Synthesis of Multipurpose Linear Control Laws of Discrete Objects under Integral and Phase Constraints

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1 ISSN , Automation and Remote Control, 2011, Vol. 72, No. 7, pp Pleiades Publishing, Ltd., Original Russian Text M.M. Kogan, L.N. Krivdina, 2011, published in Avtomatika i Telemekhanika, 2011, No. 7, pp LINEAR SYSTEMS Synthesis of Multipurpose Linear Control Laws of Discrete Objects under Integral and Phase Constraints M. M. Kogan and L. N. Krivdina Nizhni Novgorod University of Architecture and Civil Engineering, Nizhni Novgorod, Russia Received December 16, 2010 Abstract Multipurpose linear discrete control laws are synthesized, which afford a suboptimal value of the goal functional of the integral type or the kind of maximum in time from the square of the norm of a controllable output in the fulfilment of integral and phase constraints. DOI: /S INTRODUCTION This work deals with the synthesis of linear laws of control for linear dynamic objects in discrete time. These control laws, called multipurpose ones, must ensure the fulfilment of constraints involving the fact that values of the functionals of the integral type or the kind of maximum in time from the squares of norms of controllable outputs should not exceed the prescribed bounds, and also must optimize the goal functional representing the weighted sum of functionals of these kinds. The constraints on values of the state and control variables exist for certain in any practical problem and must necessarily be taken into account if the linear model of an object is taken into consideration. Integral constraints commonly correspond to the finiteness of some energy resources and also express requirements for a transient process. As a rule in the presence of constraints, the transition is made to the synthesis of nonlinear control laws (see, for example, [1] and the bibliography in [1]). The specific feature of the approach outlined is the fact that the synthesis is made of the linear control law, at which for the given initial state the phase and integral constraints are fulfilled and the suboptimal value of the goal functional is afforded. In addition, such an ellipsoid is pointed up that for all trajectories of the closed system with initial states the phase and integral constraints are fulfilled in this ellipsoid and the value of the goal functional does not exceed the above-stated value. The synthesis is carried out by means of the approach developed for continuous systems [2 4] and based on the method of Lyapunov functions and the apparatus of linear matrix inequalities [5, 6]. Specifically, sufficient conditions in terms of linear matrix inequalities are given, in the fulfilment of which the trajectory of the closed system with the prescribed initial condition satisfies the prescribed integral and phase constraints. These conditions permit us to obtain the control laws in the form of the linear stationary feedback by the state or the linear dynamic feedback by the measurable output, which afford the fulfilment of constraints and minimize the estimate of the goal functional. Let us note that the control laws obtained are suboptimal in view of the fact that the conditions forming the basis of their synthesis are obtained by the method of Lyapunov functions and are only sufficient. In the specific case of the integral goal functional, in the absence of constraints, these conditions coincide with the necessary and sufficient conditions of existence of the optimal linear-quadratic output control law [7]. As far as the authors know, the necessary and sufficient 1427

2 1428 KOGAN, KRIVDINA conditions in the problem of the optimal control of continuous objects under integral constraints are found in [8], where for the solution, generally speaking, of the nonconvex problem use is made of the theory of convex duality. In another specific case of the goal functional in the form of maximum in time of the norm of the controllable output, when the multipurpose control minimizes the maximum value of the norm of the controllable output, the recovery of necessary and sufficient conditions is so far rather problematic (see, for example, [9]). 2. STATEMENT OF THE PROBLEM Let the linear stationary discrete object be prescribed: x t+1 = Ax t + Bu t, y t = Cx t, z (i) t = C i x t + D i u t, i = 1,m, t = C j x t + D j u t, j = m +1,N, (1) in which x t R nx is the object state, u t R nu is control, y t R ny is the measurable output, z (i) t R n i are the outputs defining the integral constraints, t R n j are the outputs defining the phase constraints. We will consider the functionals J i = t=0 z (i) t 2, i = 1,m, J j =max t 0 z(j) t 2, j = m +1,N, where denotes the Euclidean norm. From the presented functionals we will compile the goal functional of the form J = λ k J k, λ k 0, λ k =1. (2) If λ s > 0, then the functional J s (s = 1,N) enters into the goal functional (2). Otherwise, when λ s = 0, it defines the constraint J s γ 2 s (3) with the prescribed γ s > 0. We note that the constraint on the control quantity is included in the problem if we prescribe the output z t = u t defining the phase constraints, i.e., if we select for it C j =0andD j = I, wherei is the identity matrix of the requisite order. In the general form, the problem for developing the multipurpose control of a discrete object (1) involves the synthesis of the stabilizing control law from the class of linear state feedbacks or measurable output feedbacks, minimizing the goal functional (2) in the fulfilment of integral and/or phase constraints (3). This means that the trajectory of the optimal closed system must satisfy the inequality J γ 2 (4) at a minimum-possible value γ>0 and the constraints (3) must be fulfilled.

