Two-step Design of Critical Control Systems Using Disturbance Cancellation Integral Controllers

International Journal of Automation and Computing 8(1), February 2011, 37-45 DOI: 10.1007/s11633-010-0552-2 Two-step Design of Critical Control Systems Using Disturbance Cancellation Integral Controllers Tadhi Ishihara 1 Takahiko Ono 2 1 Faculty of Science and Technology, Fukushima University, Fukushima 960-1296, Japan 2 Graduate School of Information Sciences, Hiroshima City University, Hiroshima 731-3194, Japan Abstract: An efficient critical control system design is proposed in this paper. The key idea is to decompose the design problem into two simpler design steps by the technique used in the clsical loop transfer recovery method (LTR). The disturbance cancellation integral controller is used a bic controller. Since the standard loop transfer recovery method cannot be applied to the disturbance cancellation controller, the nonstandard version recently found is used for the decomposition. Exogenous inputs with constraints both on the amplitude and rate of change are considered. The majorant approach is taken to obtain the analytical sufficient matching conditions. A numerical design example is presented to illustrate the effectiveness of the proposed design. Keywords: Critical control systems, principle of inequalities, principle of matching, majorants, disturbance cancellation controller, integral controller, loop transfer recovery (LTR). 1 Introduction As a new framework of control systems design, Zakian [1 6] proposed the principle of matching, which requires that the system comprising the plant and the controller should be matched with the environment generating exogenous inputs to the system. A typical problem that can be dealt with the principle is the design of critical control systems the controlled responses of interest should be kept within prescribed ranges for all possible exogenous inputs. It is worth noting that the new framework is also useful to design other control systems by regarding the environment a design object, e.g., [5]. The control system design bed on the new framework is achieved by finding a controller satisfying a set of inequalities called the practical matching condition. A controller satisfying the practical matching condition is usually found by a numerical search on tuning parameters of a fixed structure controller. The practical matching condition is usually expressed nonconvex inequality constraints on the controller parameters. Successful use of heuristic search methods, such the moving boundaries method [7], the genetic algorithm, and the simulated annealing have been reported (see [6] and the references therein), but these search methods cannot be used to conclude that the design problem h no solutions when they cannot locate a solution. The node array method [6] h been successful in achieving systematic exhaustive search. However, the method requires an enormous amount of computation the number of the tuning parameters increes. Recently, Ishihara and Ono [8] proposed a new method for designing critical control systems, which does not require extensive numerical search. The key idea is to decompose the design problem into two simpler design steps by the technique used in the clsical loop transfer recovery (LTR) method, e.g., [9, 10]. In the first step, the state Manuscript received March 9, 2009; revised June 9, 2010 This work w supported by Grants-in-Aid for Scientific Research (No. 20560209). feedback controller satisfying the matching condition is designed by using the weighting coefficient of the quadratic performance index a tuning parameter. The output feedback controller is obtained in the second step by the formal recovery procedure using the variance of the fictitious white noise process a tuning parameter. In each step, a small number of the tuning parameters are sufficient to determine the controller parameters guaranteeing the closed loop stability. Since the integral action of the controller is required to deal with rate-limited exogenous signals [5, 6], the work sumes the use of the Davison-type integral controller [11] under the linear quadratic Gaussian (LQG) framework, e.g., [12]. In this paper, a more efficient two-step design of the critical control systems is proposed. Instead of Davison-type controller, the integral controller bed on the disturbance cancellation [13 17] is adopted a bic controller structure. Although the disturbance cancellation integral controller can be designed by the LQG framework, the standard LTR procedure cannot be used due to the nonstandard controller structure. It is shown in [15] that the difficulty can be remedied by a simple modification of the standard LTR method, which makes it possible to utilize the key idea in [8] for the disturbance cancellation integral controller. The proposed design h a merit over the previous work in that the two-degree-of-freedom structure of the disturbance cancellation controller can be utilized in the first step of the design. In addition, exogenous inputs with constraints both on the amplitude and the rate of change, which are more general than that in the previous work [8], are considered in this paper. Unfortunately, the exact practical matching condition for the cls of the exogenous inputs cannot be obtained in analytical form. The majorant approach [2, 5, 6] is adopted to obtain sufficient matching conditions in analytical form. It is worth noting that, by virtue of the LTR procedure, the controller designed by the proposed method automatically h sufficient stability margins.

