ISSN 20751087, Gyroscopy and Navigation, 2010, Vol. 1, No. 2, pp. 119–125. © Pleiades Publishing, Ltd., 2010. Original Russian Text © E.I. Veremei, 2009, published in Giroskopiya i Navigatsiya, 2009, No. 4, pp. 3–15.

Synthesis of MultiObjective Control Laws for Ship Motion E. I. Veremei St. Petersburg State University, St. Petersburg, Russia Received February 5, 2009

Abstract—This study presents some aspects of a multiobjective synthesis for ship motion control systems. The proposed approach is based on a unified structure of control laws for motion in different operating modes. This control structure contains variable elements that switch on when necessary to maintain the desired dynamic performance. The key focus is on the synthesis of dynamic compensators that determine the motion dynamics at sea. A simple method for the optimal tuning of compensators is proposed for normal seas. An example illustrates the results of the simulation. DOI: 10.1134/S2075108710020069

INTRODUCTION A wealth of practical experience has been gained in the design and development of ship motion control systems for marine vehicles of different classes [1, 4]. Building on this experience will allow a wide range of conceptual problems to be solved in both traditional and new formulations, including a wellreasoned choice of the structure and parameters of the control laws for different operation modes. The mathematical formalization of the above prob lems will lead to corresponding problems of synthesis whose solution will have as their end goal the practical implementation of the laws of control on board ships. The synthesis of the control laws is performed using optimization theory for dynamic objects in metric spaces. The optimization approach is not an end in and of itself, but essentially a means of ensuring the desired properties of the system to be synthesized. Note that in some cases the novelty of conceptual problems can render the methods of control theory inapplicable. This requires new analytical and compu tational tools, together with the adaptation of known approaches to specific characteristics of mathematical and conceptual problems. Ship motion control is in particular a multifunc tion, multimode, multicriterion and multivariable problem. The synthesis of motion control systems (MCSes) is complicated by the nonlinear behavior of actuators and environmental disturbances (waves, wind, impulse actions) in which dynamic constraints must be considered. A multiobjective approach to solving the synthesis problem that considers the above specific characteris tics of ship motion control systems was first proposed in [4]. Recent efforts have resulted in the creation of a unified methodology for the development of algorith mic support for MCSes. This methodology uses advanced optimization techniques based on control theory and automated computeraided design philos

ophy. Unlike most papers published on this topic (e.g., [3–6]), which state that control laws improve individ ual dynamic characteristics, the multiobjective approach should facilitate control design. The methodology developed to some extent in [8, 9] has introduced a common control law structure for each of the various ship motion modes. The proposed control structure contains variable elements that switch on when necessary to maintain the desired dynamic characteristics. The focus of this paper is the selection of a com pensator as part of a multiobjective structure permit ting the desired stabilization of ship motion at sea. A simple adjustment procedure is proposed based on the assumption of normal sea waves. Application of this procedure is illustrated by the synthesis of a stabi lizing compensator for maintaining course. The mathematical modeling of ship dynamics [3, 9, 10] is based on the following system of nonlinear differential equations: x· = F in ( t, V, ω ) + F hd ( t, V, ω, x p, δ ) + f w ( t ),

(1)

where x = {V, ω, xp} ∈ E12 is the ship’s state vector, V = {Vx, Vy, Vz} ∈ E3 is the linear velocity vector, ω – {ωx, ωy, ωz} ∈ E3 is the angular velocity vector, and xp ∈ E6 is the ship’s linear motions and rotation angles. The available controls are defined by vector δ ∈ Em, whose dimensions and components depend on the ship’s design and function. The motion of a ship is influ enced by internal forces and moments, both inertial (Fin) and hydrodynamic (Fhd), dependent on the ship’s current state as well as on the external forces and moments fw acting on the ship. To deal with formalized problems, mathematical model (1) can be supple mented by equations for actuator dynamics:

119

· δ = F δ ( t, δ, u ),

(2)

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where u ∈ Em is the vector of control signals, and by certain metering equations: y = F y ( t, x, δ ),