3 SYNTHESIS OF MULTIPURPOSE LINEAR CONTROL LAWS ON TRAJECTORIES SATISFYING INTEGRAL AND PHASE CONSTRAINTS Let the asymptotically stable system of linear difference equations be preset: x t+1 = Ax t, z (i) t = C i x t, i = 1,m, t = C j x t, j = m +1,N, where x t R nx is the object state, z (i) t R n i are the outputs defining the integral constraints, t R n j are the outputs defining the phase constraints, A is the matrix, all eigenvalues of which lie strictly within the unit disk of the complex plane. The need arises to determine when the trajectory of this system with the initial condition x 0 satisfies the integral constraints and the phase constraints t=0 z (i) t 2 γi 2, i = 1,m, (6) t γ j t 0, j = m +1,N, (7) where γ k > 0, k = 1,N are prescribe numbers. Let V k (x) = x T Yk 1 x, where Y k = Yk T > 0, k = 1,N are the Lyapunov functions of the system (5), such that ΔV i < γi 2 z (i) t 2, i = 1,m, (8) ΔV j < 0, j = m +1,N, (9) where ΔV k = V k (x t+1 ) V k (x t ) is the increment of the Lyapunov function in one step along the trajectory of the system (5). We will denote by E(Y )={x : x T Y 1 x 1} the ellipsoid. Adding up each of the m inequalities (8) from t =0toT and passing to the limit at T,weobtain t=0 z (i) t 2 γ 2 i V i (x 0 ), i = 1,m. (10) It follows from these inequalities that all trajectories of the system (5) with initial conditions x 0 m i=1 E(Y i ) satisfy the integral constraints (6). It follows from (9) that for all trajectories of the system (5) with initial conditions x 0 in the ellipsoid E(Y j ), fitted in the field of the phase space C j x γ j, the phase constrains t γ j t 0 are met. Thus, all trajectories of the system that emerge from the set N E(Y k ) will satisfy the integral and phase constraints. The field of the phase space, defined by the unification of all such sets at various Lyapunov functions of the mentioned kinds, can be set aside in terms of the linear matrix inequalities in the following way: Lemma 1. Let Y k = Yk T > 0, k = 1,N be the solution of the system of linear matrix inequalities Y Y i 0 AY i i x 0 0, x T 0 γ i I C iy i < 0, i = 1,m, (11) Y i A T Y i Ci T Y i Y j x 0 0, Y j AY j < 0, C j Y j C x T 0 1 Y j A T j T γj 2 I, j = m +1,N. (12) Y j (5)