38 International Journal of Automation and Computing 8(1), February 2011 The organization of the paper is follows. The problem formulation is given in Section 2, the critical control problem and the disturbance cancellation controller are introduced. In Section 3, the two-step design of the critical control system using the disturbance cancellation controller is proposed. An illustrative design example is presented in Section 4. Concluding remarks are given in Section 5. 2 Problem formulation 2.1 Critical control problem Although the critical control system design proposed in this paper can be applied to multivariable plants, the design method is described for the scalar plant to simplify notations. Consider the critical control system design [1 6] for the control system shown in Fig. 1, G(s) is a scalar plant, C(s, p) is a two input controller with the tuning parameter vector p, r is a scalar reference input, and d is a scalar disturbance. 2.2 Disturbance cancellation controller The integral controller bed on the disturbance cancellation [13 17] is used a controller in Fig. 1. After a brief introduction of the disturbance cancellation controller, some closed form expressions required for the critical control system design are given. Although the plant is sumed to be scalar to simplify notation, the results are given in the form that the corresponding multivariable results can be eily obtained. Consider a state space representation of the plant with a disturbance ẋ(t) = Ax(t) + b[u(t) + d(t)], y(t) = c T x(t) (5) x(t) is an n-dimensional state vector, u(t) is a scalar control input, y(t) is a scalar output, and d(t) is a scalar disturbance. Note that the disturbance d(t) in Fig. 1 is sumed to belong to the set F (M d, D d ). To design integral controller bed on the disturbance cancellation, the disturbance d(t) is temporarily sumed to be a step signal satisfying d(t) = 0. (6) In terms of (A, b, c T ), the plant transfer function is written G(s) c T (si A) 1 b. (7) Fig. 1 Critical control system Define the set of functions with constraints on the amplitude and the rate of change F (M, D) {f : f M, f D, f(0) = 0} (1) M and D are positive scalars. It is sumed that the reference r and the disturbance d satisfy r F (M r, D r), d F (M d, D d ) (2) respectively. The responses of interest are sumed to be the tracking error e(t) r(t) y(t) (3) and the control input u. It is required that the magnitude of the tracking error and that of the control input remain within the prescribed bounds ε e and ε u, respectively, for all possible exogenous inputs. The requirement is equivalent to the peak norm condition: φ e(p) sup r, d e ε e, φ u(p) sup u ε u (4) r, d which is called the matching condition. For the inputs satisfying (2), it is difficult to express the peak norms in (4) in analytical form. Zakian [1,2,5] suggested the use of the upper bounds of the peak norms called majorants to obtain a sufficient matching condition in analytical form. The use of the majorants for the matching condition (4) with an integral controller will be discussed in Section 2.3. It is sumed that (A, b, c T ) satisfies the following conditions: Assumption 1. (A, b, c T ) is a minimal realization and G(s) is nonsingular for almost all s. Assumption 2. (A, b, c T ) is the minimum phe. Assumption 3. (A, b, c T ) h no zero at s = 0. From (5) and (6), the extended plant can be constructed ξ(t) = Φξ(t) + γu(t), y(t) = θ T ξ(t) (8) ξ(t) [ d(t) x(t) ], Φ [ 0 0 b A ], γ [ 0 b ], θ T [0 c T ]. (9) It can be eily checked that the pair (θ T, Φ) is observable, but (Φ, γ) is not stabilizable under Assumptions 1 and 3. By the observability of (θ T, Φ), it is possible to construct an observer for estimating the state vector ξ(t). A full order observer is given by ˆξ(t) = Φˆξ(t) + γu(t) + k[y(t) θ T ˆξ(t)] (10) ˆξ(t) [ ˆd T (t) ˆx T (t)] T (11) is the estimate of ξ(t), and k is an observer gain matrix. The Kalman filter theory can be used to determine the observer gain matrix k by introducing a stochtic model ξ(t) = Φξ(t) + γu(t) + γw(t) y(t) = θ T ξ(t) + v(t) (12) v(t) and w(t) are mutually independent zero-mean white noise processes with the variances given by σ v > 0 and σ w > 0, respectively, and γ is chosen such that the pair

