Stabilization of Interconnected Systems with Decentralized State and/or Output Feedback

Transcription

1 Stabilization of Interconnecte Systems with Decentralize State an/or Output Feeback Parisses Constantinos Department of Electrical Engineering, Technology Eucational Institute of Western Maceonia, Greece, bstract This paper investigates the problem of the stabilization of a system (, B ) consisting of two interconnecte subsystems, with ecentralize state or output feeback Following the initial efinition of the global system an of its two subsystems in the state-space, an base on the intercontrollability matrix D(s) of system (, B ) an on the kernel U(s) of D(s), an equivalent system {M(s), I 2 } efine in the operator omain by an appropriate polynomial matrix escription (PMD) is etermine The interconnecte system can then stabilize with a suitable local feeback, base on which, a ecentralize output feeback can be etermine as well Keywors: control systems, moeling an simulation interconnecte systems, ecentralize stabilization, local output feeback Introuction Decentralize control has been a control of choice for large-scale systems (consist of many interconnecte subsystems) for over four ecaes It is computationally efficient to formulate control laws that use only locally available subsystem states or outputs Such an approach is also economical; since it is easy to implement an can significantly reuce costly communication overhea lso, when exchange of state information among the subsystems is prohibite, ecentralize structure becomes an essential esign constraint Necessary an sufficient conitions, as well as methos an algorithms have been propose in these four ecaes, to fin ecentralize feeback controllers which stabilize the overall system (see (Ikea,98), (Sanell, 978), (Siljak, 978 ), (Wang, 973) an the references therein) In recent years, the problems of ecentralize robust stabilization for interconnecte uncertain linear systems have been stuie by many researchers Different esign approaches have been propose, such as the Riccati approach (Ge, 996), (Ugrinovsskii, 998), the LMI (Linear Matrix Inequality) approach (Liu, 24), (Souza, 999), a combination of genetic algorithms an graient-base optimization (Labibi, 23), (Patton, 994) It is the main purpose of this paper to present the stabilization problem of an interconnecte (global) system with ecentralize state or output feeback The interconnecte (global) system (, B ), consists of two local scalar subsystems, 43

2 uner the very general assumptions of the global an the local controllability It is note that only the case of two interconnecte subsystems is examine, since only then the global system will have no ecentralize fixe moes, when local static state-vector feebacks are applie ((erson, 979), (Caloyiannis, 982), (Davison, 983), (Fessas, 982,987,988) an (Wolovich, 974) itionally, we assume, without loss of generality (Davison, 983) that both input channels of system are scalar Following the initial efinition of the system in the state-space, an equivalent system - efine in the operator omain- is first etermine This is presente in the next section, together with some known results concerning (a) the intercontrollability matrix D(s) of (, B ) (b) the kernel U(s) of D(s), (c) the equivalent system {M(s), I 2 } in the operator omain, an () the stabilization of the interconnecte system with linear, local, state-vector feeback (LLSVF), introucing linear programming methos for computing them (Parisses, 998) In case the values of these feebacks are consiere to be large for practical implementation, an algorithm for esigning optimal ecentralize control can be applie (Parisses, 26) In section 3, the main result on the stabilizing local output feebacks, an on a metho to esign a suitable output matrix C, is presente s a corollary, the ecentralize version of all theses is given To emonstrate this illustrative example is given, in section 4 2 Preliminaries 2 Form of Matrices an B We consier the interconnecte system (, B ) efine by x x B u () where x is the n-imensional state of (, B ), u is the 2-imensional input vector, is the nxn system matrix, an B is its nx2 input matrix Matrices an B amits the following partitioning: 2 n an n b B b with n=n +n 2 System (, B ) consists of the interconnecte n i -imensional subsystems ( ii, b ii ) -i=,2- of local state vectors x an x 2, with x=[x ' x 2 ']', u an u 2 being, respectively, the scalar inputs of these subsystems, with u=[u u 2 ]' We further assume that the global system (, B ), as well as its two subsystems ( ii, b ii ) -i=,2- are controllable In that case, an in orer to have some analytical results, subsystems ( ii, b ii ) are suppose to be in their companion controllable form (Kailath, 98) b 22 (2) 432

3 ii a ii (n i xn i ) an b ii = (n i x) (3) where a ii ' enotes the last row elements of ii When ( ii, b ii ) are in the above form, the n i xn j sub matrices ij assume no particular form; for them we use the notation = ij o ij ' a ij (i j) (4) where a ij ' enotes the last row elements of ij, an o ij the others It is obvious that when the various sub matrices of (, B ) are in the above form, system (,B ) is calle Canonical Interconnecte Form (Fessas, 982) an is as: 2 a a 2 2 a 2 a 22 B (5) Finally, with the elements of rows n an n +n 2 (=n) of, we form matrix m : m a a 2 a a 2 22 (6) 22 The intercontrollability matrix D(s) an its kernel The following (n-2)xn polynomial matrix is the intercontrollability matrix of system (,B ) : 433