4 1430 KOGAN, KRIVDINA Then the trajectory of the system (5) with the initial condition x 0 satisfies the integral and phase constraints (6), (7). The proofs of this lemma and consequent theorems are given in the Appendix. Further we will find the upper estimate of the goal functional (2) on trajectories of the system (5) in fulfilling the constraints (3). For this, along with the constrains (3), we will introduce artificial constraints on the functionals entering into the goal functional (2), i.e., for which λ k 0: J k γ 2 k, (13) where γ k > 0 are auxiliary variables. Then on fulfilling the prescribed and introduced constraints, we have J = λ k J k λ k γk. 2 Thus the problem of finding the estimate of the goal functional can be stated in such a way: to find a minimum value of γ 2, such that at some values of the auxiliary variables γ 2 k (λ k 0)there exists an inequality λ k γk 2 γ2 and the constraints (3), (13) are fulfilled. The solution of this problem is presented in the next lemma. Lemma 2. Let γ 2 be a minimum value of γ 2, such that the system of linear matrix inequalities (11), (12), in which Y k = Y, k = 1,N,and λ k γk 2 γ 2, (14) is solvable relative to Y = Y T > 0, γ 2 > 0, andallγk 2 > 0, for which in the goal functional (2) we have λ k 0. Then the trajectory of the system (5) with the initial condition x 0 satisfies the inequality and the constraints (3) are fulfilled. J γ 2 (15) The assertion of Lemma 2 immediately follows from Lemma 1 in the case when the matrices Y k = Y, k = 1,N coincide. We will make a few remarks. First, if there is at least one integral constraint, then the second inequality (12) is the consequence of the first inequality (11). In addition, it follows from Lemma 2 that if for the preset initial condition x 0 we found the matrix Y satisfying the inequalities (11), (12), and (14), then for all initial conditions x 0 E(Y ) the integral and/or phase constraints (3) are fulfilled and values of the goal functional do not exceed the found value γ 2. Finally, we note that the procedure of minimization of the linear function under the constraints prescribed by linear matrix inequalities is carried out with the aid of the standard command mincx in the LMI toolbox of the package Matlab. Hereafter, we will synthesize the suboptimal control laws at which the estimate γ 2 of the goal functional for the closed system, which is obtained on the basis of Lemma 2, is minimum.

5 SYNTHESIS OF MULTIPURPOSE LINEAR CONTROL LAWS MULTIPURPOSE STATE CONTROL Let an object of control be described by the system of equations: x t+1 = Ax t + Bu t, z (i) t = C i x t + D i u t, i = 1,m, t = C j x t + D j u t, j = m +1,N, (16) in which x t R nx is the object state, u t R nu is the control, z (i) t R n i are the outputs defining the integral constraints, t R n j are the outputs defining the phase constraints. We will determine the multipurpose state control law of the form u t =Θx t, (17) at which in the closed system (16), (17) with the initial condition x 0, the integral and/or phase constraints (3) are fulfilled, and for the goal functional (2) the inequality J γ 2 (18) is fulfilled with a minimum possible value γ 2, obtained according to Lemma 2. Theorem 1. Let γ 2 be a minimum value of γ 2, such that the system of linear matrix inequalities Y x Y 0 AY + BZ 0 0, x T 0 γ i I C iy + D i Z < 0, YA T + Z T B T YCi T + Z T Di T Y Y AY + BZ Y YCj T < 0, + ZT Dj T (19) YA T + Z T B T Y C j Y + D j Z γj 2I 0, λ k γk 2 γ2, i = 1,m, j = m +1,N, is solvable relative to Y = Y T > 0, Z, γ 2 > 0, andallγk 2 > 0, for which in the goal function (2), we have λ k 0. Then for the initial state x 0 the control law (17), in which Θ=Θ=Z Y 1, while Y and Z are the solutions corresponding to the value γ 2, is the multipurpose state control law for the system (16). It follows from Theorem 1 that if for the prescribed initial condition x 0 the multipurpose state controller is found, then for all initial conditions x 0 E(Y ) in this closed system the integral and/or phase constraints (3) are fulfilled and values of the goal functional do not exceed the found value γ 2. The multipurpose control includes both already known control laws (for example, the goal integral functional conforms to the optimal linear-quadratic control in the absence of integral and phase constraints) and also new problems. By way of illustration let us consider the following two problems and formulate for them the requisite assertions. Problem 1. The suboptimal bounded linear-quadratic control: min z t 2 at u 1. This is a multipurpose control if in the object (16) m =1,N =2,γ 2 = 1 and in the goal functional (2) λ 1 = 1. From Theorem 1 we will obtain the following assertion. t=0

