¯ M˜ S )c(W2 )} min{v (N Copyright q. ⎤ −1.0000 ⎥ ⎢ 0 0.0705 ⎥ ⎢ 0 −0.0538 −0.1712 ⎥ ⎢ ⎥ ⎢ 0 0 1.0000 0 A = ⎢0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0.0485 0 −0.8556 −1.0130 ⎦ ⎣ ⎡. 0. 0. 1.1320. ⎡. 0 ⎢ ⎢ −0.1200 ⎢ ⎢ 0 B =⎢ ⎢ ⎢ 4.4190 ⎣. 0. 0. 0. 0. 1. 0. 1.0532. −0.6859. ⎤. ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ 0 −1.6650⎥ ⎦. 0 −0.0732 ⎡ ⎤ ⎤ ⎡ 0 0 0 1 0 0 0 0 ⎥ ⎥ ⎢ ⎢ C = ⎣ 0 1 0 0 0 ⎦ D = ⎣ 0 0 0⎦ 1.5750. ¯ S )c(W1 ), 1+v ( ¯ N˜ S )c(W1 )} min{v (M −1. The state-space matrices of the aircraft model are. 0 −0.2909 (B3). (K ¯ (I +GK)−1 G) ¯ S )c(W1 ), v ( ¯ N˜ S )c(W1 )} min{1+v (M. APPENDIX C. 2009 John Wiley & Sons, Ltd.. (B7). 0 0 1 0 0. 0 0 0. Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(21) 1194. A.-C. LEE, Y.-R. PAN AND Y.-Y. HUANG. APPENDIX D 1. Realization of X r with [A, B, C, D] for Example 1 ⎡. −42.7680. ⎢ ⎢ ⎢ −1.4270 ⎢ ⎢ ⎢ 2.0524 ⎢ ⎢ ⎢ ⎢ 96.2007 ⎢ A=⎢ ⎢ 168.9301 ⎢ ⎢ ⎢ ⎢ 16.0264 ⎢ ⎢ ⎢ 1.9211 ⎢ ⎣ 31.9141 ⎡. −1.4270. 3.1844. 0. −1. −17.4481. 3.1461. 0. 0.0705. 3.3173. −69.2723. 1. 0. 49.0611. 0. 0. 0. −0.3718 2.1909 0. 0. −632.9635 −.8556 −1.0130. 13.6918. 0. 21.6927. −217.1787. 1.0532. −0.6859. 4.8780. 0. 1.5231. −11.0802. 0. 0. 0. 0. −2.8493. −1.2976. 0. 0. 0. 0. .0024. 22.6255. 0. 0. 0. 0. 0. 0. 0. 0. 0. 3.0984. ⎤. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ −5.1588⎥ ⎥ ⎥ −0.2268⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎦ 0 0. ⎤. ⎥ ⎢ ⎥ ⎢ −2.8800 12.0000 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎢ 106.0560 0 −39.9600⎥ ⎥ ⎢ ⎥ B =⎢ ⎥ ⎢ ⎢ 37.8000 0 −1.7568 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 3.0984 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 2.1909 0 ⎦ ⎣ ⎡. −0.6030 −0.0916. 0.1276. −0.7958 −0.0313 −1.0234 −0.1524 0.3435. 0.0002. ⎡. 1 0 0. 0.9904. 0.0708. 0.0236. 0.0082. ⎤ −0.0006 −0.0015 ⎥ 0.0234 −0.0007 0.1790 −0.0017⎥ ⎦ 0.1474. ⎢ C =⎢ ⎣ −0.0557. 0.6053. 0.0006. 0.1287. ⎤. ⎢ ⎥ ⎥ D=⎢ ⎣ 0 1 0⎦ 0 0 1. Copyright q. 2009 John Wiley & Sons, Ltd.. Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

(22) 1195. ROBUST OBSERVER–CONTROLLER COMPENSATOR DESIGN. ⎡. 2. Realization of Yr with [A, B, C, D] for Example 1. A. 42.7680. 1.4270. −2.0524. ⎤. ⎢ ⎥ ⎢ ⎥ ⎢ 1.4270 17.3943 −3.3173 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −2.0524 −3.3173 69.27230 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −96.2007 −49.0126 632.9635 ⎥ ⎢ ⎥ B =⎢ ⎥ ⎢ −168.9301 −21.9836 217.1787 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −16.0264 −1.5231 11.0802 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −1.9211 2.8493 1.2976 ⎥ ⎢ ⎥ ⎣ ⎦ −31.9141 −0.0024 −22.6255 C. 0. 0. 0. 0. 3.0984. ⎢ ⎥ ⎢ −2.8800 ⎥ 12 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 ⎢ ⎥ ⎢ ⎥ 0 −39.9600⎥ ⎢ 106.0560 ⎥ B =⎢ ⎢ ⎥ 0 −1.7568 ⎥ ⎢ 37.8000 ⎢ ⎥ ⎢ 3.0984 ⎥ 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 2.1909 0 ⎣ ⎦. is the same as A of Appendix D.1. ⎡. ⎤. 0. C. is the negative C of Appendix D.1 ⎤ ⎡ 1 0 0 ⎥ ⎢ D = ⎣ 0 1 0⎦ 0 0 1. is the same as C of Appendix D.1.. ⎤ 0 0 0 ⎥ ⎢ ⎥ ⎢ D = ⎢ 0 0 0⎥ ⎦ ⎣ ⎡. ACKNOWLEDGEMENTS. This study was supported by the Republic of China National Science Council under contract number NSC 95-2221-E009-181.. 0 0 0 3. Realization of M S with [A, B, C, D] for Example 1 ⎡ 0 0 1.1320 0 ⎢ ⎢ 1.2885 0.1056 ⎢ −1.0689 −12.2025 ⎢ ⎢ ⎢ 0 0 0 1 ⎢ ⎢ ⎢ ⎢ 32.1579 8.5154 −105.1063 −14.4611 ⎢ A=⎢ ⎢ 21.3971 3.1176 −24.6786 −1.9826 ⎢ ⎢ ⎢ ⎢ 1.8685 0.2839 −1.8754 −0.2195 ⎢ ⎢ ⎢ 0.1219 −2.1699 −0.0518 −0.0180 ⎢ ⎣ 2.4656 0.09697 3.1722 0.4723 Copyright q. 2009 John Wiley & Sons, Ltd.. −1 0.2145 0 −2.9165 −5.6525 −0.4566 −0.0512 −1.0641. 0. 0. 0.0038. 0.0412. 0. ⎤. ⎥ ⎥ 0.1498 ⎥ ⎥ ⎥ ⎥ 0 0 0 ⎥ ⎥ ⎥ 0.1674 0.0920 0.1498 ⎥ ⎥ ⎥ 0.05659 0.02497 0.0579 ⎥ ⎥ ⎥ ⎥ −0.3954 0.0020 0.0048 ⎥ ⎥ ⎥ 0.0015 −0.3921 0.0037 ⎥ ⎥ ⎦ −0.0008 −0.0019 −0.3989. Int. J. Robust Nonlinear Control 2010; 20:1176–1196 DOI: 10.1002/rnc.

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