Partially Decentralized Design Principle in Large-Scale System Control

A number of problems that arise in state control can be reduced to a handful of standard convex and quasi-convex problems that involve matrix inequalities. It is known that the optimal solution can be computed by using interior point methods (Nesterov & Nemirovsky (1994)) which converge in polynomial time with respect to the problem size, and efficient interior point algorithms have recently been developed for and further development of algorithms for these standard problems is an area of active research. For this approach, the stability conditions may be expressed in terms of linear matrix inequalities (LMI), which have a notable practical interest due to the existence of powerful numerical solvers. Some progres review in this field can be found e.g. in Boyd et al. (1994), Hermann et al. (2007), Skelton et al. (1998), and the references therein. Over the past decade, H∞ norm theory seems to be one of the most sophisticated frameworks for robust control system design. Based on concept of quadratic stability which attempts to find a quadratic Lyapunov function (LF), H∞ norm computation problem is transferred into a standard LMI optimization task, which includes bounded real lemma (BRL) formulation (Wu et al. (2010)). A number of more or less conservative analysis methods are presented to assess quadratic stability for linear systems using a fixed Lyapunov function. The first version of the BRL presents simple conditions under which a transfer function is contractive on the imaginary axis of the complex variable plain. Using it, it was possible to determine the H∞ norm of a transfer function, and the BRL became a significant element to shown and prove that the existence of feedback controllers (that results in a closed loop transfer matrix having the H∞ norm less than a given upper bound) is equivalent to the existence of solutions of certain LMIs. Linear matrix inequality approach based on convex optimization algorithms is extensively applied to solve the above mentioned problem (Jia (2003), Kozakova & Veselý (2009)), Pipeleers et al. (2009). For time-varying parameters the quadratic stability approach is preferable utilized (see. e.g. Feron et al. (1996)). In this approach a quadratic Lyapunov function is used which is independent of the uncertainty and which guarantees stability for all allowable uncertainty values. Setting Lyapunov function be independent of uncertainties, this approach guarantees uniform asymptotic stability when the parameter is time varying, and, moreover, using a parameter-dependent Lyapunov matrix quadratic stability may be established by LMI tests over the discrete, enumerable and bounded set of the polytope vertices, which define the uncertainty domain. To include these requirements the equivalent LMI representations of 16