6 1432 KOGAN, KRIVDINA Assertion 1. If γ 2 1 is a minimum value γ2 1 at which the system of linear matrix inequalities Y x Y 0 AY + BZ 0 0, x T 0 γ I C 1Y + D 1 Z < 0, YA T + Z T B T YC1 T + ZT D1 T Y (20) Y ZT 0, Z I is solvable relative to Y = Y T > 0, Z, andγ1 2 > 0, then the control law (17), whereθ=θ= Z Y 1,whileY and Z are solutions corresponding to the value γ 2 1, is the suboptimal bounded linear-quadratic law of control by the state for the system (16). Problem 2. The suboptimal bounded control in the problem for minimization of the maximum deviation of the controllable output: min max t 0 z t 2 at u 1. This is the multipurpose control when in the object (16) m =0,N =2,γ 2 = 1, while in the goal functional (2) λ 1 =1. Assertion 2. If γ 2 1 is a minimum value of γ2 1 at which the system of linear matrix inequalities Y x 0 Y AY + BZ 0, < 0, x T 0 1 YA T + Z T B T Y C 1 YC1 T γ2 1 I, Y (21) ZT 0, Z I is solvable relative to Y = Y T > 0, Z, andγ1 2 > 0, then the control law (17), whereθ=θ= Z Y 1,whileY and Z are the solutions corresponding to the value γ 2 1, is the suboptimal bounded state control law for the system (16) in the problem for minimization of the maximum deviation of the controllable output. 5. MULTIPURPOSE OUTPUT CONTROL The stated method of synthesis can be applied to objects with the nonmeasurable state where there is no more than one integral functional, and the control does not directly enter into the outputs defining the phase constraints. According to this, let us describe the object of control by the equations x t+1 = Ax t + Bu t, y t = Cx t, z (1) t = C 1 x t + D 1 u t, t = C j x t, j = 2,N, where y t is the measurable output. There is a need to synthesize the multipurpose control law of the form of the linear dynamic output feedback (22) x (r) t+1 = A rx (r) t + B r y t, x (r) 0 =0, u t = C r x (r) t + D r y t, (23)

7 SYNTHESIS OF MULTIPURPOSE LINEAR CONTROL LAWS 1433 at which in the closed system (22), (23) with the initial state x 0 the condition (18) for the goal functional (2) and the integral and/or phase constraints (3) are fulfilled. We will denote by W S the matrix whose columns form the basis of the matrix kernel S, i.e., SW S =0,whereW S isthematrixofthemaximumrank. Theorem 2. Let γ 2 be the minimum value of γ 2, such that the system of linear matrix inequalities W(C T 0) AT X 11 A X 11 C1 T C 1 γ1 2I W (C 0) < 0, W(B T T D T ) AY 11A T Y 11 AY 11 C1 T C 1 Y 11 A T C 1 Y 11 C1 T γ2 1 I W (B T D T ) < 0, (24) X 11 I 0, C j Y 11 Cj T γj 2 I, j = 2,N, I Y 11 x T 0 X 11x 0 1, (25) λ k γk 2 γ 2, (26) is solvable relative to (n x n x )-matrices X 11 = X T 11 > 0, Y 11 = Y T 11 > 0, γ2 > 0 and all γ 2 k > 0, for which in the goal functional (2) we have λ k 0. Then for the system (22) there exists a multipurpose output control law of the form (23). If the integral functional is unavailable in the problem, the first two inequalities in Theorem 2 should be changed for W T C (AT X 11 A X 11 )W C < 0, W T B T (AY 11 A T Y 11 )W B T < 0. It follows from Theorem 2 that if for the given initial condition x 0 the multipurpose output controller is found, then for all initial conditions x 0 belonging to the set {x : x T X 11 x 1} all constraints are fulfilled in this closed system, and values of the goal functional do not exceed the found value γ 2. The parameters of the multipurpose output controller are found in such a way as it is described in the proof of Theorem CONCLUSIONS For linear discrete systems a certain universal method is suggested for synthesis of linear control laws affording the fulfilment of integral and phase constraints and delivering a suboptimal value to the goal functional that represents the weighted sum of integral functionals and maxima of the norms of controllable outputs. The basis for the synthesis is the method of Lyapunov functions and the apparatus of linear matrix inequalities, which permit us to synthesize the multipurpose control laws by the state and the output in modern packages of programs. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, projects nos and and the Federal Goal-oriented Program Scientific and Educational Personnel of Innovative Russia.

