A Case Study for the Delay-type Nehari Problem

Transcription

1 Proceedings of the 44th EEE Conference on Decision and Control, and the European Control Conference 5 Seville, Spain, December -5, 5 MoA A Case Study for the Delay-type Nehari Problem Qing-Chang Zhong Abstract n this paper, a case study is given for the delaytype Nehari problem so that it can be better understood ndex Terms Delay-type Nehari problem, algebraic Riccati equation, H control, Smith predictor NTRODUCTON The H control of processes with delay(s) has been an active research area since the mid 8 s t is well known that the Nehari problem played an important role in the development of H control theory This is still true in the case for systems with delay(s) Some papers, eg 3, 4, were devoted to calculate the infimum of the delay-type Nehari problem in the stable case t was shown in 3 that this problem in the stable case is equivalent to calculating an L,h-induced norm For the unstable case, Tadmor 5 presented a state-space solution using the differential/algebraic matrix Riccati equation-based method and Zhong 6 proposed a frequency-domain solution using the J-spectral factorization The solvability condition of the delay-type Nehari problem is formulated in terms of the nonsingularity of a delay-dependent matrix in 6 The optimal value γ opt is the maximal γ, ) such that this matrix becomes singular when γ decreases from All sub-optimal compensators are parameterized in a transparent structure incorporating a modified Smith predictor The solution is mathematically elegant However, it seems not trivial to find the solution for a given system, even for a simple system n order to better understand this problem, a case study for a first-order system is given here SUMMARY OF THE THEORETCAL RESULTS The Delay-type Nehari Problem(NP h ) is described as follows Given a minimal state-space realization G β (s) = A B = C(s A) C B, which is not necessarily stable and h, characterize the optimal value γ opt =inf{ Gβ (s)+e sh K(s) L : K(s) H } () This work was supported by the EPSRC under Grant No EP/C5953/ Q-C Zhong is with the Department of Electrical Engineering and Electronics, The University of Liverpool, Brownlow Hill, Liverpool L69 3GJ, UK zhongqc@ieeeorg, URL: and for a given γ>γ opt, parameterize the suboptimal set of proper K(s) H such that Gβ (s)+e sh K(s) L <γ () The result about this problem is summarized as follows See 6, for more details Assume that G β has neither jω-axis zeros nor jω-axis poles Define the following two Hamiltonian matrices: A γ H c = BB A A, H o = C C A and A γ H = BB C C A, Σ Σ Σ= =Σ(h) =e Σ Σ Hh, (3) The algebraic Riccati equations (ARE) Lc H c = (4) L c and Lo Lo Ho = (5) always have unique solutions L c and L o such that A + γ BB L c = H c and A + L L o C C = c Lo Ho are stable, respectively Then, the optimal value γ opt is given by where γ opt =max{γ :detˆσ =}, ˆΣ = L c Σ Lo (6) Furthermore, for a given γ > γ opt, the suboptimal set of proper K H such that () holds is given by K = H r ( W,Q) (7) Z where Q(s) H <γ is a free parameter and Gβ Z = π h {F u (,γ G β )}, (8) /5/$ 5 EEE 386

2 W = A + γ BB Lc ˆΣ (Σ + LoΣ )C ˆΣ B C γ B (Σ Σ Lc) Here, π h, called the completion operator, is defined as π h (G) = A B A B Ce Ah e sh () C D A B for G = = D + C(s A) C D B tiseasytoseethatγ opt satisfies ΓGβ γopt G β (s) L () because, on one hand, the delay-type Nehari problem is solvable 7 iff γ>γ opt = Γ e sh G β, () where Γ denotes the Hankel operator and, on the other hand, it can be seen from () that γ opt G β (s) L (at least K can be chosen as ) The symbol e sh G β in () is non-causal and, possibly, unstable Hence, γ opt ΓGβ Consider with a minimal realization A CASE STUDY G β (s) = s a G β = a ie, A = a, B =and C = According to the assumptions, a Thisgives a γ a H c =, H a o = a, and a γ H = a The ARE (4) and (5) can be re-written as L c γ +al c = and L o +al o = When a<, ie, G β is stable, the stabilizing solutions are L c = and L o =; When a>, ie, G β is unstable, these are L c = and L o = a n this case, A + γ BB L c = a a = a and A + L oc C = a a = a are stable (9) t is easy to find that the eigenvalues of H are λ, = ±λ with λ = a γ Note that λ maybeanimaginary number Assume that γ / a temporarily With two similarity transformations with S = a + λ and S = λ H can be transformed into λ S λ S HS S = λ Hence, the Σ-matrix defined in (3) is Σ=e Hh e λh = S S e λh = S = S S e λh (eλh e λh ) e λh + a λ (eλh e λh ) (eλh e λh ) e λh, ie, S, a λ (eλh e λh ) e λh a λ (eλh e λh ) When γ =/ a, ie, λ =, the above Σ still holds if the limit for λ is taken on the right-hand side, which gives + a Σ λ= = h h n the sequel, Σ isassumedtobedefinedasaboveforλ = A The stable case (a <) n this case, G β is stable and L c = L o = Hence, ˆΣ defined in (6) is ˆΣ =Σ = e λh a λ (eλh e λh ) When γ / a, the number λ is positive and hence the eigenvalues of H are real t is easy to see that ˆΣ is always positive (nonsingular) According to Section, there is γ opt < / a When <γ</ a, the number λ = ωi, whereω = γ a, is an imaginary and hence the eigenvalues of H are imaginaries However, ˆΣ is still a real number because ˆΣ = e ωhi a ωi (e ωhi e ωhi ) ωi =cos(ωh) a ω sin(ωh) Substitute ω = γ a into it, then ˆΣ = when γ =/ a = /a 387