T. Ishihara and T. Ono / Two-step Design of Critical Control Systems Using Disturbance 39 (Φ, γ) is stabilizable. Then, the observer gain matrix can be determined by solving the Riccati equation. In the previous works [13 16], the disturbance cancellation control law h been given a priori without optimality consideration. Recently, it is shown in [17] that, under the perfect observation of x(t) and d(t), the optimal control law for the quadratic performance index J ξ 0 {ρy 2 (t) + [d(t) + u(t)] 2 }dt (13) ρ is a positive weighting coefficient, and is obtained u(t) = f T x(t) d(t) (14) f T is an optimal gain for the standard quadratic performance index J x = 0 [ρy 2 (t) + u 2 (t)]dt. (15) Note that the optimal control (14) stabilizes the plant but not the extended plant (8), which is obviously unstabilizable. By the separation principle, the output feedback controller including the term for the reference input can be constructed u(t) = f Tˆx(t) ˆd(t) + t f r(t) (16) ˆx(t) and ˆd(t) are the estimates of x(t) and d(t), respectively, generated by the observer (10), and t f is a precompensation gain for the reference input given by t f = [c T ( A + bf T ) 1 b] 1. (17) The structure of the disturbance cancellation controller is shown in Fig. 2. k(s) s 1 bk d + k x. (21) It follows from (16) (21) that the control input can be written u(s) = C r(s)r(s) + C y(s)y(s) (22) C r(s) [1 + s 1 k d c T (si A + k xc T ) 1 b] (23) [1 + f T (si A + k xc T ) 1 b] 1 t f C y(s) [1 + s 1 k d c T (si A + k xc T ) 1 b] [1 + f T (si A + k xc T ) 1 b] 1 f T (si A + k xc T ) 1 k x s 1 k d [1 c T (si A + k xc T ) 1 k x]. (24) Note that the Laplace transform of the output is expressed y(s) = G(s)[u(s) + d(s)]. (25) From (21), (24), and (25), the transfer functions from the exogenous inputs r(t) and d(t) to the tracking error e(t) and the control input u(t), with obvious notations, can be expressed G er(s) 1 G(s)[1 C y(s)g(s)] 1 C r(s) (26) G ed (s) G(s)[1 C y(s)g(s)] 1 (27) G ur(s) [1 C y(s)g(s)] 1 C r(s) (28) G ud (s) [1 C y(s)g(s)] 1 C y(s)g(s). (29) Note that the term [1 C y(s)g(s)] 1 can be expressed [1 C y(s)g(s)] 1 =[1 + f T (si A) 1 b] 1 [1 + f T (si A + k xc T ) 1 b] [1 + s 1 k d c T (si A + k xc T ) 1 b] 1 (30) the lt term [1 + s 1 k d c T (si A + k xc T ) 1 b] 1 h zero at s = 0 under the Assumptions 1 3 provided k d 0. In terms of the feedback gain f, the observer gains k x and k d and the pre-compensator gain t f ; the transfer functions (26) (29) can be explicitly expressed Fig. 2 Disturbance cancellation controller Define the partition of the observer gain k k [k T d k T x ] T (18) k d is a scalar, and k x is an n-dimensional vector. It can be eily checked that the Laplace transform of the estimates generated by the observer can be written ˆx(s) =[si A + k(s)c T ] 1 [bu(s) + k(s)y(s)] (19) ˆd(s) = s 1 k d c T [si A + k(s)c T ] 1 bu(s)+ s 1 k d { 1 c T [si A + k(s)c T ] 1 k(s) } y(s) (20) G er(s) =1 c T (si A + bf T ) 1 bt f (31) G ed (s) = c T (si A + bf T ) 1 b [1 + f T (si A + k xc T ) 1 b] [1 + s 1 k d c T (si A + k xc T ) 1 b] 1 (32) G ur(s) =[1 + f T (si A) 1 b] 1 t f (33) G ud (s) =[1 + f T (si A) 1 b] 1 [1 + f T (si A + k xc T ) 1 b] [1 + s 1 k d c T (si A + k xc T ) 1 b] 1 1. (34) Some important observations are obtained from the above expressions, which are summarized follows. Lemma 1. The transfer functions G er(s) and G ur(s) are independent of the observer gain k x and k d. Under the Assumptions 1 3 and k d 0, G er(s) and G ed (s) have a zero

40 International Journal of Automation and Computing 8(1), February 2011 at s = 0, G ur(s) h a zero at s = 0 if the plant h a pole at s = 0 and G ud (s) h no zero at s = 0, but G ud (s) + 1 h a zero at s = 0. Note that the disturbance cancellation controller (16) is determined by the state feedback gain f T and the observer gain k. The LQG method is used for determining both gains. On the sumption that the variance of the observation noise v(t) is given by σ v = 1, the tuning parameter p of the controller is defined p = [ρ γ T σ w] T (35) ρ is the weighting coefficient of the quadratic performance index, σ w is the variance of the white noise w(t), and γ is the input vector for w(t) in the stochtic model (12). If the parameter p is specified, the optimal feedback gain f T and the Kalman filter gain k are determined by solving the Riccati equations. The design of the critical control system is to find p, which gives a controller satisfying the matching condition (4). In Section 3, efficient determination of the parameter p in two steps is proposed. 2.3 Majorants with integral action The zero structure given by Lemma 1 is introduced by the integral action of the disturbance cancellation controller, which is now used to obtain majorants for the critical control system design follows. Proposition 1. Assume that the tuning parameter p is chosen such that A bf T and Φ kθ T are stable matrices. 1) Define ˆφ e(p) g er(h, p) 1D r + g ed (h, p) 1D d (36) g er(h, p) and g ed (h, p) are unit step responses of the transfer function G er(s) and G ed (s). Then, ˆφ e(p) is a majorant of φ e(p), i.e., 2) Define φ e(p) ˆφ e(p). (37) ˆφ u(p) g ur(p) M r + g ur(h, p) g ur(p) 1D r+ M d + g ud (h, p) + 1 1D d (38) g ur(h, p) and g ud (h, p) denote unit step responses of the transfer function G ur(s) and G ud (s), respectively, and g ur(p) [1 f T ( A + bf T ) 1 b]t f (39) is the steady state value of g ur(h, p). Then, ˆφ u(p) is a majorant of φ u(p), i.e., φ u(p) ˆφ u(p). (40) If the plant (5) h a pole at s = 0, the majorant (38) is simplified to ˆφ u(p) = M d + g ur(h, p) 1D r + g ud (h, p) + 1 1D d. (41) Proof. See Appendix. Remark 1. In the early work [1] on the critical systems, Zakian obtained the