Introduction
A number of problems that arise in state control can be reduced to a handful of standard convex and quasi-convex problems that involve matrix inequalities.It is known that the optimal solution can be computed by using interior point methods (Nesterov & Nemirovsky (1994)) which converge in polynomial time with respect to the problem size, and efficient interior point algorithms have recently been developed for and further development of algorithms for these standard problems is an area of active research.For this approach, the stability conditions may be expressed in terms of linear matrix inequalities (LMI), which have a notable practical interest due to the existence of powerful numerical solvers.Some progres review in this field can be found e.g. in Boyd et al. (1994), Hermann et al. (2007), Skelton et al. (1998), and the references therein.Over the past decade, H ∞ norm theory seems to be one of the most sophisticated frameworks for robust control system design.Based on concept of quadratic stability which attempts to find a quadratic Lyapunov function (LF), H ∞ norm computation problem is transferred into a standard LMI optimization task, which includes bounded real lemma (BRL) formulation (Wu et al. (2010)).A number of more or less conservative analysis methods are presented to assess quadratic stability for linear systems using a fixed Lyapunov function.The first version of the BRL presents simple conditions under which a transfer function is contractive on the imaginary axis of the complex variable plain.Using it, it was possible to determine the H ∞ norm of a transfer function, and the BRL became a significant element to shown and prove that the existence of feedback controllers (that results in a closed loop transfer matrix having the H ∞ norm less than a given upper bound) is equivalent to the existence of solutions of certain LMIs.Linear matrix inequality approach based on convex optimization algorithms is extensively applied to solve the above mentioned problem (Jia (2003), Kozáková & Veselý (2009)), Pipeleers et al. (2009).For time-varying parameters the quadratic stability approach is preferable utilized (see.e.g.Feron et al. (1996)).In this approach a quadratic Lyapunov function is used which is independent of the uncertainty and which guarantees stability for all allowable uncertainty values.Setting Lyapunov function be independent of uncertainties, this approach guarantees uniform asymptotic stability when the parameter is time varying, and, moreover, using a parameter-dependent Lyapunov matrix quadratic stability may be established by LMI tests over the discrete, enumerable and bounded set of the polytope vertices, which define the uncertainty domain.To include these requirements the equivalent LMI representations of Robust BRL for continuous-time, as well as discrete-time uncertain systems were introduced (e.g.see Wu and Duan (2006), and Xie (2008)).Motivated by the underlying ideas a simple technique for the BRL representation can be extended to state feedback controller design, performing system H ∞ properties of quadratic performance.When used in robust analysis of systems with polytopic uncertainties, they can reduce conservatism inherent in the quadratic methods and the parameter-dependent Lyapunov function approach.Of course, the conservativeness has not been totally eliminated by this approach.In recent years, modern control methods have found their way into design of interconnected systems leading to a wide variety of new concepts and results.In particular, paradigms of LMIs and H ∞ norm have appeared to be very attractive due to their good promise of handling systems with relative high dimensions, and design of partly decentralized schemes substantially minimized the information exchange between subsystems of a large scale system.With respect to the existing structure of interconnections in a large-scale system it is generally impossible to stabilize all subsystems and the whole system simultaneously by using decentralized controllers, since the stability of interconnected systems is not only dependent on the stability degree of subsystems, but is closely dependent on the interconnections (Jamshidi (1997), Lunze (1992), Mahmoud & Singh (1981)).Including into design step the effects of interconnections, a special view point of decentralized control problem (Filasová & Krokavec (1999), Filasová & Krokavec (2000), Leros (1989)) can be such adapted for large-scale systems with polytopic uncertainties.This approach can be viewed as pairwise-autonomous partially decentralized control of large-scale systems, and gives the possibility establish LMI-based design method as a special problem of pairwise autonomous subsystems control solved by using parameter dependent Lyapunov function method in the frames of equivalent BRL representations.The chapter is devoted to studying partially decentralized control problems from above given viewpoint and to presenting the effectiveness of parameter-dependent Lyapunov function method for large-scale systems with polytopic uncertainties.Sufficient stability conditions for uncertain continuous-time systems are stated as a set of linear matrix inequalities to enable the determination of parameter independent Lyapunov matrices and to encompass quadratic stability case.Used structures in the presented forms enable potentially to design systems with the reconfigurable controller structures.The chapter is organized as follows.In section 2 basis preliminaries concerning the H ∞ norm problems are presented along with results on BRL, improved BRLs representations and modifications, as well as with quadratic stability.To generalize properties of non-expansive systems formulated as H ∞ problems in BRL forms, the main motivation of section 3 was to present the most frequently used BRL structures for system quadratic performance analyzes.Starting work with such introduced formalism, in section 4 the principle of memory-less state control design with quadratic performances which performs H ∞ properties of the closed-loop system is formulated as a feasibility problem and expressed over a set of LMIs.In section 5, the BRL based design method is outlined to posse the sufficient conditions for the pairwise decentralized control of one class of large-scale systems, where Lyapunov matrices are separated from the matrix parameters of subsystem pairs.Exploring such free Lyapunov matrices, the parameter-dependent Lyapunov method is adapted for pairwise decentralized controller design method of uncertain large-scale systems in section 6, namely quadratic stability conditions and the state feedback stabilizability problem based on these conditions.Finally, some concluding remarks are given in the end.However, especially in sections 4-6, Proposition 1. .L e tQ > 0, R > 0, S are real matrices of appropriate dimensions, then the next inequalities are equivalent Proof.Let the linear matrix inequality takes the starting form in (3), det R = 0 then using Gauss elimination principle it yields and it is evident that (4) implies (3).This concludes the proof.
Note that in the next sections the matrix notations Q, R, S, can be used in another context, too.

