# Dissipative Analysis and Control of State-Space Symmetric Systems

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2 useful later in the proofs of the main theorems of the paper are introduced. A. Definition of Symmetric Systems Consider the following state-space representation for a linear time-invariant system ẋ(t) = Ax(t) + Bw(t), x() = z(t) = Cx(t) + Dw(t) (1) where x(t) R n is the state vector, w(t) R m is the vector of exogenous inputs, z(t) R p is the vector of controlled outputs, and {A,B, C, D} denote the state-space matrices. We say that the state-space representation (1) is symmetric if the following conditions hold A = A T, B = C T, D = D T. (2) The above system property is often referred to as internal or state-space symmetry to make a distinction from external or system symmetry that requires G(s) = G T (s), where G(s) = C(sI A) 1 B + D is the transfer function representation of the system. Obviously, state-space symmetry (2) implies external symmetry, but the converse is not true, that is, there exist symmetric transfer matrices for which there is no symmetric realization. B. Dissipativity of Dynamical Systems In this section, we review the concept of dissipativity in dynamical system and we present the LMI formulation for quadratic dissipativity for linear time invariant (LTI) systems. Consider a dynamical system represented by ẋ(t) = f(x(t),w(t)), x() = x z(t) = g(x(t),w(t)) (3) where x(t) R n is the state vector, w(t) R m is the input, z(t) R p is the output, and f and g are smooth real vector functions. Let us introduce the following quadratic energy supply function E associated with the system (3). E(w, z, T) = z,qz T +2 z,sw T + w, Rw T (4) where Q, S and R are real matrices of appropriate dimensions with matrices Q and R symmetric, and u,v T = T ut vdt for u,v L n 2e. Next, we recall the notion of quadratic dissipativeness [12]. Definition 1: Given matrices Q, S and R where Q and R are symmetric, the system (3) with energy supply function E is called (Q,S, R)-dissipative if for some real function β(.) with β() =, E(w, z, T) + β(x ), w L q 2e, T (5) Furthermore, if for some scalar α > E(w, z, T)+β(x ) α w,w T, w L q 2e, T (6) then (3) is called strictly (Q,S, R)-dissipative. Given a system in input/state/output form and a supply rate, the question that arises is if there exists a storage such that the dissipation inequality is satisfied. If such a nonnegative storage exists, we call the system dissipative with respect to the supply rate. The problem of constructing a nonnegative storage has been extensively studied for general systems and, in particular, for linear systems with a supply rate that is a quadratic function of the input and output variables. In the case of linear systems with quadratic supply rates, it has been shown that both the constructed storage and the required supply are quadratic. In this case, the dissipation inequality becomes a linear matrix inequality (LMI) as we will discuss later in this section. Remark 1: The notion of strict (Q,S, R)-dissipativity includes performance and passivity as special cases represented in one of the following categories: 1) When Q = γ 1 I, S = and R = γi, strict (Q,S, R)-dissipativity reduces to the norm constraint. 2) When Q =, S = I, R =, (6) reduces to the strict positive realness. 3) When Q = γ 1 I, S = (1 )I, R = γi, (, 1), (6) represents a mixed and positive real performance. In this case represents a weighting parameter that defines the trade-off between and positive real performance. We make the following assumptions concerning the system (1) and the weighting matrices Q, S and R. R + D T S + S T D + D T QD > (7) Q Q (8) The following result presents necessary and sufficient conditions for the state-space representation (1) to be strictly (Q,S, R)-dissipative [14]. Theorem 1: Let Q, S, R be given matrices where Q and R are symmetric, and consider the system (1) subject to assumptions (7) and (8). Then the system (1) is asymptotically stable and strictly (Q,S, R)-dissipative, if there exists a matrix P > such that A T P + PA PB C T S C T Q 1 2 (R + D T S + S T D) D T Q 1 2 < (9) I where Q = Q >. The next lemmas will be useful in the proofs of the main results of the paper. Lemma 1: [3] Consider matrices Γ and Λ such that Γ has full column rank, and Λ is symmetric positive definite. Then Λ ΓΓ T if and only if λ max (Γ T Λ 1 Γ) 1. Lemma 2: (Finsler s Lemma) [7]. Consider matrices M and Z such that M has full column rank and Z = Z T. Then the following statements are equivalent: 1) There exists a scalar µ such that µmm T Z >. (1) 414