8 1434 KOGAN, KRIVDINA APPENDIX Proof of Lemma 1. It follows from the first inequality of (11) by the Schur lemma [10, p. 559] that x T 0 Y i 1 x 0 1. From the second inequality of (11) it follows that A T Yi 1 A Yi 1 + γi 2 Ci T C i < 0. This means that V i (x t ) = x T t Y i 1 x t is the Lyapunov function of the system (5), for which at all x t 0weget ΔV i (x t ) < γ 2 i C i x t 2, i = 1,m. Summing these inequalities from t =0toT and passing to the limit at T we obtain the inequality (10). Considering that x T 0 Y i 1 x 0 1, we get (6). It follows from the first inequality of (12) by the Schur lemma that x T 0 Y j 1 x 0 1. From the second inequality of (12) it follows that A T Yj 1 A Yj 1 < 0. This means that V j (x t )=x T t Y j 1 x t is the Lyapunov function of the system (5) and that x T t Y j 1 x t 1 for all t 0. According to the uninjury of the S-procedure, at one constraint [11] for each j = m +1,N,the inequality t 2 γj 2 is fulfilled for all x t satisfying the inequality x T t Y j 1 x t 1 if and only if for a certain μ j > 0andallx t the following inequality is met: x T t C T j C j x t γ 2 j μ j (x T t Y 1 j x t 1) 0. Hence it follows that μ j γj 2 and Cj TC j μ j Yj 1. We find from these two inequalities that Cj TC j γj 2Y j 1, which is equivalent by the Schur lemma to the inequality ( I Cj C T j γ 2 j Y 1 j ) 0. Applying once again the Schur lemma to the last inequality, we will obtain the third inequality in (12). Lemma 1 is proved. Proof of Theorem 1. Equations of the closed system (16), (17) have the form where t z (i) t x t+1 = A c x t, = C (i) c x t, i = 1,m, = C (j) c x t, j = m +1,N, A c = A + BΘ, C (i) c = C i + D i Θ, C (j) c = C j + D j Θ.

9 SYNTHESIS OF MULTIPURPOSE LINEAR CONTROL LAWS 1435 According to Lemma 2, if γ 2 is a minimum value of γ 2, at which the system of linear matrix inequalities ( ) Y 0 (A + BΘ)Y Y x0 x T 0, γi 2I (G i + D i Θ)Y < 0, Y (A + BΘ) T Y (C i + D i Θ) T Y (A.1) Y (A + BΘ)Y < 0, (C Y (A + BΘ) T j + D j Θ)Y (C j + D j Θ) T γj 2 Y I, λ k γk 2 γ2, i = 1,m, j = m +1,N, (A.2) is solvable relative to Y = Y T > 0, γ 2 > 0andallγk 2 > 0, for which in the goal functional (2) we have λ k 0, then the trajectory of the closed system (16), (17) with the initial condition x 0 satisfies the inequality J γ 2 (A.3) and the constraints (3) are fulfilled. The system of inequalities (A.1), (A.2) is not linear relative to the variables Y,Θ,γ 2 > 0and all γ 2 k > 0(λ k 0). We will turn from this system to the system of linear matrix inequalities. For this we will denote Z =ΘY, so that the second and the third inequality in (A.1) will be written in the form of the second and the third inequality in (19). In view of the designation Z =ΘY, it follows from the last inequality in (A.1) that (C j Y + D j Z)Y 1 (C j Y + D j Z) T γ 2 j I, which, according to the Schur lemma, is equivalent to the fourth inequality in (19). The inequality (A.2) means that (A.3) is fulfilled. Theorem 1 is proved. Proof of Theorem 2. Equations of the closed system (22), (23) have the form where t x t+1 = A c x t, z (1) t = C c (1) x t, = C c (j) x t, j = 2,N, ( ) ( ) ( ) x0 xt A + BDr C BC r x 0 =, x t = 0 x (r),t 1, A c =, t B r C A r ( ) ( ) C c (1) = C 1 + D 1 D r C D 1 C r, C c (j) = C j 0 nj n x. (A.4) We will present the matrices of the closed system in the form A c = A 0 + BΘ C, C c (1) = G 1 + where ( ) ( ) ( ) Ar B r A 0 nx nx Θ=, A 0 =, B 0nx n x B =, C r D r 0 nx nx 0 nx nx I nx 0 nx nu ( ) 0nx n x I nx C =, D ( ) ( ) = 0 n1 n x D 1, G 1 = C 1 0 n1 n x, C 0 ny nx DΘ C, (A.5)