Bounded real lemma
Proposition 2. System (1), ( 2) is stable with quadratic performance C(sI− A) −1 B +D 2 ∞ ≤ γ if there exist a symmetric positive definite matrix P > 0, P ∈ I R n×n and a positive scalar where I r ∈ I R r×r , I m ∈ I R m×m are identity matrices, respectively.
Hereafter, * denotes the symmetric item in a symmetric matrix.
ii.Since H ∞ norm is closed with respect to complex conjugation and matrix transposition (Petersen et al. (2000)), then and substituting the dual matrix parameters into i. of (2) implies ii. of (2).
iii.Defining the congruence transform matrix and pre-multiplying left-hand side and right-hand side of i. of (2) by ( 16) subsequently gives ii. of (16).
iii.Analogously, substituting the matrix parameters of the dual system description form into iii. of (2) implies iv. of (2).
Note, to design the gain matrix of memory-free control law using LMI principle only the condition ii. and iii. of (2) are suitable.Preposition 2 is quite attractive giving a representative result of its type to conclude the asymptotic stability of a system which H ∞ norm is less than a real value γ > 0, and can be employed in the next for comparative purposes.However, its proof is technical, which more or less, can brings about inconvenience in understanding and applying the results.Thus, in this chapter, some modifications are proposed to directly reach applicable solutions.

Improved BRL representation
As soon as the representations (2) of the BRL is given, the proof of improvement BRL representation is rather easy as given in the following.

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ii.Using duality principle, substituting the dual matrix parameters into i. of (17) implies ii. of (17).

Basic modifications
Obviously, the aforementioned proof for Theorem 1 is rather simple, and connection between Theorem 1 and the existing results of Preposition 2 can be established.To convert it into basic modifications the following theorem yields alternative ways to describe the H ∞ -norm.
Theorem 2. System (1), ( 2) is stable with quadratic performance Proof.i.SinceS 1 , S 2 are arbitrary square matrices selection of S 1 can now be made in the form S 1 = −P, and it can be supposed that det(S 2 ) = 0. Thus, defining the congruence transform matrix and pre-multiplying right-hand side of i.of(1 7)b yL 2 , and left-hand side of i. of ( 17) by L T 2 leads to i. of (24).
ii. Analogously, selecting S 1 = −P, and considering det(S 2 ) = 0 the next congruence transform matrix can be introduced and pre-multiplying right-hand side of ii.of( 17)byL 3 , and left-hand side of ii. of ( 17) by L T 3 leads to ii.of( 24).

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Partially Decentralized Design Principle in Large-Scale System Control 7

Associate modifications
Since alternate conditions of a similar type are also available, similar to the proof of Theorem 2 the following conclusions can be given.
Corollary 1.Similarly, setting S 2 = −δP,whereδ > 0, δ ∈ I R the inequality ii.given in ( 24) reduces to respectively, and using Schur complement property then ( 28) can now be rewritten as where Choosing δ as a sufficiently small scalar, where (28) be negative definite for a feasible P of ii. of (2).
Remark 1. Associated with the second statement of the Theorem 2, setting and ( 34) can be written as ( 29), with ( 30) and with Thus, satisfying ( 32), ( 33) then (34) be negative definite for a feasible P of iii. of (2).
Note, the form (34) is suitable to optimize a solution with respect to both LMI variables γ, δ in an LMI structure.Conversely, the form (28) behaves LMI structure only if δ is a prescribed constant design parameter, and only γ can by optimized as an LMI variable if possible, or to formulate design task as BMI problem.
Remark 2. By a similar procedure, setting S 2 = −δI n ,whereδ > 0, δ ∈ I Rtheni. of ( 24) implies the following ⎡ ⎢ ⎢ ⎣ It is evident that (39) yields with the same Λ 1 as given in ( 37) and Thus, this leads to the equivalent results as presented above, but with possible different interpretation.