3 2) The following condition holds M ZM T <. (11) If the above statements hold, then all scalars µ satisfying (1) are given by µ > λ max [M (Z ZM T (M ZM T ) 1 M Z)M T ]. (12) Lemma 3: Consider the following LMI with a matrix parameter P m (A i PB i + Bi T PA T i ) < (13) i=1 where A 1,...,A m and B 1,...,B m are given matrices. Suppose that for every P > that is a positive definite solution of (13), P 1 is also a solution of (13). Then, the identity matrix is a solution of (13), that is m (A i B i + Bi T A T i ) < i=1 Proof. Consider the eigenvalue decomposition (EVD) of P > as follows P = UΣ U T, U T = U 1, Σ = diag(σ 1,...,σ n ) >. Then P 1 = UΣ 1 UT, where Σ 1 = diag( 1 1 σ 1,..., σ n ) and since σ 1 >, there exists α 1 1 such that α 1 σ 1 + (1 α 1 )σ 1 = 1. Define P 1 = α 1 P + (1 α 1 )P 1 = UΣ 1 U T, where Σ 1 = diag(1, σ 2,..., σ n ) with σ i = α 1 σ i + (1 α 1 )σ 1 i for i = 2,...,n. Then, because of convexity of (13), P 1 > satisfies (13) and hence P1 1 also satisfies (13). Since σ 2 >, there exists α 2 1 such that α 2 σ 2 + (1 α 2 ) σ 2 1 = 1. Defining P 2 = α 2 P 1 + (1 α 2 )P1 1 = UΣ 2 U T where Σ 2 = diag(1, 1, σ 3,..., σ n ) with σ i = α 2 σ i + (1 α 2 ) σ 1 i for i = 3,...,n we obtain that P 2 > satisfies (13). By repeating this process we obtain that P n = UΣ n U T = I for Σ n = I is a solution of (13) and this concludes the proof. In the next section, we propose a simple formulation to determine whether or not the symmetric system (1)-(2) is asymptotically stable and strictly dissipative, without having to solve the above linear matrix inequality (9). Subsequently, we provide an explicit solution of the mixed and positive real performance analysis problem for symmetric systems. III. STABILITY AND DISSIPATIVITY OF SYMMETRIC SYSTEMS For simplicity of the formulations in this section, we assume D = in (1). The results will be extended to the case where D later in this section. Lemma 4: Consider the system (1) that satisfies the statespace symmetry conditions (2) with D =. Also, assume that Q = Q T, S, and R = R T are given weighting matrices in (4) subject to assumptions (7) and (8). The system (1)- (2) is asymptotically stable and strictly (Q,S,R)-dissipative if and only if there exists a positive scalar α satisfying the following inequality 2αA + B(Q + (αi S)R 1 (αi S T ))B T < (14) Proof. Let us consider the following Lyapunov matrix P = αi n (15) where α is a positive scalar and I n is the identity matrix. Taking (15) into account, the LMI (9) results in 2αA αb BS BQ 1 2 (R + DS + S T D) DQ 1 2 < (16) I Then, (14) is obtained by applying Schur complement formula to the latter inequality assuming that D =. Proof of the necessity of (16) follows similar lines as the proof of Theorem 2, that will be presented next, and it is omitted here for brevity. Remark 2: It is noted that the problem of checking asymptotic stability and dissipativity of the system (1) requires solving the LMI problem (9). Lemma 4 states that for symmetric systems this condition collapses to an LMI that includes only one scalar decision variable α. In the following, we study the special case of Lemma 4 for the mixed and positive real performance problem stated in Remark 1. Note that the extreme values of are interpreted as which corresponds to positive realness of the system, and 1 which corresponds to the performance. Theorem 2: Consider the following stable symmetric system ẋ(t) = Ax(t) + Bw(t), x() = z(t) = Cx(t) (17) where A = A T and B = C T. Let (, 1) be a