analytical formul for computing upper bounds on the control and the error and called them majorants in [2] before he gave the definition of critical control systems in [3]. Since the structure of the system in [1] is different from that considered in this paper, the upper bounds obtained in [1] cannot directly be used for the present problem. Remark 2. The response g ud (h, p) + 1 (t 0) in ˆφ u(p) is the step response of the transfer function G ud (s, p) + 1, which h a zero at s = 0 stated in Lemma 1. Note that G ud (s, p) + 1 can be regarded the transfer function from the disturbance d to the plant input d + u. Remark 3. The step responses required to compute the majorants ˆφ e(p) and ˆφ u(p) are obtained the impulse responses of strictly proper transfer functions since the Laplace transform of the step signal is cancelled by the zero at s = 0. The L 1 norms of the impulse response of strictly proper transfer functions can be computed by the state space method proposed by Rutland and Lane [18]. 3 Two-step design In this section, the two-step design is proposed for the efficient determination of the controller parameters satisfying the matching conditions. The key idea is, in our previous work [8], to decompose the design problem into two simpler steps. However, the standard LTR method used in [8] cannot be applied to the disturbance cancellation controller. The nonstandard LTR method for disturbance cancellation controllers [13 17] is used for the decomposition. In the first step, the state feedback version of the disturbance cancellation controller is designed a target. The feibility of the design specifications (4) can be checked in this step. The second step obtains the output feedback controller satisfying the design specifications by the formal procedure. 3.1 First step The target controller is constructed follows: sume that the plant state x(t) is meurable. The plant and the disturbance model described by (5) and (6) can be rewritten d(t) = 0, ẋ(t) Ax(t) bu(t) = bd(t). (42) It follows from the above expression that the estimator for the disturbance d(t) bed on the observation of the plant state x(t) can be constructed d(t) = k T d [ẋ(t) Ax(t) bu(t) b d(t)] (43) d(t) is an estimate of the disturbance d(t), and k d T is an estimator gain. Note that the number of the observed variables is larger than that of the variable to be estimated. It can be eily confirmed that the behavior of the estimation error is determined by k d T b rather than k d T, and that the estimation error converges to zero if and only if the gain k d T is chosen such that the scalar k d T b is positive. The target controller with the reference input term is defined u(t) = f T x(t) d(t) + t f r(t) (44) the structure of which is shown in Fig. 3. Remark 4. The target controller is used only in the design stage and never used a controller in real world.

T. Ishihara and T. Ono / Two-step Design of Critical Control Systems Using Disturbance 41 In the numerical evaluation of the target responses using a software like Matlab/Simulink, the derivative ẋ(t) required on the right side of (43) can be obtained an input signal to an integrator. Note that the target controller is determined by the weighting coefficient ρ in the quadratic performance index (15) and λ in (54). From Lemma 2, the majorants for the target controller design can be eily obtained by replacing the responses in Proposition 1 with appropriate target responses. The result is summarized follows. Proposition 2. For the target controller (46) with the tuning parameters ρ and λ, denote the performance indices in (4) by φ e(ρ, λ) and φ u(ρ, λ). Define φ e(ρ, λ) = ḡ er(h, ρ) 1D r + ḡ ed (h, ρ, λ) 1D d (55) φ u(ρ, λ) = ḡ ur(ρ) M r + ḡ ur(h, ρ) ḡ ur(ρ) 1D r+ M d + ḡ ud (h, ρ, λ) + 1 1D d (56) Fig. 3 Target controller Remark 5. It can be shown that the target controller in Fig. 3 h large stability margins the standard LQ regulators provided k T d b is positive. It follows from (43) that the Laplace transform of d(t) can be written d(s) = (s + k T d b) 1 kt d [(si A)x(s) bu(s)]. (45) From (44) and (45), the Laplace transform of the control input can be expressed u(s) = C r(s)r(s) + C x(s)x(s) (46) C r(s) = s 1 (s + k T d b)t f (47) C x(s) = s 1 (s + k T d b)[f T + (s + k T d b) 1 kt d (si A)]. (48) Note also that x(s) = (si A) 1 b[u(s) + d(s)], y(s) = c T x(s). (49) From the expressions (46) (49), the target transfer functions corresponding to (31) (34) for the output feedback ces are obtained Ḡ er(s) = 1 c T (si A + bf T ) 1 bt f (50) Ḡ ed (s) = s(s + λ) 1 c T (si A + bf T ) 1 b (51) Ḡ ur(s) = [1 + f T (si A) 1 b]t f (52) Ḡ ud (s) = s(s + λ) 1 [1 + f T (si A) 1 b] 1 1 (53) λ k T d b. (54) From the above expressions, the result corresponding to Lemma 1 is obtained follows. Lemma 2. The transfer functions Ḡer(s) and Ḡur(s) are independent of the estimator gain k d and equal to G er(s) given by (31) and G ur(s) given by (33), respectively. Under the Assumptions 1 3 and k d 0, Ḡ er(s) and