Problem description
Through this section the task is concerned with the computation of a state feedback u(t), which control the linear dynamic system given by ( 1), (2), i.e. q(t)=Aq(t)+Bu(t) ( 41) Problem of the interest is to design stable closed-loop system with quadratic performance γ > 0 using the linear memoryless state feedback controller of the form where matrix K ∈ I R r×n is a gain matrix.Then the unforced system, formed by the state controller (43), can be written as

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y(t)=(C − DK)q(t) (45)
The state-feedback control problem is to find, for an optimized (or prescribed) scalar γ > 0, the state-feedback gain K such that the control law guarantees an upper bound of √ γ to H ∞ norm of the closed-loop transfer function.Thus, Theorem 2 can be reformulated to solve this state-feedback control problem for linear continuous time systems.
Theorem 3. Closed-loop system ( 44), ( 45) is stable with performance C c (sI The control law gain matrix is now given as Proof.Considering that det S 1 = 0, det S 2 = 0 the congruence transform L 4 can be defined as follows and multiplying left-hand side of i. of (17) by L 4 , and right-hand side of ( 17) by L T 4 gives and with W = KV T (53) (50) implies (47).
Fig. 1.System output and state response 24) and denoting

Corollary 3. Following the same lines of that for Theorem 2 it is immediate by inserting
Thus, using Schur complement equivalency, and with

Illustrative example
The approach given above is illustrated by an example where the parameters of the ( 41), ( 42) are 57), ( 58) with respect to the next LMI variables X, Y, Z,a n dδ using SeDuMi (Self-Dual-Minimization) package for Matlab (Peaucelle et al. (1994)) given task was feasible with
as inserting the same into (34) and setting where feasible X, Y, γ, ξ implies the gain matrix (48).

Illustrative example
Considering the same parameters of ( 41), ( 42) and desired output values as is given above then solving ( 59 Fig. 3. System output and state response Remark 4. The closed-loop system (44), ( 45) is stable with quadratic performance γ > 0 and the inequalities (15) are true if and only if there exists a symmetric positive definite matrix X > 0, X ∈ I R n×n ,amatrixY ∈ I R r×n ,andascalarγ > 0, γ ∈ I Rsuchthat Recent Advances in Robust Control -Novel Approaches and Design Methods www.intechopen.com

Illustrative example
Using the same example consideration as are given above then solving (61), ( 62) with respect to LMI variables X, Y,andγ given task was feasible with i. γ = 6.8386 ii.γ = 17.6519The simulation results are shown in Fig. 3, and are concerning with i. of (62).
It is evident that different design conditions implying from the equivalent, but different, bounded lemma structures results in different numerical solutions.

Dependent modifications
Similar extended LMI characterizations can be derived by formulating LMI in terms of product ξP,w h er eξ is a prescribed scalar to overcome BMI formulation (Veselý & Rosinová (2009)).
Theorem 4. Closed-loop system (1), ( 2) is stable with quadratic performance for given ξ > 0 there exist a symmetric positive definite matrix X > 0, X ∈ I R n×n ,ar e g u l a rs q u a r em a t r i xZ ∈ I R n×n ,am a t r i xY ∈ I R r×n ,a n das c a l a rγ > 0, where K is given in (48).

Proof. i. Inserting
Fig. 4. System output and state response Note, other nontrivial solutions can be obtained using different setting of S l , l = 1, 2.

Illustrative example
Considering the same system parameters of (1), (2), and the same desired output values as are given above then solving ( 63 The same simulation study as above was carried out, and the simulation results concerning ii. of (64) for the states and output variables of the system are shown in Fig. 4.
It also should be noted, the cost value γ will not be a monotonously decreasing function with the decreasing of ξ,ifδ = ξ −1 is chosen.

Uncertain continuous-time systems
The importance of Theorem 3 is that it separates T from A, B, C,andD,i.e.therearenoterms containing the product of T and any of them.This enables to derive other forms of bonded real lemma for a system with polytopic uncertainties by using a parameter-dependent Lyapunov function.

Problem description
Assuming that the matrices A, B, C,andD of (1), ( 2) are not precisely known but belong to a polytopic uncertainty domain O, where Q is the unit simplex, A i , B i , C i ,a n dD i are constant matrices with appropriate dimensions, and a i , i = 1, 2, . . ., s are time-invariant uncertainties.
Since a is constrained to the unit simplex as (66) the matrices (A, B, C, D) (a) are affine functions of the uncertain parameter vector a ∈ I R s described by the convex combination of the vertex matrices The state-feedback control problem is to find, for a γ > 0, the state-feedback gain matrix K such that the control law of u(t)=−Kq(t) (67) guarantees an upper bound of √ γ to H norm.
By virtue of the property of convex combinations, ( 48) can be readily used to derive the robust performance criterion.
Theorem 5. Given system ( 65), ( 66) the closed-loop H ∞ norm is less than a real value √ γ > 0,i f there exist positive matrices T i ∈ I R n×n , i = 1, 2, . . ., s, real square matrices U, V ∈ I R n×n ,andareal matrix W ∈ I R r×n such that γ > 0 (68) If the existence is affirmative, the state-feedback gain K is given by Proof.It is obvious that (47), (48) implies directly ( 69), (70).Theorem 6.Given system ( 65), ( 66) the closed-loop H ∞ norm is less than a real value √ γ > 0,i f there exist positive symmetric matrices T i ∈ I R n×n , i = 1,2,...,n, a real square matrices V ∈ I R n×n , a real matrix W ∈ I R r×n , and a positive scalar δ > 0, δ ∈ I Rsuchthat

Remark 5. Thereby, robust control performance of uncertain continuous-time systems is guaranteed by a parameter-dependent Lyapunov matrix, which is constructed as
If the existence is affirmative, the state-feedback gain K is given by Proof.i. Setting U = δV then (69) implies i.of( 73).

Illustrative example
The approach given above is illustrated by the numerical example yielding the matrix parameters of the system D(t)=D = 0 where the time varying uncertain parameter r(t) lies within the interval 0.5, 1.5 .In order to represent uncertainty on r(t) it is assumed that the matrix parameters belongs to the polytopic uncertainty domain O, Thus,solving (72) and i. of (73) with respect to the LMI variables T 1 , T 2 , V, W,a n dδ given task was feasible for a 1 = 0.2, δ = 20.Subsequently, with Fig. 5. System output and state response and including into the state control law the were obtained the closed-loop system matrix eigenvalues set ρ(A c0 )={−2.0598,−22.2541, −24.7547}Solving (72) and ii. of (73) with respect to the LMI variables T 1 , T 2 , V, W,a n dδ given task was feasible for a 1 = 0.2, δ = 20, too, and subsequently, with γ = 10.5304 Fig. 6.System output and state response closed-loop dominant eigenvalues, as well as in the control law gain matrix norm, giving together closed-loop system matrix eigenstructure.To prefer any of them is not as so easy as it seems at the first sight, and the less gain norm may not be the best choice.Fig. 5 illustrates the simulation results with respect to a solution of i. of ( 73) and ( 72).The initial state of system state variable was setting as [q 1 q 2 q 3 ] T =[ 0.5 1 0] T ,t h ed e s i r e d steady-state output variable values were set as [y 1 y 2 ] T =[ 1−0.5]T , and the system matrix parameter change from p = 1t op = 0.54 was realized 5 seconds after the state control start-up.The same simulation study was carried out using the control parameter obtained by solving ii. of ( 73), ( 72), and the simulation results are shown in Fig. 6.It can be seen that the presented control scheme partly eliminates the effects of parameter uncertainties, and guaranteed the quadratic stability of the closed-loop system.

Problem description
Considering the system model of the form (1), (2), i.e. q(t)=Aq(t)+Bu(t) ( 75) but reordering in such way that where i, l = 1, 2, . . ., p, and all parameters and variables are with the same dimensions as it is given in Subsection 2.1.Thus, respecting the above give matrix structures it yields

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where q h (t Problem of the interest is to design closed-loop system using a linear memoryless state feedback controller of the form u(t)=−Kq(t) (80) in such way that the large-scale system be stable, and Lemma 1. Unforced (autonomous) system ( 75)-( 77) is stable if there exists a set of symmetric matrices where Proof.Defining Lyapunov function as follows v(q(t)) = q T (t)Pq(t) > 0 (87) where P = P T > 0, P ∈ I R n×n , then the time rate of change of v(q(t)) along a solution of the system (75), (77 Considering the same form of P with respect to K,i.e. and considering that for unforced system there are u l (t)=0, l = 1,...p then ( 91) implies ( 85).Subsequently, with ( 90), (91) the inequality ( 88) implies (84).