10 1436 KOGAN, KRIVDINA and the matrices X and Y, in the block form: X = X 11 X 12, Y = Y 11 Y 12. X12 T X 22 Y12 T Y 22 If γ 2 is a minimum value of γ 2, at which the system of matrix inequalities ( ) Y 0 A c Y Y x0 x T 0, 0 γ I C(1) c Y < 0, Y YA T c YC (1)T c (A.6) C c (j) YC c (j)t γj 2 I, j = 2,N, λ k γk 2 γ 2, is solvable relative to Y = Y T > 0, γ 2 > 0andallγk 2 > 0, for which in the goal functional (2) we have λ k 0, then according to Lemma 2 the trajectory of the closed system (A.4) with the initial condition x 0 satisfies the inequality J γ 2 (A.7) and the prescribed constraints are fulfilled. The first inequality in (A.6), with due regard for the block structure of the matrix Y and the structure of the initial condition x 0, is equivalent to the inequality (25). The second inequality in (A.6), with due regard for the Schur lemma, is equivalent to the following inequality: Y A c 0 A T c Y 1 C c (1)T < 0. 0 C c (1) γ1 2I We will substitute the matrices A c, C (1) c into this inequality and write it in the form Ψ+P T Θ T Q + Q T ΘP<0, (A.8) where Y A 0 0 Ψ= A T 0 X G T ( 1, P = 0 C ) ( 0, Q = B T 0 D ) T, X = Y 1. 0 G 1 γ1 2I According to the lemma of exclusion [6, p. 27], the inequality (A.8) is solvable relative to the matrix Θ if and only if the following inequalities are solvable: W T P ΨW P < 0, W T Q ΨW Q < 0, i.e., Y A 0 0 WP T A T 0 X G T 1 W P < 0, 0 G 1 γ1 2I Y A 0 0 WQ T A T 0 X G T 1 W Q < 0. 0 G 1 γ1 2I (A.9)

11 SYNTHESIS OF MULTIPURPOSE LINEAR CONTROL LAWS 1437 We will transform the inequalities (A.9) in accordance with the block structure of the matrices A 0, G 1, X and Y. In the block form the matrix Ψ will be written as Y 11 Y 12 A 0 0 Y T 12 Y Ψ= A T 0 X 11 X 12 C1 T, 0 0 X12 T X C 1 0 γ1 2I and the matrices P and Q will take the form ( 0nx n x 0 nx nx 0 nx nx I nx 0 nx n1 P = 0 ny nx 0 ny nx C 0 ny nx 0 ny n1 ( 0nx n x I nx 0 nx nx 0 nx nx 0 nx n1 Q = B T 0 nu nx 0 nu nx 0 nu nx D T ), ). Then we directly verify that as the matrices W P and W Q of the maximum rank, satisfying the equations PW P =0andQW Q = 0, respectively, we can have I W I W P = 0 0 W C 0, W Q = 0 I 0, I I W where the columns of the matrix W C form the basis of the matrix kernel C, while the columns of the matrix col (W 1,W 2 ) form the basis of the matrix kernel (B T D T ). In view of the matrices Ψ, W P,andW Q the inequalities (A.9) are brought to the form Y 11 Y 12 AW C 0 Y12 T Y W C TAT 0 WC TX 11W C WC T < 0, (A.10) CT C 1 W C γ1 2I W1 T Y 11W 1 γ1 2W 2 TW 2 (A T W 1 + C1 TW 2) T 0 A T W 1 + C1 TW 2 X 11 X 12 < 0. (A.11) 0 X12 T X 22 According to the Schur lemma, the inequality (A.10) is equivalently written as (Y 11 Y 12 Y22 1 Y 12 T) AW C 0 Y 22 > 0, W T C AT WC TX 11W C WC TCT 1 < 0. 0 C 1 W C γ1 2I The first of these inequalities is fulfilled, and the second, with due regard for Frobenius formula on requisite blocks X 11 and Y 11 of the reciprocal matrices X and Y, i.e., Y 11 Y 12 Y 1 22 Y T 12 = X 1 11,