(3)

where y ∈ Ek is the vector of the measured dynamic variables. Note that nonlinear functions Fin, Fhd, and Fy usu ally have continuously differentiable components, but the righthand sides of Fδ in (2) almost always contain significant nonlinearities. In particular, the following actuator equations are widely used in certain practical problems: · (4) δ u = σ, σ = f u ( u ), δ = f δ ( δ u ), where σ ∈ Em is the deviation rate of control actuators, δu ∈ Em is an auxiliary vector, fu and fδ are the functions which satisfy sat(x, x0) = x if x ≤ x 0 ; sat(x, x0) = x 0 sgn ( x ) if x > x 0 : f u ( u ) = sat ( u, u 0 )

and

f δ ( δ u ) = sat ( δ, δ 0 ),

(5)

where vectors u0 and δ0 define the values of the corre sponding constraints. Note that if control is implemented within the lin ear part of functions fu and fδ, the actuators can be rep resented by the linear model · (6) δ = u. For the stabilization mode of a ship, we shall use the results from linearization of Eqs. (1), (3), and (6) at a constant longitudinal component of ship velocity around a zero equilibrium point for all other dynamic variables: x· = Ax + Bδ + Hf w ( t ), (7) y = Cx, where x ∈ En is the ship’s state vector, which defines the deviation from equilibrium position; A, B, H, C, and D are matrices with some constant components. MULTIOBJECTIVE CONTROL SYNTHESIS PROBLEM A control law of ship motion is generally developed using equations of linear regulators, u = W ( p )y + W 0 ( p )δ,

p = d/dt,

(8)

written in the operator form. Transfer matrices W and W0 of the regulator (8) are a priori unknown and can then be found during syn thesis, which is well based on the optimization approach associated with the following problem: J = J ( W, W 0 )

inf

( W, W 0 ) ∈ Ω*

.

(9)

Here, J is the performance functional of the dynamics over the motions of closedloop nonlinear system (1)–(3), (8); Ω* ⊂ Ω is the admissible set of the sought matrices contained in domain Ω, whose char acteristic polynomial of closedloop linear system (7), (6), (8) is strictly Hurwitz. The set Ω* is defined by conditions which have to be met by synthesized control laws (8), which include stability, astatism for any controlled condition, and dynamic constraints. The set Ω* and functional J are given in a certain manner for each specific control design situation. In any case, however, the proposed formulation of opti mization problem (9) is very complex and thus not amenable to practical application, due to the narrow ness of the set Ω* and the use of a simple algorithmic notation for the functional to be minimized. The logical answer to this difficulty is to fix the structure of control law (8) and to simplify the prob lem (9) in some way or other by proper selection of adjustable elements. Note that ship motion control systems of tradi tional structure [3–6] can provide stability and satis factory control performance only within a number of substantial constraints. Such systems are usually used at speeds of 5–15 knots in waves of up to 3 on the WMO Sea State Code. For more severe sea states, dif ficulties emerge [11] that cannot be overcome using the traditional control structure by virtue of its limita tions. In this context, we propose a specialized structure for a multiobjective control system containing adjust able elements that can be chosen based on optimiza tion approaches, thus offering a means for improving dynamic performance. Let us consider a particular structure of control law (8), hereafter referred to as AOCS (Asymptotic Observer + Compensator + Speed). We assume that the linear model for a given plant with actuators can be expressed by (7) and (6). The AOCS structure contains the following ele ments: z· = Az + Bδ + G ( y – Cz ) (10) is the observer equation; ξ = F ( p ) ( y – Cz ) is the compensator equation; u = μz· + νy + ξ

(11) (12)

is the speed control signal equation. Here, z ∈ En and ξ ∈ Em are the observer state vec tor and the output of the compensator, respectively. One problem of multiobjective control synthesis is selecting the optimal constant matrices G, μ, ν and transfer matrix F of the compensator, based on the desired dynamic performance of the ship motion.

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Several issues, such as problem decomposition and componentwise optimization of closedloop matrices G, μ, ν, were examined in detail in [4, 9, 11]. We shall therefore consider below the optimization problem of searching for a compensator transfer matrix, assuming it is these matrices that are given (11): Jc = Jc ( F )

inf .