Ḡed(s) have a zero at s = 0, Ḡ ur(s) h a zero at s = 0 if the plant h a pole at s = 0, and Ḡud(s) h no zero at s = 0, but Ḡ ud (s) + 1 h a zero at s = 0. ḡ er(h, ρ), ḡ ur(h, ρ), ḡ ed (h, ρ, λ), and ḡ ud (h, ρ, λ) are the unit step responses of Ḡ er(s), Ḡ ur(s), Ḡ ed (s), and Ḡ ud (s), respectively, and ḡ ur(ρ) is the steady state value of ḡ ur(h, ρ). Then, φ e(ρ, λ) and φ u(ρ, λ) are the majorants satisfying φ e(ρ, λ) φ e(ρ, λ), φ u(ρ, λ) φ u(ρ, λ). (57) If the matrix A in (5) h a zero eigenvalue, the majorant (56) can be simplified to φ u(ρ, λ) = M d + ḡ ur(h, ρ) 1D r+ ḡ ud (h, ρ, λ) + 1 1D d. (58) Note that the majorants (55), (56), and (58) can be eily evaluated for given ρ and λ by computing the L 1 norms of the step responses. From (57), a matched target controller is obtained by finding the tuning parameters ρ and λ satisfying φ e(ρ, λ ) ε e, φu(ρ, λ ) ε u. (59) The parameters satisfying (59) can be found by a direct numerical search. However, it is more efficient to make use of the two-degree-freedom structure of the target controller with the ymptotic behavior of the responses with respect to the parameters λ. The procedure is given follows. Proposition 3. Consider the inequalities ḡ er(h, ρ) 1D r < ε e (60) ḡ ur(ρ) M r + ḡ ur(h, ρ) ḡ ur(ρ) 1D r + M d < ε u. (61) Assume that ρ satisfies the inequalities (60) and (61). Then, there exists sufficiently large λ such that ρ and λ satisfy the matching conditions (59). Proof. The inequalities (60) and (61) are obtained by deleting the terms ḡ ed (h, ρ, λ) 1 and ḡ ud (h, ρ, λ) + 1 1 from the inequalities in (59). Note that, for the fixed ρ, ḡ ed (h, ρ, λ) 1 0 and ḡ ud (h, ρ, λ) + 1 1 0 λ, which follows from (51) and (53). By using these ymptotic properties together with (55), (56), (60), and (61), it can be eily shown that ρ and λ given by the procedure satisfy the matching conditions (59). Remark 6. Intuitively, ρ increes, ḡ er(h, ρ) 1 in (55) decrees, while ḡ ur(h, ρ) ḡur(ρ) 1 in (56) increes. In addition, it follows from the well-known cheap control result [8] that ḡ er(h, ρ) 1 0 and ḡ ur(h, ρ) ḡur(ρ) 1

42 International Journal of Automation and Computing 8(1), February 2011 ρ. Therefore, for sufficiently small bounds ε e and ε u, ρ satisfying (60) and (61) fails to exist. If the inequalities (60) and (61) have no solution, a matched target controller does not exist. It is worth noting that the feibility of the design specifications can be checked by the inequalities (60) and (61), which includes the responses only for the reference input r. 3.2 Second step In the second step of the design, the output feedback controller satisfying the matching condition (4) is constructed using the parameters ρ and λ determined in the first step and simple one-dimensional search. Note that Proposition 2 guarantees that the tuning parameter p satisfying ˆφ e(p) ε e, ˆφu(p) ε u (62) ˆφ e(p) and ˆφ u(p) are majorants given by (36) and (38), respectively, providing an output feedback controller satisfying the matching condition. The feedback gain matrix f corresponding to ρ is used in the output feedback controller. The Kalman filter gain is determined by the stochtic model (12) with γ = [λ b T ] T. (63) The variance σ w of the white noise process w(t) in (12) is taken the tuning parameter in the second step suming that the variance σ v of the observation noise v(t) is unity. The tuning parameter vector for the output feedback controller is written p = [ρ ( γ ) T σ w] T (64) the tuning parameter in the second step is only the lt element σ w. The Kalman filter gain is written k(σ w) = Πθ (65) Π is a solution of the Riccati equation ΠΦ T + ΦΠ Πθθ T Π + σ w γ ( γ ) T = 0. (66) As shown in [14, 15] for the discrete-time ce, under the Assumptions 1 3, the pair (Φ, γ ) with λ > 0 is stabilizable and (θ T, Φ) is detectable and that the invariant zeros of the realization (Φ, γ, θ T ) are in the open left plane. These results guarantee that the Kalman filter gain (64) satisfies the ymptotic property: k(σ w) σ 1 2 w γ. (67) The matching condition (62) for the majorants is rewritten ˆφ e(σ w) = ḡ er(h, ρ ) 1D r+ g ed (h, ρ, γ, σ w) 1D d ε e (68) ˆφ u(σ w) = ḡ ur(ρ ) M r + ḡ ur(h, ρ ) ḡ ur(ρ ) 1D r+ M d + g ud (h, ρ, γ, σ w) + 1 1D d ε u (69) input r are independent of the parameter σ w and already fixed by ρ determined in the first step. Define φ e(σ w) ˆφ e(σ w) φ e(ρ, λ ) (70) φ u(σ w) ˆφ u(σ w) φ u(ρ, λ ). (71) Then, it follows from (55), (56), (68), and (69) that φ e(σ w) = g ed (h, ρ, γ, σ w) 1D d ḡ ed (h, ρ, λ ) 1D d (72) φ u(σ w) = g ud (h, ρ, γ, σ w) + 1 1D d g ud (h, ρ, λ ) + 1 1D d. (73) The transfer functions related to φ e(σ w) and φ u(σ w) have the following ymptotic properties. Lemma 3. Let G ed (s, σ w) and G ud (s, σ w) denote the transfer functions defined in (32) and (34), respectively, corresponding to the feedback gain f T determined by the weight ρ and the observer gain given by (64). Then, the transfer functions satisfy G ed (s, σ w) Ḡ ed(s) G ud (s, σ w) Ḡ ud(s), σ w (74) Ḡ ed(s) and Ḡ ud(s) are defined in (51) and (53), respectively, with λ = λ and f T determined by ρ. Proof. Define the partition of the Kalman filter gain (64) k(σ w) = [kd T (σ w) kx T (σ w)] T. (75) It follows from (63) and (66) that k d (σ w) σ 1 2 w λ, k x(σ w) σ 1 2 w b (76) for sufficiently large σ w. It follows from (76) that two matrices common in G ed (s, σ w) and G ud (s, σ w) satisfy the following ymptotic properties: [si A + k x(σ w)c T ] 1 b = (si A) 1 b[1 + σ 1 2 w c T (si A) 1 b] 1 0, σ w s 1 k d (σ w)c T [si A + k x(σ w)c T ] 1 b = (77) s 1 σ 1 2 w λ c T (si A) 1 b[i + σ 1 2 w c T (si A) 1 b] 1 s 1 λ, σ w. (78) The ymptotic properties in (74) are obtained by using (77) and (78) in (32) and (34). From the above lemma, the second step in achieving the design specification (4) is stated follows. Proposition 4. Assume that the parameters ρ and λ are determined such that the matching conditions (59) for the state feedback ce are satisfied. Consider the performance indices ˆφ e(σ w) and ˆφ u(σ w) given by the left sides of (68) and (69), respectively, for the output feedback controller with the feedback gain f T determined by the weight ρ and the observer gain given by (64). Then, which explicitly shows the dependency on the tuning parameters. Note that the responses related to the reference ˆφ e(σ w) φ e(ρ, λ ) ˆφ u(σ w) φ u(ρ, λ ), σ w (79)

T. Ishihara and T. Ono / Two-step Design of Critical Control Systems Using Disturbance 43 φ e(ρ, λ ) and φ u(ρ, λ ) satisfy the matching condition (59) for the state feedback ce. Consequently, the output feedback controller ymptotically satisfies the matching conditions. Proof. First, consider φ e (σ w) defined in (70). From (72), φ e (σ w) includes the unit step responses g ed (h, ρ, γ, σ w) and ḡ ed (h, ρ, λ ), which are the inverse Laplace transforms of G ed (s, σ w)/s and that of Ḡ ed(s)/s, respectively. From Lemma 3, G ed (s, σ w)/s converges to Ḡ ed(s)/s σ w pointwise in s. In addition, both Laplace transforms are strictly proper and stable since G ed (s, σ w) and Ḡed(s) have a zero at s = 0 seen from Lemm 1 and 2. It follows from these results that the step response g ed (h, ρ, γ, σ w) converges uniformly to ḡ ed (h, ρ, λ ) σ w. The uniform convergence implies the convergence of the norm g ed (h, ρ, γ, σ w) 1 to ḡ ed (h, ρ, λ ) 1, which implies that φ e(σ w) 0 σ w. Similarly, for φ u(σ w) defined in (71), it can be shown that φ u(σ w) 0 σ w. The ymptotic properties (79) follow from φ e(σ w) 0 and φ u(σ w) 0. 4 Design example A simple design example is presented to illustrate the procedure and the effectiveness of the proposed design method. An earth scanning satellite antenna control problem discussed by Whidborne and Liu [19] is considered. The plant transfer function is given by G(s) = 27697 s(s 2 + 1429s + 42653). (80) Since the plant h a pole at s = 0, the amplitude constraint on the reference input r is not required in the matching condition bed on the majorants pointed out in Propositions 1 and 2. The reference input r is sumed to belong to the set of the rate limited functions with D r = 1. The amplitude and rate constraints of the disturbance d are sumed to be M d = 12 and D d = 3, respectively. The design specifications are given by ε e = 0.020944 rad, ε u = 19.5 V. (81) The first step of the design follows the procedure given in Proposition 3. Note that ḡ ur(ρ) = 0 since the plant (80) h a pole at the origin so that the inequality (61) is rewritten ḡ ur(h, ρ) 1D r + M d < ε u. (82) It follows from (59), (81), and (82) that the inequalities (55) and (56) for the determination of ρ are reduced to the norm constraints Fig. 4 Target L 1 norms versus ρ For the fixed ρ, λ h to be determined to satisfy the matching condition (62). It is confirmed numerically that the norms ḡ ed (h, ρ, λ) 1 and ḡ ud (h, ρ, λ)+1 1 are monotone decreing functions of λ. For example, for λ = 1, the norms are computed ḡ ed (h, ρ, λ ) 1 = 0.0016 ḡ ud (h, ρ, λ ) + 1 1 = 0.0101 (85) which are obviously sufficient for the matching condition (59) with (81). The first design step is completed with ρ = 4 10 5 and λ = 1. To check the performance of the target system, the following test signals satisfying the sumptions are considered: r(t) = 1 sin(2t), 0 t 15 (86) 2 0, 0 t 5 d(t) = 3(t 5), 5 t 9 (87) 12, 9 t 15. For the target system, the response of error to the above test signals is shown in Fig. 5 together with the response for the choice λ = 0, which corresponds to the response without the integral action. Note that the disturbance (87) is inserted at t = 5 s. It is seen that the response of the target system is within the admissible bound ε e 0.02, while the state feedback controller without the integral action fails to satisfy the design specification on the error against the disturbance. ḡ er(h, ρ) 1 < 0.02094, ḡ ur(h, ρ) 1 < 7.5. (83) The plot of ḡ er(h, ρ) 1 and that of ḡ ur(h, ρ) 1 are shown in Fig. 4, the L 1 norms are computed by using the result of [18] for the minimal realizations of Ḡ er(s)/s and Ḡ ur(s)/s. From the plots in Fig. 4, it turns out that the norm constraints (83) are satisfied for a wide range of ρ, roughly, 1.5 10 5 < ρ < 2 10 6. As an example, choose ρ = 4 10 5, then the norms are computed ḡ er(h, ρ ) 1 = 0.015, ḡ ur(h, ρ ) 1 = 4.9927. (84) Fig. 5 Errors of the target system for the test signal