4 D(s) s s s 2 s s the following lemma inicates, D(s) expresses the conitions for the controllability of (, B ): Lemma 2: (Caloyiannis, 982) System (, B ) is controllable if an only if rank D(s) = n-2 for all complex numbers s (Caloyiannis, 982) Thus the matrix D(s) of a controllable system is a full-rank matrix Its kernel U(s) is an nx2 polynomial matrix of rank 2, such that D(s) U(s) = The analytical etermination of U(s) is as follows: P is the matrix representing the column permutations of matrix D(s), which brings it to the form of the matrix pencil: D(s) = D(s) P = [s I n-2 - F G] (8) In (8) G is an (n-2)x2 (constant) matrix, consisting of columns n an n +n 2 =n of D(s), F is an (n-2)x(n-2) constant matrix, an I n-2 is the unity matrix of orer n-2 Since D(s) is a full rank matrix, the pair (F,G) is controllable, an can be brought to its Multivariable Controllable Form (MCF) ((Kailath, 98), (Wolovich, 974)) (, F G ) by a similarity transformation T; let, 2 be the controllability inices of (F,G), S(s) be the associate structure operator, let δ(s) be the characteristic (polynomial) matrix of F, an (in case rank[g]=2) let G m be the 2x2 matrix consisting of rows an + 2 =n-2 of G The precise form of U(s) is the content of the following lemma: Lemma 22 Let D(s) be the intercontrollability matrix of (, B ) as in (7), an suppose that rank[g]=2, for G as in (8) Then the kernel U(s) of D(s) is equal to TS( s) U ( s) P ˆ G ( s) m ( where P, T, S(s), G m, an δ(s) are as previously explaine (7) (9) 23 n equivalent system efine by a PMD Consier the interconnecte system (, B ), with an B as in (5) In that case, the corresponing ifferential equation in the state space is: x(t)=x(t)+b u(t) () In the operator omain, this equation correspons to the equation 434

5 (si - ) x(s) = B u(s) () this, in its turn, reuces to the equations: D(s) x(s) = an (se - m ) x(s) = u(s) (2a) (2b) In these equations, D(s) is as in (7), E is a 2xn (constant) matrix, of the form: E = iag{e ' e 2 '}, the n i -imensional vector e i being equal to : e i = [ ]' -for i=,2-, an m is the matrix efine in (6) From (2a) it follows that x(s) must satisfy the relation: x(s) = U(s) ξ(s) (3) where U(s) is the kernel of D(s), an ξ(s) is any two-imensional vector It follows that ξ(s) must satisfy the equation: M(s) ξ(s) = u(s) (4) The matrix M(s) appearing in (4) is terme Characteristic Matrix of the interconnecte system (, B ) (Fessas, 982) an is efine by the relation: M(s) = (se - m ) U(s) (5) The three systems efine respectively (i) in the state space by the pair of matrices (, B ), (ii) in the operator omain by {si-, B }, an (iii) by the polynomial matrix escription (PMD): M (D) ξ (t) = u (t) (6a) X (t) = U (D) ξ (t) (6b) are equivalents (Chen, 984), (Fessas, 987), (Kailath, 98) It is note that in (6) ξ(t) is the pseuo state vector of the system, an is relate to the state vector x(t) of (, B ), by the relation x(t) = U(D) ξ(t) (7) (in the relations (6), (7), the symbol D enotes the ifferential operator /t) 24 Stabilizability with local state-vector feeback We present analytically Theorem 2, on the stabilizability of the interconnecte system (, B ) with LLSVF, as well as a result, which is neee in the proof of it Lemma 23 : Let h(s) be a polynomial of the form: h(s)=r(s)p(s)+q(s), for which the following assumptions hol: (i) The polynomials r(s), p(s), q(s) are monic (ii) r(s) is arbitrary, (iii) egree r(s)p(s) > egree q(s) (iv) p(s) is a stable polynomial Then, the arbitrary polynomial r(s) can be chosen so, that h(s) is stable (Seser, 978) Theorem 2: Consier the interconnecte system (, B ) as in (), an suppose that the global system (, B ), an the local ones ( ii, b ii ) -i=,2- are controllable 435