(13)

F ∈ Ωc

Here, Jc is a functional that describes the behavior of a closedloop system moving in waves. Admissible set Ωc is defined by the stability of the closedloop sys tem, the astatism for the controlled conditions, and the condition limitations imposed by the constraints value of the conditions for a system with stepwise dis turbances (such as wind). Some particular versions of problem (13) that can be reduced to H–optimization theory were discussed in [11]. Just as such an approach to synthesis has its own advantages, however, it also has certain draw backs. There are two main drawbacks related to the practical application of this approach. First, sophisti cated algorithmic support for H–optimization makes it difficult to achieve online adaptive control reconfig uration on board a ship. Second, the synthesized com pensator structure might also be difficult to implement in practice. In this context, a quasioptimal approach based on an idealized version of (13) for normal seas can be used for practical synthesis. This approach will be discussed in the next section. Note that compensator Eq. (11) can then be writ ten in the normal form of Cauchy’s equation p· = αp + β ( y – Cz ), (14) ξ + γp, n1

where p ∈ E is the compensator state vector, and α, β, γ are constant matrices (s is the Laplace variable) satisfying the identity –1

γ ( Es – α ) β ≡ F ( s ) given as a minimal realization. Considering (14) and (15), the multiobjective control law equations for the AOCS structure are given by z· = Az + Bδ + G ( y – Cz ), p· = αp + β ( y – Cz ), u = μz· + νy + γp.

transfer matrix F, obtained as a solution to optimiza tion problem (13). Let us consider the simplest version, which can be interpreted as an auxiliary tool for synthesizing the H2 and H∞ quasioptimal compensators. Let us assume that an external disturbance takes the form fw(t) = f0 sinω0t, where f0 is the specified constant vector and ω0 is the specified fixed frequency. We determine functional Jc defining the system of dynamic performance by the degree of control transfer matrix deviation (6), (15) from the prescribed matrix over the wave frequency. It will be shown below that this degree can under certain conditions be zero. The solution to the prob lem under consideration (13) requires that the choice of a compensator transfer matrix define the desired components of the control transfer matrix at point s = jω0. In particular, if the compensator operates as a dynamic filter, the desired components should be zero. To solve this problem, let us first specify the funda mental characteristics of an AOCS structure, based on the known properties of asymptotic observers, by introducing some auxiliary notations: K = μ(A – GC), K0 = μB, Kx = μA + νC and ν0 = μG + ν. ⎛ ⎞ Theorem 1. If matrices A – GC and ⎜ A B ⎟ are ⎝ Kx K0 ⎠ Hurwitz (i.e., the observer and the statecontrolled sys tem are asymptotically stable), then closedlooped sys tem (7), (6), (15) is asymptotically stable if dynamic compensator matrix α is Hurwitz. Theorem 2. If the matrices of Theorem 1 are Hur witz, any control command u0(t) applied to the actuators, the responses of closedloop system (7), (6) with control u = μx· + νy + u0(t) or u = μz· + νy + γp + u0(t) are identical, provided the initial conditions for vectors x and z are similar and p(0) = 0. For defining the solution statement of the problem, suppose that a mathematical model for the basic part of the control channel is given by a set of equations for the actuator, observer and control signal · δ = u, z· = Az + Bδ + G ( y – Cz ), u = μz· + νy + ξ, or

(15)

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(16)

z· = Az + Bδ + G ( y – Cz ), · δ = Kz + K 0 δ + ν 0 y + ξ,

It is in this form that the control laws should be implemented on a ship. COMPENSATOR PARAMETER TUNING FOR REGULAR SEA CASE The key unknown element that controls the motion performance of the ship at sea is compensator

121

where K = μ(A – GC), K0 = μB, ν0 = μG + ν. Equations (16) define the linear system with input (y ¦ ξ)', state (z ¦ δ)' and output (e ¦ ζ)', where e = δ, ζ = y – Cz. Then, using the Laplace transform, we can easily formulate the input–output relationship when