44 International Journal of Automation and Computing 8(1), February 2011 The response of the control input for the target system and that for λ = 0 are shown in Fig. 6, the two responses are indistinguishable in the scale of the figure satisfying the design specification ε u = 19.5. Fig. 7 Errors of the output feedback system Table 1 Performance indices versus σ w Fig. 6 Control input for the test signal The second step of the design provides the matched output feedback controller by the formal procedure given in Proposition 4. For the plant (80), the performance indices (68) and (69) for the output feedback controller are given by ˆφ e(σ w) = ḡ er(h, ρ ) 1 + 3 g ed (h, ρ, γ, σ w) 1 (88) ˆφ u(σ w) = ḡ ur(h, ρ ) 1+ 3 g ud (h, ρ, λ, σ w) + 1 1 + 12. (89) Note that the first terms in (88) and (89) are given by (84). On the other hand, the norms g ed (h, ρ, γ, σ w) 1 and g ud (h, ρ, λ, σ w) + 1 1 in the second terms can be computed by using the result of [14] for the minimal realizations of G ed (s)/s and [G ud (s) + 1]/s, respectively. For several values of the tuning parameter σ w, the performance indices are summarized in Table 1, which shows that the output feedback controller satisfying the design specification (81) is obtained by choosing σ w 10 10. The error responses of the output feedback controller to the test signals (86) and (87) are shown in Fig. 7 for σ w = 10 5, 10 7, and 10 10. It is confirmed by the test signal that the choice σ w = 10 7 is not sufficient to keep the response within the admissible bound ε e 0.02. The response for σ w = 10 10 is indistinguishable from the target response due to the scale of the figures. The control input is within the bound ε u = 19.5 for σ w = 10 5, 10 7, and 10 10 and the control histories are indistinguishable from those in Fig. 7 due to the scale of the figure expected from the small difference of ˆφ u(σ w) in Table 1. It is worth noting that the output feedback controller h ymptotic stability margins large the target controller. Table 2 compares the stability margins of the output feedback controller for several values of σ w with that of the target controller. It is seen that the stability margins of the output feedback controller approach those of the target controller σ w is increed. Note that the target controller h the theoretically guaranteed stability margins the standard LQ controllers. σ w ˆφe(σ w) ˆφu(σ w) 10 5 0.0387 17.1022 10 7 0.0246 17.0527 10 10 0.0207 17.0293 10 12 0.0201 17.0259 Table 2 Stability margins versus σ w σ w Gain margin (db) Phe margin (degree) 10 5 10.2 46.3 10 7 10.8 44.3 10 10 18.8 58.8 10 12 25.1 63.9 Target 68.8 5 Conclusions The two-step design of critical control systems using the disturbance cancellation controller h been proposed. The proposed method utilizes the fruits of the LQG/LTR technique to decompose the original design problem into two steps. As a desirable by-product of the proposed design, the designed controller provides large stability margins. It is an interesting future topic to sess the effectiveness of the majorants with the numerical evaluation methods, such that found in [6] and the convex optimization approach [20]. Appendix Proof of Proposition 1 1). Note that the time functions g er(h, p) and g ed (h, p) are the inverse Laplace transforms of G er(s, p)/s and that of G ed (s, p)/s, respectively. Since G er(s, p) and G ed (s, p) are stable and have a zero at s = 0 by Lemma 1, the step responses g er(h, p) and g ed (h, p) approach zero time tends to infinity. It follows the Zakian s inequality (e.g., pp. 57 of [1]) that sup e g er(h, p) 1D r r, d=0 e g ed (h, p) 1D d. sup r=0, d The definition of φ e(p) and the linearity imply that φ e(p) = sup e + sup e. r, d=0 r=0, d (A1) (A2)

T. Ishihara and T. Ono / Two-step Design of Critical Control Systems Using Disturbance 45 It follows from (A1) and (A2) that ˆφ e(p) defined in (36) is the majorant of φ e(p). Proof of Proposition 1 2). The time functions g ur(h, p) and g ud (h, p) are the inverse Laplace transforms of G ur(s, p)/s and that of G ud (s, p)/s, respectively. Note that Proposition 1 does not guarantee that G ur(s, p) h a zero at s = 0. It follows from the Zakian s inequality that sup u gur(p) M r + g ur(h, p) gur(p) 1D r. (A3) r, d=0 The steady value of g ur(h, p) is obtained (39) by applying the final value theorem to (33) with the identity [1+f T (si A) 1 b] 1 = 1 f T (si A + bf T ) 1 b. Since G ud (s), which is now written G ud (s, p), h no zero at s = 0 from Lemma 1, it follows (34) that the steady state value of the step response of g ud (h, p) is 1. Therefore, the