6 Then, there exists a static LLSVF of the form u=k x, so that the resulting closeloop system is stable Proof: For the proof we consier the equivalent system {M(s), I 2 } an examine the stability of the polynomial matrix M (s)=(se- m -K )U(s) We assume that the feeback matrix K has the form: α α n α K = = β β n 2 where α i (i=,,n ), an β j (j=,,n 2 ) are some unknown, real numbers We shall eal with the case where rank [G]=2, which is the usual one for the matrix G Then the matrix M (s) takes the form: TS( s) M (s)=(sε- m -K )U(s)=(sΕ- m -K ) P ˆ G ( s) m ( -(s-a n -a,n)[ ]-a(s)+a,n [ 2 ] -(s-α α n -a,n )[ 2 ]-α(s)+a,n [ 22 ] = = -(s-β β -a )[ 2 ]-β(s)+a [ ] -(s-β -a )[ 22 ]-β(s)+a [ 2 ] n2 2,n 2,n n2 2,n 2,n β = (8) = M M 2 ( s) ( s) M M 2 22 ( s) ( s) (9) where α(s)=[α +a, α n - +a,n - a,n + a,n- ]TS (s) α (s)=[α +a, α n - +a,n - a,n + a,n- ]TS 2(s) β(s)=[a 2, a 2,n - β +a 2,n + β n 2 - +a 2,n- ]TS 2(s) β (s)=[a 2, a 2,n - β +a 2,n + β n 2 - +a 2,n- ]TS (s) are scalar polynomials, not monic, TS (s) = T [ s s - ]' TS 2 (s) = T [ s s 2- ]' (ie, TS(s)=[TS (s) TS 2 (s)],an [ij] -for i,j=,2- are the entries of the polynomial matrix G m (s ) Then the matrix in (9) is equivalent to the following matrix: (s)+m (s)+m Μ (s)= M 2(s) M 2 22(s) M (s) M (s) 2 22 h(s)=etm ' (s)={m (s)+m 2 (s)}m 22 (s)-{m 2 (s)+m 22 (s)}m 2 (s)= ={M (s)+m 2 (s)}{-β(s)-(s-β n2-a 2,n) [22]+a 2,n [2]}-{Μ 2 (s)+m 22 (s)}m 2 (s)= (2) 436

7 =-[22]{M (s)+m 2 (s)}(s-β n2-a 2,n) -{M 2 (s)+m 22 (s)}m 2 (s)+{m (s)+m 2 (s)} {-β(s)+a 2,n [2]}=r(s)p(s)+q(s) (2) The eterminant of this matrix is actually a monic polynomial of egree n, by ientifying r(s) as the polynomial -[22][M (s)+m 2 (s)], which is of egree (n-), arbitrary an monic, p(s) as the polynomial (s-βn2-a 2,n), which is stable by choice of β n2, an q(s) as the polynomial - {M 2 (s)+m 22 (s)}m 2 (s)+{m (s)+m 2 (s)}{-β(s)+a 2,n [2]}, of egree (n-) Then, accoring to lemma, the arbitrary polynomial r(s) can be chosen so that the polynomial h(s) is stable QED This proof is complete with an iterative metho (Parisses, 998), in orer to compute the feeback coefficients The central iea is to compute the feeback parameters by solving a linear programming problem (Luenberger, 984) corresponing to choosing positive the coefficients of the polynomials that shoul be stable set of such polynomials (with positive coefficients) is generate They are then examine whether they are stable or not LGORITHM Step Choose the feeback parameter β n2 +α 2,n < so that a stable p(s) results Step2 Write the polynomial r(s) in the following form: r(s) =s n- +kρ(s)=s n- +k(s n-2 +k s n-3 ++k n-2 ) By viewing the egrees of the polynomials α(s) an β(s), it is seen that k-the leaing coefficient of the polynomial ρ(s)- contains only the parameters β n2 an α n It follows that by giving a value to k, we can also compute α n Step3 Form n-2 inequalities with the n-2 unknown feeback parameters, by setting positive the coefficients k i of the polynomial ρ(s) (k i >, for i=,n-2) Step4 Solve the linear programming problem, by putting an objective function with unity weighting coefficients, an fin all feeback parameters α i an β j Step5 Evaluate the polynomial ρ(s), an check if it is stable If it is not, go back to Step, an select another β n2 Step6 Evaluate the polynomial r(s), an check if it is stable If it is not, go back to Step 2, an select another k Step7 Evaluate the polynomial h(s), an check if it is stable If it is not, go back to Step, an select another β n2 Step8 The feeback matrix K can be evaluate from steps, 2, an 4 END OF THE LGORITHM 3 Main Result Theorem 3 Consier the interconnecte system (, B ) as in (), uner the usual assumptions of the global an the local controllability Then, this system can 437