12 1438 KOGAN, KRIVDINA will take the form X 1 11 AW C 0 W T C AT W T C X 11W C W T C CT 1 0 C 1 W C γ 2 1 I < 0. Because X 11 > 0, then by the Schur lemma we will obtain the first of the inequalities (24). In a similar way, the inequality (A.11) is brought to the following two inequalities: X 22 > 0, W 1 TY 11W 1 γi 2W 2 TW 2 W1 TA + W 2 TC 1 A T W 1 + C1 TW 2 (X 11 X 12 X22 1 XT 12 ) < 0. Because Y 11 > 0, then by the Schur lemma we obtain the second of the inequalities (24). Thus, the solvability of the inequalities (A.9) relative to the two reciprocal (2n x 2n x )-matrices X = X T > 0andY = Y T > 0 is equivalent to the solvability of the first and the second inequality in (24) relative to the two (n x n x )-matrices X 11 and Y 11, which are the requisite blocks of the reciprocal matrices X and Y. So that the matrices X 11 and Y 11 be the requisite blocks of the reciprocal matrices X and Y, it is necessary and sufficient, according to Lemma A.7 from [6], to fulfil the third inequality in (24). The third inequality in (A.6) will take the form ( ) ( ) ( ) Y11 Y 12 C T j C j 0 γ Y12 T j 2 Y 22 0 I, which is equivalent to the fourth inequality in (24). Finally, the inequality (26) means that (A.7) is fulfilled. On fulfilling the conditions of Theorem 2, the parameters of the required controller Θ are deduced in the following way. For the found matrices X 11 and Y 11 we set up the matrix ( Y = Y 11 Y 11 X 1 11 Y 11 X11 1 Y 11 X11 1 and then substitute it in the left side of the second inequality (A.6), which is solved as a linear matrix inequality relative to the matrix Θ. Theorem 2 is proved. REFERENCES 1. Hu,T.andLin,Z.,Control Systems with Actuator Saturation, Norwell: Birkhauser, Balandin, D.V. and Kogan, M.M., Linear Control Design Under Phase Constraints, Autom. Remote Control, 2009, no. 4, pp Balandin, D.V. and Kogan, M.M., The Method of Lyapunov Functions in Synthesis of Control Laws under Integral and Phase Constraints, Differ. Uravn., 2009, vol. 45, no. 5, pp Balandin, D.V. and Kogan, M.M., LMI Based Multi-Objective Control under Multiple Integral and Output Constraints, Int. J. Control, 2010, vol. 83, no. 2, pp Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Synthesis of Control Laws on the Basis of Linear Matrix Inequalities), Moscow: Fizmatlit, ),

13 SYNTHESIS OF MULTIPURPOSE LINEAR CONTROL LAWS Balandin, D.V. and Kogan, M.M., Optimal Linear-Quadratic Control in the Class of Output Feedbacks, Dokl. Akad. Nauk, 2007, vol. 415, no. 6, pp Yakubovich, V.A., Nonconvex Optimization Problems: The Infinite-Horizon Linear-Quadratic Problems with Quadratic Constraints, Syst. Control Lett., 1992, vol. 16, pp Dahleh, M.A. and Pearson, J.B., l 1 -Optimal Feedback Controllers for MIMO Discrete-Time Systems, IEEE Trans. Automat. Control, 1987, vol. AC-32, no. 4, pp Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge: Cambridge Univ. Press, Translated under the title Matrichnyi analiz, Moscow: Mir, Gelig, A.H., Leonov, G.A., and Yakubovich, V.A., Ustoichivost nelineinykh system s needinstvennym sostoyaniem ravnovesiya (Stability of Nonlinear Systems with the Nonunique Equilibrium State), Moscow: Nauka, This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board