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F ( jω 0 )

the initial conditions are zero for the basic part of the control channel: ⎛ e⎞ = T ( s ) ⎛ y⎞ = ⎛⎜ T 11 ( s ) T 12 ( s ) ⎝ ζ⎠ ⎝ ξ⎠ ⎝ T 21 ( s ) T 22 ( s )

⎞ ⎛ y⎞ ⎟ ⎝⎠ , ⎠ ξ

(17)

–1

–1

× [ R – T 11 ( jω 0 ) ]T 21 ( jω 0 ), we find transfer matrix M = F( jω0) at s = jω0. 3. We choose n1 ≥ 2 to be a dimension of the com pensator state vector and set an arbitrary Hurwitz

where transfer matrix T(s) is written ⎛ 0 ⎛ ⎞⎞ E ⎞⎛ T ( s ) = ⎜ m × n m ⎟ ⎜ E n + m s – ⎜ A – GC B ⎟ ⎟ ⎝ –C 0k × m ⎠ ⎝ ⎝ K K0 ⎠ ⎠

–1

= { T 12 ( jω 0 ) + [ R – T 11 ( jω 0 ) ]T 21 ( jω 0 )T 22 ( jω 0 ) }

–1

n1

polynomial, αp(s) = s + α n1 – 1 s

n1 – 1

+ … + α1 s + α0 . ij

4. We find the complex of numbers fij = ε ( jω 0 ) = m ij α p ( jω 0 ) , where mij is the ijth component of the

⎛ G 0n × m ⎞ ⎛ 0m × k 0m × m ⎞ ×⎜ ⎟ +⎜ ⎟. ⎝ ν0 Em ⎠ ⎝ Ek 0k × m ⎠

obtained matrix M, i = 1, m , j = 1, k and choose poly

If system (17) is a local plant, it is closed via a local compensator: ⎛ e⎞ = ⎛⎜ T 11 ( s ) T 12 ( s ) ⎝ ζ⎠ ⎝ T 21 ( s ) T 22 ( s )

⎞ ⎛ y⎞ ⎟ ⎝⎠ , ⎠ ξ

ij

ij

n1 – 1

ij

ij

nomials ε ( s ) = ε n1 – 1 s + … + ε 1 s + ε 0 of degree not greater than n1 – 1, with real coefficients such that ij

Let us introduce notation Fyδ(s, F) for the control transfer matrix as a local closedloop system (18) from input y to input e. Theorem 3. For

ε ( jω 0 ) = fij, i = 1, m , j = 1, k . 5. For obtained transfer matrix F(s) = ij { ε ( s )/α p ( s ) } i = 1, m ; j = 1, k , we derive a minimal real ization written as state–space equations p· = αp + βζ, ξ = γp

detT 21 ( jω 0 ) ≠ 0,

which can be taken as a mathematical model of the target dynamic compensator.

then over set ΩF of matrices of dimensions m × k with proper linear fractional components having a Hurwitz denominator, there exists the compensator matrix F, such that

EXAMPLE OF SYNTHESIS Let us consider a linear model representing the dynamic characteristics of a ship heading control for a 6000 dwt vessel, written in the form of Eq. (7), where

ξ = F ( s )ζ.

(18)

(19) –1 det { T 12 ( jω 0 ) + [ R – T 11 ( jω 0 ) ]T 21 ( jω 0 )T 22 ( jω 0 ) } ≠ 0,

F yδ ( jω 0, F ) = R,

(20)

where ω0 is real and nonnegative (frequency) and R is specified matrix of dimensions m × k with constant com plex components. According to Theorem 1, any matrix F ∈ ΩF that satisfies conditions (20) is a solution to problem (13) in the formulation under consideration, and Theorem 3 enables us to develop an algorithm that solves this formulation. Let the input data be matrices A, B, and C of the object, matrix G, μ, ν of elements (10), (12) of the AOCS structure, and matrix R defining the desired properties of the control channel over fixed frequency range ω0. To find a compensator transfer matrix F(s) whose components are Hurwitz and that satisfies Eq. (20), the following operations are needed. 1. We build numerical matrices K = μ(A – GC), K0 = μB, ν0 = μG + ν and generate transfer matrix T(s) of local object (17) by resolving into blocks T11, T12, T21, T22 dimensions m × k, m × m, k × k, and k × m, respectively. 2. If the Eq. (19) is satisfied, then using formula