8 stabilize with the feeback u=ly, where the output feeback matrix L is arbitrary, an the output matrix C is: C=L - K, matrix K being the feeback stabilizing matrix Proof: Since system (, B ) satisfies the assumptions of the global an the local controllability, there exists a local feeback stabilizing matrix K, such that +B K is stable ccoring to lemma 2 of (Fessas, 994), system (, B, C) can stabilize with the output feeback u=ly, when the output matrix C is given by the relation C=L - K Remark 3 It is remarke that, while the 2x2 output matrix L is arbitrary, it is the 2xn matrix C that takes care of the stabilization s an extreme case consier L=I 2 (the unity matrix); it is follows that the output matrix C is ientical to the stabilizing local state feeback matrix K Corollary 3 We consier matrix L as a iagonal L matrix (corresponing to the control with local feebacks) It follows that the output matrix C is also blockiagonal C =L - K corresponing, thus, to the case where the measurements are also ecentralize 4 n illustrative example The controllable system (, B ) is: B = The system is unstable, since the eigenvalues of are: { j, 385 5j}, an it is aske to be stabilize by the -control u=k x Subsystems ii - i=,2- are transforme into their companion forms, by the transformation matrices: while matrix T, as in (5), is: 2 5 T The intercontrollability matrix D(s), of system (,B ) is: D(s) It follows that system (F, G) is given by: s s

9 F= 43 G= The controllability inices an 2 of (F, G) are: =, 2 = Obviously, rank [G]=2 The canonical form of matrix F is: The kernel U(s), of D(s), is: ˆF= s U(s)= s-268 t this point begins the search for stable polynomials, by applying simultaneously the linear programming metho, as escribe by the algorithm We use the same notation as in the text, an give the final results: β n2 =-8, α n =-2822 Polynomial ρ(s)=s s+26 (roots of ρ(s):-84, -4) Polynomial r(s)=s 3 +25s s+ 65 (roots of r(s) : -233, -24, -27), an finally h(s) = s s s 2 +84s+624 The roots of this polynomial are the numbers {-26, j, -66}, which are the eigenvalues of the close-loop system, ie, of system B K The matrix of the feeback parameters is: K = It is remarke that the above values of K are in the transforme system of coorinates (use to apply the metho base on the equivalent system efine by a PMD) For a given matrix the output matrix C is If we suppose, as corollary 3 iagonal L 3 L= C= L = 5 the corresponing matrix C is block-iagonal, where the measurements are inee ecentralize C =

10 5 Conclusion In this paper we consiere the stabilization of a global system (, B ), resulting from the interconnection of subsystems ( ii, b ii ) -i=,2-, with ecentralize state an/or output feeback control We stuie initially the problem of the stabilization of (, B ) with linear, static feeback of the local state-vectors, uner the weak conitions of the global an the local controllability lthough the problem was efine in the state-space, it was transforme into the frequency omain an stuie therein The existence of a local, feeback stabilizing matrix was formally proven an it is complete by a numerical proceure -base on linear programming methos- for the numerical computation of the feeback parameters (K ) It is suppose the output feeback matrix L is arbitrary, an one wishes to etermine the appropriate output matrix C which realizes the ecentralize feeback u=k x, by the matrix C=L - K That correspons to what (Zheng, 989 ) refers to as the esigner s possibility to choose the output matrix C References nerson, BD an Clements, DJ (979) lgebraic characterizations of fixe moes in ecentralize control, utomatica, 5, p Caloyiannis, J an Fessas, P (982) Fixe moes an local feebacks in interconnecte systems, Int J of Control, 37, p Chen, CT (984) Linear System Theory an Desig, Holt, Reinhar an Winston, New York 4 Davison, E an Ozguner, O (983) Characterizations of ecentralize fixe moes for interconnecte systems, utomatica, 9, p Fessas, P (982) Stabilization with LLSVF: The State of the rt, Fachbericht Nr 82-4, IE/ETH 6 Fessas, P (987) Stabilizability of two interconnecte systems with local statevector feeback, Intern J of Control, 46, p Fessas, P (988) generalization of some structural results to interconnecte systems, Int J of Control, 43, p Fessas, P an Parisses, C (994) When can the ecentralize state feeback be realize as a static output feeback?, IEE Proc-Control Theory ppl 4(2), p4-6 9 Ge, J, Frank, PM an Lin, C (996) Robust H state feeback control for linear systems with state elay an parameter uncertainty, utomatica, 32, p83-85 Ikea an Siljak, DD (98) Decentralize stabilization of linear time-varying systems, IEEE Trans on utomatic Control, 25, p 6-7 Kailath, T (98) Linear systems, Prentice Hall, Englewoo Cliffs 2 Liu, X Yang, Y Qin, S an Chen, X (24), Robust ecentralize stabilization for uncertain interconnecte elaye systems using reuction metho, in Proc of 44

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