given scalar weight representing a trade-off between and positive real performances. The norm of the system can be explicitly obtained from the following relation. γ = f() λ max (B T A 1 B) (18) where f() is given by 2 + ( 1) f() = (19) Proof. Considering the symmetry conditions A = A T, B = C T, and substituting Q = γ 1 I, S = (1 )I, and R = γi into (9) result in AP + PA PB (1 )B γ B B T P (1 )B T γi < γ BT I (2) Now, we consider the Lyapunov matrix P > to be P = αp 1, with α and P 1 being positive scalar and matrix, respectively. After substituting P in the above inequality and using Schur complement, (2) can be written as αap 1 + αp 1 A + α2 γ P 1BB T P 1 (1 )α γ (1 )α BB T P 1 γ P 1 BB T + (1 )2 + 2 BB T < (21) γ 415

4 Pre- and Post-multiplying (21) by P1 1, we obtain a similar Hence, the solvability condition (11) in Finsler s Lemma 2 is condition with respect to P1 1, that is, satisfied since it results in 2αA <. It is also noted that the norm of the symmetric system is given by (12). Since αp1 1 A + αap1 1 + α2 (1 )α γ BBT P1 1 BB T [ ] γ 1 (1 )α BB T P1 1 + (1 M I = )2 + 2 I P1 1 BB T P1 1 < γ γ the formula in (12) provides the following expression for the (22) norm of the symmetric system [ ] Setting α 2 = 2 + ( 1) 2, (21) and (22) will result in the (α+1) 2 same inequality with respect to P 1 and P1 1 γ = λ, respectively. max 2α Ω + 2(1) D α+1 2α Ω α+1 2α From Lemma 3 we obtain that P 1 = I is a solution to the Ω 2α Ω (29) above inequalities. Substituting P = αi into (21), we obtain where Ω = B T A 1 B. The right hand side of the above equality can be rewritten as f()bb T γa (23) where f() is defined as in (19). Now, making use of Lemma 1, (23) can be written as γ f() λ max (B T A 1 B) (24) and this concludes the proof. Remark 3: For the case of strict performance (i.e., for = 1), the bound obtained from Theorem 2 is determined to be γ = λ max (B T A 1 B) (25) which is the same explicit formula as given in [11]. In the next theorem we consider the symmetric system in (1)-(2), where now D is a non-zero matrix. Theorem 3: Let (, 1) be a given scalar weight representing a trade-off between and positive real performances. The stable state-space symmetric system (1)-(2) has an norm given by γ = max{λ max [f() 1 D],λ max [f()b T A 1 B +(2 2 + f()1 )D]} (26) Proof. Similar to the proof of Theorem 2 we consider P = αi. Substituting the system matrices, the corresponding weighting matrices, and P = αi into the matrix inequality (9) leads to 2αI (α + 1)BT B γi 2(1 )D D < (27) γ I This inequality can be rewritten in the form γmm T Z > where M and Z are defined as following 2αA (α + 1)B B M = I, Z = 2( 1) D. 1 I (28) Note that M = [ I ]. 416 [ α+1 ] λ max ( I I I α+1 2α I 2α I I 2(1) 2(1) α+1 I α+1 I α+1 I α+1 I Ω Ω D ) D T Taking into account the fact that λ i (AB) = λ i (BA) for all nonzero eigenvalues and any pair of matrices A and B of compatible dimensions, we obtain that [ α+1 Ω + (2 2 γ = λ + max ( α+1 α+1) )D 2α Ω + α+1 )D α+1 D α+1 D ]. (3) Similar to the proof of the previous theorem, setting α 2 = 2 +(1) 2 makes the latter matrix an upper triangular one, and consequently the maximum eigenvalue is determined from (26). IV. THE MIXED AND POSITIVE REAL CONTROL SYNTHESIS PROBLEM Now consider the following state-space system representation ẋ(t) = Ax(t) + B 1 w(t) + B 2 u(t) z(t) = C 1 x(t) + D 11 w(t) (31) y(t) = C 2 x(t) where x(t) R n is the state vector, w(t) R m1 is the vector of exogenous inputs, u(t) R m 2 is the vector of control inputs, z(t) R p 1 is the vector of controlled outputs, and y(t) R p 2 is the vector of measured outputs. We call this system state-space symmetric if the system state-space data satisfy the following symmetry conditions A = A T, B 1 = C T 1, B 2 = C T 2, D 11 = D T 11. (32) The static symmetric output feedback mixed and positive real control synthesis problem is to design a symmetric static output feedback gain K such that the control law u(t) = Ky(t) (33) renders the closed-loop system stable and guarantees a mixed and positive real performance.