⎛ a a 0⎞ ⎜ 11 12 ⎟ A = ⎜ a a 0 ⎟, ⎜ 21 22 ⎟ ⎝ 0 1 0⎠

⎛ b ⎞ ⎜ 1⎟ B = ⎜ b ⎟, ⎜ 2⎟ ⎝ 0 ⎠

⎛ h ⎞ ⎜ 1⎟ H = ⎜ h ⎟, ⎜ 2 ⎟ (21) ⎝ 0 ⎠

C = ( 0 0 1 ). The components of the state vector would be drift angle x1, angular rate of course changing x2, deviation from heading x3. Let rudder angle δ be the control input and simple harmonic motion fw(t) = sin0.5t described by regular waves be the disturbance. At constant forward speed V = 8 m/s, the compo nents of matrices (21) assume the following values: a11 = –0.0454, a12 = –0.560, b1 = –0.0132, a21 = ⎯0.0267, a22 = –0.0408, b2 = –0.0074, h1 = –0.108, h2 = 0.00760. Based on the desired control performance in response to automatic steering corrections, we obtain the following coefficients for control law (12): μ1 = 3.29, μ2 = –39.8, μ3 = –20.5, ν = –1.23. Assuming that only the heading deviation is acces sible for measurement, and to achieve the desired per

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x3, deg (a)

2 1 0 −1 −2 0 δ, deg 10

50

100

150

200

250

300

350

400

450

500

300

350

400

450

500

(b)

5 0 −5 −10 0

50

100

150

200

250 t, s

Fig. 1. The process of maintaining course in waves: (a) course deviation x3, deg, (b) rudder angle δ, deg.

formance under the action of a wind gust, we designed an observer (10) with a matrix G = (g1 g2 g3)', where g1 = –0.0335, g2 = 0.00450, g3 = 0.0944. According to step 1 of the compensator tuning algorithm, we have: K 0 = – 0.339,

K = ( 0.912 – 6.11 2.22 ),

K x = ( 0.912 – 6.11 – 1.22 ),

compensator. Let γ = 0.8 to define a compromise between the disturbance dynamics and the frequency response of the system around ω0. According to step 4, find numerator ε(s) = 0.0648s2 + 5.809s of the compensator transfer function and build its model in the Frobenius normal form: p· 1 = p 2 , p· 2 = p 3 + β 1 ( x 3 – z 3 ),

ν 0 = – 3.45, 3

T 11 ( s ) = ( –3.45s – 1.74s

p· 3 = – α 0 p 1 – α 1 p 2 – α 2 p 3 + β 2 ( x 3 – z 3 ),

2

ξ = p2 ,

– 0.0918s – 0.00176 )/Δ a ( s ), 3

where β1 = 0.0647, β2 = 5.65, α0 = 0.512, α1 = 1.92, α2 = 2.40. The performance of the synthesized closedloop system is illustrated in Figs. 1 and 2. The closedloop dynamics when maintaining course in waves is depicted in Fig. 1.

2

T 12 ( s ) = ( s + 0.548s + 0.0509s + 0.00144 )/Δ a ( s ), 4

3

T 21 ( s ) = ( s + 0.793s + 0.215s

2

–4

+ 0.0232s + 8.44 × 10 )/Δ a ( s ), T 22 ( s ) = ( –0.00742s – 0.000690 )/Δ a ( s ), 4

3

2

Δ a ( s ) = s + 0.887s + 0.294s + 0.0205s. The compensator can be synthesized to mitigate rudder deflection induced by wave action in normal seas, thus maximizing the effectiveness of the motion control systems. Here we assume that R = 0 and, according to step 2, verify for given frequency range ω0 = 0.5 that condition (19) is satisfied. We then obtain M = F( jω0) = 3.44 – 0.343j. We then set dimension n1 = 3 and choose binomial αp(s) = (s + γ)3 as the characteristic polynomial of the GYROSCOPY AND NAVIGATION

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x3, deg 3

3

2

2

1

1

0

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150

200 0

50

100

150

Fig. 2. Course deviation x3 (deg) under wind effects.