Pairwise system description
Supposing that there exists the partitioned structure of K as is defined in ( 81), ( 82) then it yields where for l = 1, 2, . . ., p, i = h, k Defining with and combining (92) for h and k it is obtained Recent Advances in Robust Control -Novel Approaches and Design Methods respectively.Then substituting (97) in (91) gives Using the next notations where where ω hk (t) can be considered as a generalized auxiliary disturbance acting on the pair h, k of the subsystems.
On the other hand, if Now, taking ( 103), ( 106) considered pair of controlled subsystems is fully described as  75), ( 77), controlled by control law ( 97) is stable with quadratic performances C ∞ ≤ ε hkl if for h = 1,2...,p−1,k= h+1, h+2...,p, l = 1,2...,p, l = h, k, there exist a symmetric positive definite matrix X , and positive scalars γ hk , ε hkl ∈ I Rsuchthat where 99), ( 101), ( 104), respectively, and where A h• hk , A k• hk ,a sw e l la sC h• hk , C k• hk are not included into the structure of ( 110).Then K • hk is given as Note, using the above given principle based on the the pairwise decentralized design of control, the global system be stable.The proof can be find in Filasová & Krokavec (2011).
Proof.Considering ω • hk (t) given in (100) as an generalized input into the subsystem pair (107), (108) then using ( 83) -( 86), and (107) it can be written respectively.Introducing the next notations (114) can be written as Matrix K structure implies evidently that the control gain is diagonally dominant.

Pairwise decentralized design of control for uncertain systems
Consider for the simplicity that only the system matrix blocks are uncertain, and one or none uncertain function is associated with a system matrix block.Then the structure of the pairwise system description implies Analogously it can be obtained equivalent expressions with respect to B hk r(t), C hk r(t), respectively.Thus, it is evident already in this simple case that a single uncertainty affects p−1fromq = ( p 2 ) linear matrix inequalities which have to be included into design.Generally, the next theorem can be formulated.
Robust T In the same sense as given above, the control laws are realized in the partly-autonomous structure ( 94), ( 95), too, and as every subsystem pair as the large-scale system be stable.Only for comparison reason, the composed gain matrix (defined as in ( 81)), and the resulting closed-loop system matrix eigenvalue spectrum, realized using the nominal system matrix parameter A n and the robust and the nominal equivalent gain matrices K, A n , respectively, were constructed using the set of gain matrices K hk , k = 1, 2, 3, h = 2, 3, 4, h = k.A si tc a n see hki where V • hk needs not be symmetric and positive definite.This enables a robust BRL can be obtained for a system with polytopic uncertainties by using a parameter-dependent Lyapunov function, and to deal with linear systems with parametric uncertainties.Although no common Lyapunov matrices are required the method generally leads to a larger number of linear matrix inequalities, and so more computational effort be needed to provide robust stability.However, used conditions are less restrictive than those obtained via a quadratic stability analysis (i.e. using a parameter-independent Lyapunov function), and are more close to necessity conditions.It is a very useful extension to control performance synthesis problems.

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Concluding remarks
The main difficulty of solving the decentralized control problem comes from the fact that the feedback gain is subject to structural constraints.At the beginning study of large scale system theory, some people thought that a large scale system is decentrally stabilizable under controllability condition by strengthening the stability degree of subsystems, but because of the existence of decentralized fixed modes, some large scale systems can not be decentrally stabilized at all.In this chapter the idea to stabilize all subsystems and the whole system simultaneously by using decentralized controllers is replaced by another one, to stabilize all subsystems pairs and the whole system simultaneously by using partly decentralized control.In this sense the final scope of this chapter are quadratic performances of one class of uncertain continuous-time large-scale systems with polytopic convex uncertainty domain.It is shown how to expand the Lyapunov condition for pairwise control by using additive matrix variables in LMIs based on equivalent BRL formulations.As mentioned above, such matrix inequalities are linear with respect to the subsystem variables, and does not involve any product of the Lyapunov matrices and the subsystem ones.This enables to derive a sufficient condition for quadratic performances, and provides one way for determination of parameter-dependent Lyapunov functions by solving LMI problems.Numerical examples demonstrate the principle effectiveness, although some computational complexity is increased.

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