5 The closed-loop system of the open-loop system (31) and the controller (33) becomes ẋ(t) = (A B 2 KB T 2 )x(t) + B 1 w(t) z(t) = C 1 x(t) + D 11 w(t). (34) Note that the closed-loop system (34) is also symmetric. The following result provides explicit expressions for the norm of the closed-loop system and the corresponding controller gain for the special case of mixed and positive real performance described earlier. Theorem 4: Consider the symmetric system represented by (31) where D 11 =. The achievable level of performance can be computed from γ bound = f() λ max [B T 1 B T 2 (B 2 AB T 2 ) 1 B 2 B 1 ] (35) where (, 1) represents the trade-off between performance and positive real performance and f() is defined in (19). For any γ γ bound, a static symmetric output feedback control gain which makes the closed-loop system stable with norm less than γ can be selected as K B 2 [Σ ΣB T 2 (B2 ΣB2 T ) 1 B2 Σ]B T 2 (36) where Σ is defined by Σ = A + f() γ B 1B T 1. (37) Proof. Substituting the closed-loop system matrices (34) into (9), and taking Q = γ 1 I, S = (1 )I, and R = γi into account along with applying Schur complement while assuming D 11 =, we obtain B 2 KB T 2 > A + f() γ B 1B T 1 (38) Note that f() is defined in (19), where the trade-off parameter is known. Now, applying Finsler s lemma on (38) results in the following solvability condition B 2 (A + f() γ B 1B T 1 )B T 2 < (39) Applying Lemma 1 on (39) results in (35) which provides a bound on the norm γ. Inequality (36) is also obtained as the result of Finsler s lemma. Remark 4: It should be noted that the solution of a static output feedback control problem, in general, results in a non-convex formulation with no efficient solution available [1]; however, the results of Theorem 4 provides an explicit solution without having to solve a bilinear-matrix-inequality (BMI) problem. Remark 5: The solution of the symmetric strictly positive real output feedback control problem is obtained when =. For =, we obtain f() = and hence γ bound =, and a static symmetric output feedback gain that renders the closed-loop system strictly positive real is given by K B 2 [A AB T 2 (B2 AB2 T ) 1 B2 A]B T 2. V. NUMERICAL EXAMPLES In this section, we validate our mixed and positive real performance analysis and static output feedback synthesis results using numerical examples. Example 1: We consider a state-space symmetric system with the randomly generated data given in [6]. Let us consider the mixed and positive real performance analysis problem for the system represented by (A,B 1,C 1, D). We vary the weight in the interval (, 1) and plot the norm of the open-loop system versus determined from Theorem 3 in comparison with the norm of the system computed using MATLAB. This comparison is illustrated in Figure 1 where it is observed that the proposed formulation for the norm computation matches the actual norm of the symmetric system. Example 2: As a second example, we consider the RL circuit network shown in Figure 2. We choose the currents of the inductors L 1, L 2 and L 3 as state variables x i, i = 1, 2, 3, the disturbance voltage V d as the disturbance input w, and the current I as the output z. Assume that L 1 = L 2 = L 3 = 1, and R 1 = 1, R 2 = 2,R 3 = 3,R 4 = 4. The open-loop system we obtain is a symmetric system as in (1) with the following data A = A T = , B = C T = 3 7 1, D = 1. We plot the norm of the open-loop system determined from Theorem 3 and the norm of the system computed using MATLAB versus in Figure 3. It is shown that the two profiles are identical. Example 3: In order to validate the synthesis condition results presented in Theorem 4, we consider the symmetric system in (31) with system matrices A and B 1 given in the Example 1 and the control input matrix given in [6]. We vary the trade-off parameter in the interval (, 1) and compute the best achievable level of the performance of the closed-loop system γ opt using (35). Then, we design the output feedback control (33) with the control gain computed using (36) which guarantees the closed-loop system performance for any γ > γ opt. For each (, 1), we assume the desired level of the closed-loop system norm to be γ = 1.1γ opt. Comparison between the actual norm of the closed-loop system and the desired γ is illustrated in Figure 4. It is observed that for any the closedloop system norm is exactly the desired value. It should be noted that the controller designed for each results in improved disturbance attenuation for the closed-loop system compared to the open-loop system. This comparison can be found in [6]. VI. CONCLUSION We addressed the dissipative analysis and output feedback control synthesis problem for LTI state-space symmetric systems. We have derived explicit necessary and sufficient conditions for quadratic dissipativeness and we have developed an explicit expression for the norm of such 417

6 1 Norms norm computed from Thm. 3 Norm computed from (9) Norms norm computed from Thm. 3 norm computed from (9) Fig. 1. Profiles of the norms vs. for Example 1 Fig. 3. Profile of the norms vs. for Example I L1 L2 L3 3 Closedloop desired norm Closedloop actual norm + V d R1 R2 R3 R4 Norms Fig. 2. RL circuit network Fig. 4. Profiles of the closed-loop actual and desired norms vs. for Example 3 systems for the special case of mixed and positive real performance. Then, we developed explicit formulas for the mixed and positive real static output feedback control law, along with an explicit expression for the achievable norm of the closed-loop LTI symmetric system. REFERENCES [1] B. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern Systems Theory, Prentice-Hall, Englewood Cliffs, NJ, [2] D.J. Hill and P.J. Moylan, Dissipative Dynamical Systems: Basic Input-Output and State Properties, in Journal Franklin Inst., 39: , 198. [3] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, [4] C.-H. Huang, P.A. Ioannou, J. Maroulas, and M.G. Safonov, Design of Strictly Positive Real Systems Using Constatnt Output Feedback, in IEEE Trans. on Automatic Control, 44(3): , [5] R. Lozano, B. Brogliato, O. Egeland, and B. Maschke, Dissipative Systems Analysis and Control: Theory and Applications, Springer- Verlag, London, 2. [6] M. Meisami-Azad, Control Design Methods for Large Scale Systems Using an Upper Bound Formulation, Masters Thesis, University of Houston, TX, Aug. 27. [7] R.E. Skelton, T. Iwasaki, and K.M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor & Francis, [8] B. Srinivasan and P. Myszkorowski, Model Reduction of Systems with Zeros Interlacing the Poles, in System and Control Letters, 3(1): 19-24, [9] W. Sun, P.P. Khargonekar, and D. Shim, Solution to the Positive Real Control Problem for Linear Time-Invariant Systems, in IEEE Trans. on Automatic Control, 39(1): , [1] V. Syrmos, C. Abdallah, P. Dorato, and K. M. Grigoriadis, Static Output Feedback: A Survey, in Automatica, 33(2): , [11] K. Tan and K.M. Grigoriadis, Stabilization and Control of Symmetric Systems: an Explicit Solution, in Systems and Control Letters, 44(1): 57-72, 21. [12] J.C. Willems, Dissipative Dynamical Systems - Part 1: General Theory, Part 2: Linear Systems with Quadratic Supply Rates, Archive for Rational Mechanics and Analysis, no. 45, pp and , [13] J.C. Willems, Average Value Stability Criteria for Symmetric Systems, Ricerche di Automatica, no. 4, pp , [14] S. Xie, L. Xie, and C.E. de Souza, Robust Dissipative Control for Linear Systems with Dissipative Uncertainty, in International Journal of Control, 7(2): , [15] K. Zhou, J.C. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, Inc., Upper Saddle River, NJ,

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