Zakian s inequality implies that sup u M d + g ud (h, p) + 1 1D d. r=0, d (A4) It follows from (A3) and (A4) with the definition of φ u(p) and the linearity that ˆφ u(p) defined in (38) is the majorant of φ u(p). If the plant h a pole at s = 0, G ur(s, p) h a zero at s = 0 stated in Lemma 1. Then, gur(p) = 0 and the majorant (38) is obviously simplified to (41). References [1] V. Zakian. New formulation for the method of inequalities. Proceedings of the Institution of Electrical Engineers, vol. 126, no. 6, pp. 579 584, 1979. [2] V. Zakian. A Framework for Design: Theory of Majorants. Control Systems Centre Report, 604, University of Manchester Institute of Science and Technology, Manchester, UK, 1984. [3] V. Zakian. Critical systems and tolerable inputs. International Journal of Control, vol. 49, no. 4, pp. 1285 1289, 1989. [4] V. Zakian. Well matched systems. IMA Journal of Mathematical Control and Information, vol. 8, no. 1 pp. 29 38, 1991. [5] V. Zakian. Perspectives on the principle of matching and the method of inequalities. International Journal of Control, vol. 65, no. 1, pp. 147 175, 1996. [6] V. Zakian. Control Systems Design: A New Framework, Berlin, Germany: Springer, 2005. [7] V. Zakian, U. Al-Naib. Design of dynamical and control systems by the method of inequalities. Proceedings of IEE, vol. 120, no. 11, pp. 1421 1427, 1973. [8] T. Ishihara, T. Ono. Design of critical control systems for non-minimum phe plants via LTR technique. IEEJ Transactions on Electronics, Information and Systems, vol. 127, no. 5, pp. 733 740, 2007. [9] G. Stein, M. Athans. The LQG/LTR procedure for multivariable feedback control design. IEEE Transactions on Automatic Control, vol. 32, no. 2, pp. 105 114, 1987. [10] A. Saberi, B. M. Chen, P. Sannuti. Loop Transfer Recovery: Analysis and Design, New York, USA: Springer-Verlag, 1993. [11] E. J. Davison, I. J. Ferguson. The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods. IEEE Transactions on Automatic Control, vol. 26, no. 1, pp. 93 110, 1981. [12] B. D. O. Anderson, J. B. Moore. Optimal Control: Linear Quadratic Methods, New Jersey, USA: Prentice Hall, 1990. [13] G. F. Franklin, J. D. Powell, A. Emami-Naeini. Feedback Control of Dynamic Systems, 2nd ed., USA: Addison- Wesley, 1993. [14] H. J. Guo, T. Ishihara, H. Takeda. Design of discrete-time servo systems using disturbance estimators via LTR technique. Transactions of the Society of Instrument and Control Engineers, vol. 31, no. 5, pp. 646 652, 1996. [15] H. J. Guo, T. Ishihara, H. Takeda. LTR design of discretetime integral controllers bed on disturbance cancellation. In Proceedings of IFAC 13th Triennial World Congress, pp. 295 300, 1996. [16] T. Ishihara, H. J. Guo, H. Takeda. Integral controller design bed on disturbance cancellation: Partial LTR approach for non-minimum phe plants. Automatica, vol. 41, no. 12, pp. 2083 2089, 2005. [17] T. Ishihara, H. J. Guo. LTR design of integral controllers for time-delay plants using disturbance cancellation. International Journal of Control, vol. 81, no. 2, pp. 1027 1034, 2008. [18] N. K. Rutland, P. G. Lane. Computing the 1-norm of the impulse response of linear time-invariant systems. Systems & Control Letters, vol. 26, no. 3, pp. 211 221, 1995. [19] J. F. Whidborne, G. P. Liu. Critical Control Systems: Theory, Design, and Applications, England: Research Studies Press, 1993. [20] W. Silpsrikul, S. Arunsawatwong. Computation of peak output for inputs restricted in and norms using finite difference schemes and convex optimization. International Journal of Automation and Computing, vol. 6, no. 1, pp. 7 13, 2009. Tadhi Ishihara received the Ph. D. degree in electrical engineering from Tohoku University, Sendai, Japan in 1979. From 1977 to 1987, he h been a research sociate at the Department of Electrical Engineering, Tohoku University. Form 1987 to 1993, he h been an sociate professor in the Department of Mechanical Engineering, Tohoku University. From 1993 to 2003, he h been an sociate professor in Graduate School of Information Sciences, Tohoku University. Currently, he is a professor in the Faculty of Science and Technology, Fukushima University, Fukushima, Japan. His research interests include robust control, stochtic adaptive control, and control system design bed on the principle of matching. E-mail: ishihara@sss.fukushima-u.ac.jp (Corresponding author) Takahiko Ono received B. Eng. degree in mechanical engineering from Tohoku University, Japan in 1994 and M. Info. Sc. and Ph. D. degrees in information science from Tohoku University in 1996 and 1999, respectively. He h been a research sociate at Tohoku University from 1999 to 2004. Currently, he is an sociate professor at Graduate School of Information Sciences, Hiroshima City University, Japan. His research interests include robust control, critical control, optimal filtering theory and analysis, and modelling of human reactions to acceleration in vehicle transportation. E-mail: ono@hiroshima-cu.ac.jp