200 t, s

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For comparison, the dynamic compensator is not activated at once but at the 260th second after initia tion of the process. Before it is started, ship stabiliza tion is performed by the observer’s output (in Eq. (12), ξ = 0). It can be seen that the oscillatory component of rudder deflection is damped by the compensator when performance improves in maintaining course. Figure 2 characterizes the dynamics of the same closedloop system with and without a compensator (in the first and second plot, respectively) for ship motion under wind effect fw(t) = 1(t). Since the ship’s heading is rendered astatic, a rud der angle of ~8° degrees fully compensates for the effect of the wind. The influence of the compensator could degrade the ship dynamics by 30% with respect to deviation from the desired course, which we find acceptable. CONCLUSION In this paper, we have presented the structure of a multiobjective ship motion control system designed to meet a combination of performance requirements for closedloop dynamics in different operating modes. A generalized formulation of the synthesis of stabilizing compensators as part of the above structure for shaping the behavior of a motion control system at sea was described. Detailed consideration was given to the ver sion of the problem for normal waves that correspond to simple harmonic motions with a fixed frequency. The goal here is to tune the compensator to achieve the desired components of a complex transfer matrix for the control channel over the wave frequency. If these components are set to zero, the compensa tor operates in a dynamic filter mode to ensure eco nomical motion with minimal loads on the rudder and steering machinery. A simple algorithm was developed to synthesize a stabilizing compensator and to guarantee the har monic tuning of the compensator to achieve the desired closedloop dynamics according to the pre scribed requirements for control system performance. Our synthesis of the compensator operating in a filter mode for mainitaining course is an example of appli cation of the proposed algorithm. APPENDIX Proof of Theorem 1. It is easily seen that the charac teristic polynomial of the closedloop system takes the form ⎛ ⎜ ⎜ Δ(s) = ⎜ ⎜ ⎜ ⎝

Es – A –ν0 C – GC – βC

⎞ –B 0 0 ⎟ Em s – K0 –K –γ ⎟ ⎟ ⎟ –B Es – A + GC 0 ⎟ 0 βC E n1 s – α ⎠

⎛ –B 0 0 ⎜ Es – A ⎜ –Kx Em s – K0 –K –γ = ⎜ ⎜ 0 0 Es – A + GC 0 ⎜ 0 0 βC E n1 s – α ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

= Δ k ( s )A g ( s )Δ α ( s ). Here, Δk(s) =

Es – A –B , Ag(s) = |Es – –Kx Em s – K0

A + GC|, Δα(s) = E n1 s – α are characteristic polyno mials of the given matrices, thereby proving the theo rem. Proof of Theorem 2. The stability of the observer implies that for any vector function δ(t), if x(0) = z(0), it holds true that x(t) = z(t) for any t ≥ 0. Due to the stability of the compensator, we then obtain identity p(0) = 0 from equation p(t) ≡ 0. This ensures a similar response of the actuator to either control u = μx· + νy + u0(t) ≡ Kxx + K0δ + u0(t) with respect to the state vector or control u = μz· + νy + γp + u0(t) ≡ Kz + K0δ + ν0Cx + γp with respect to the estimates obtained from observer (10) and com pensator (11). Proof of Theorem 3. According to (18), we obtain an expression for transfer matrix Fyδ(s, F) of the control channel in relation to F(s): F yδ ( s, F ) = T 11 ( s ) + T 12 ( s )F ( s ) (A.1) –1 × ( E m – T 22 ( s )F ( s ) ) T 21 ( s ). Using (A.1), we obtain (20) with respect to unknown matrix F( jω0). By omitting the notation for jω0 dependence we obtain –1

T 11 + T 12 F ( E m – T 22 F ) T 21 = R.

(A.2)

Since the relations (19) are the case under the the orem’s conditions, then we have F ( jω 0 ) = { T 12 ( jω 0 ) + [ R – T 11 ( jω 0 )T 22 ( jω 0 ) ] }

–1

(A.3)

–1

× [ R – T 11 ( jω 0 ) ]T 21 ( jω 0 ). We now show that there is a matrix within set ΩF that satisfies (A.3). Let α be a n1 × n1 Hurwitz matrix in Frobenius normal form, where n1 ≥ 2. We then –1

introduce transfer matrix Fp(s) ≡ γ ( E n1 s – α ) β of system (A.4) p· = αp + βζ, ξ = γp, n1

with state vector p ∈ E . It is easily seen that the ijth component of matrix M is rational function GYROSCOPY AND NAVIGATION

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SYNTHESIS OF MULTIOBJECTIVE CONTROL LAWS FOR SHIP MOTION ij

ij

ij

F p ( s ) = ε ( s )/α p ( s ), ij

ij

ε ( s ) = ε n1 – 1 s

n1 – 1

ij

ij

+ … + ε1 s + ε0 ,

where αp(s) = det( E n1 s – α ), εij(s) = [ α p ( s )γ(E n1 s – α)–1β]ij, i = 1, m , j = 1, k . ij

Let us introduce notation mij = F ( jω 0 ) for the ijth component of matrix M = F( jω0) defined by formula (A.3). Even in the simplest case of n1 = 2, condition Fp( jω0) = M is satisfied, provided that the equation ij

ij

can be always unambiguously defined as ε 1 = ij

Im[ m ij α p(jω 0) ]/ω0, ε 0 = Re [ m ij α p ( jω 0 ) ] . There then exist real row vectors γi = (γi1 γi2) and column vectors βj = (βj1/βj2) that make up matrices γ and β, respectively, so that identity εij(s) = ij ij [ α p ( s )γ(E n1 s – α)–1β]ij = γ i B j ( s ) ≡ ε 1 s + ε 0 holds ij

ij

true for any real numbers ε 1 and ε 0 , Bj(s) = α p ( s )(E n1 s – α)–1βj for the simplest case of expression (A.4) when n1 = 2. The chosen matrix in this case becomes α = ⎛ 0 1 ⎞ ⎜ ⎟ , where α0 and α1 are positive real num ⎝ –α0 –α1 ⎠ bers. By Cramer’s formula, we obtain Bj(s) = ( B j1 ( s )/B j2 ( s ) ) , where Bj1(s) = β j1 s + β j1 α 1 + β j2 , Bj2(s) = β j2 s – β j1 α 0 . Then, given εij(s) = γ i1 B j1 ( s ) + γ i1 B j1 ( s ) or ij

ε ( s ) = ( γ i1 β j1 + γ i2 β j2 )s + γ i1 β j2 ij

ij

+ ( γ i1 α 1 – γ i2 α 0 )β j1 ≡ ε 1 s + ε 0 , the linear system for finding βj1 and βj2 can be derived as follows: ⎛ ij ⎛ γ i1 γ i2 ⎞ ⎛ β j1 ⎞ ε ⎜ ⎟⎜ ⎟ = ⎜⎜ 1 ij ⎝ γ i1 α 1 – γ i2 α 0 γ i1 ⎠ ⎝ β j2 ⎠ ⎝ ε0

⎞ ⎟. ⎟ ⎠

(A.5)

If we choose γi1 = 1 and γi2 = 0, the system determi 2

nant Δij = γ i1 – γ i2 (γ i1 α 1 – γ i2 α 0 ) is obviously non

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ij

zero. Then for any α1, α0, ε 1 and ε 0 the system (A.5) has a real solution βj1, βj2. Thus, if we choose n1 = 2, there then exists a Hur witz system (A.4) so that its transfer matrix Fp(s) ≡ –1

γ(E n1 s – α) β satisfies Fp( jω0) = F( jω0); hence, it defines the dynamic compensator for which Eq. (20) is true. This statement is obviously more valid for n1 > 2. Thus, the theorem is completely proved. REFERENCES

ij

ε 1 jω 0 + ε 0 = m ij α p ( jω 0 ) holds true for any i and j and

125

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