3 (γ ) ˆΣ γ=/ a γ=+ a a Fig 3 The locus of the hidden poles of Z vs γ Fig The surface of ˆΣ with respect to and (a <) opt Fig The contour ˆΣ =on the - plane (a <) ˆΣ =cos( a γ ) a γ sin( a γ ) This can be shown as the surface in Fig, with respect to the normalized delay and the normalized performance index This surface crosses the plane ˆΣ =many times, as can be seen from the contours of ˆΣ =shown in Fig The top curve in Fig characterizes the normalized optimal performance index opt with respect to the normalized delay On this curve, ˆΣ becomes singular the first time when γ decreases from + (or actually, G β L )to Since ΓGβ =and Gβ L =/ a, the optimal value γ opt satisfies γ opt / a, ie, opt This coincides with the curve opt shown in Fig Now discuss the (sub-optimal) compensator for γ>γ opt According to (8) and (), there is a γ Z(s) = π h ( a ) γ = γ γ Σ +(e sh Σ )(s a) s + γ a (3) The locus of the hidden poles of Z(s), ie the eigenvalues of H, is shown in Fig 3 When γ / a, Z(s) has two hidden real poles symmetric to the jω-axis When <γ</ a, Z(s) has a pair of hidden imaginary poles n either case, the implementation of Z needs to be careful; see 8 The W given in (9) is well defined for γ>γ opt as a Σ W Σ Σ = γ Σ and = Z W is = Z a Σ γ Σ =+ Σ s a (Z γ Σ ) =+γ Σ e sh Σ Σ (s + a) s + γ a t is easy to see that is stable As a matter of fact, as required, is bistable for γ > γ opt The Nyquist plots of for different values of are shown in Fig 6 for = The optimal value γ opt is between 44/a and 45/a This corresponds to the transition from Fig 6(b) to Fig 6(c), in which the number of encirclements changes accordingly to the change of the bi-stability of : the 388

4 Nyquist plot encircles the origin when γ<γ opt and hence is not bistable but the Nyquist plot does not encircle the origin when γ>γ opt and hence is bistable B The unstable case (a >) n this case, G β is unstable and L c =, L o = a Hence, ˆΣ defined in (6) is ˆΣ = e λh 4a γ e λh +(e λh e λh ) λ a 3 γ +4λa γ + a + λ = λ a 3 γ + a + λ 4λa γ e λh λ a 3 γ + a λ +4λa γ e λh = λ a 3 γ + a (e λh e λh )+ e λh + e λh ( 4a γ ) = 4a3 γ +3a (e λh e λh )+ e λh + e λh ( 4a γ ) (4) ˆΣ can be re-arranged as ˆΣ = 4a(a γ ) + a (e λh e λh ) 4(a γ ) + 3 (e λh + e λh ) Hence, ˆΣ is always negative (nonsingular) 3 when γ / a, noting that a and λ are positive According to Section, the optimal performance index is less than / a, ie, ˆΣ Fig 4 The surface of ˆΣ with respect to and (a >) opt Fig 5 The contour ˆΣ =on the - plane (a >) γ opt < / a When <γ</ a, the eigenvalues ±λ of H are on the jω-axis Substitute λ = ωi with ω = γ a into (4), then ˆΣ =( 4a γ )cos(ωh)+(3 4a γ ) a ω sin(ωh) Similarly, with the substitution of ω = γ a = γ a γ,then ˆΣ =( 4a γ )cos( a γ )+ (3 4a γ ) sin( a γ a γ ) 3 ˆΣ = 3 when γ =/ a =/a This surface is shown in Fig 4 and the contours of ˆΣ = on the - plane are shown in Fig 5 The top curve in Fig 5 characterizes the normalized optimal performance index opt with respect to the normalized delay On this curve, ˆΣ becomes singular the first time when γ decreases from + to Since L c L o = 4a γ,thereis Γ Gβ = a As a result, the optimal value γ opt satisfies a γ opt a, ie, 5 opt This coincides with the curve opt shown in Fig 5 Now discuss the (sub-optimal) compensator for γ>γ opt n this case, the FR block Z remains the same as in (3) and the form is not affected because of the sign of 389

5 a However,W is changed to a ˆΣ W (Σ aσ ) ˆΣ = γ Σ +aσ and = Z W is = Z a ˆΣ γ Σ +aσ ˆΣ =+ s + a (Z γ Σ aσ ) s a =+γ ˆΣ s+a e sh (s a)σ +(a γ s)σ s +γ a s a =+γ ˆΣ s+a e sh (s a)(σ + Σ ) Σ s +γ a t is easy to see that is stable and invertible As a matter of fact, as required, is bistable for γ>γ opt The Nyquist plots of for different values of are shown in Fig 7 for = The optimal value γ opt is between 73/a and 74/a This corresponds to the transition from Fig 7(b) to Fig 7(c), in which the number of encirclements changes accordingly to the change of the bi-stability of V CONCLUSONS n this paper, a case study is given to show the delaytype Nehari problem using a first-order system The stable case and the unstable case are all discussed The system is normalized so that the (normalized) optimal value can be shown as a function of the delay t has been shown that solving the problem involving only a first-order system is not easy at all When the system order gets higher, the computation of the optimal value becomes very complicated One might have to use () for a compromise REFERENCES BA Francis, A Course in H Control Theory, vol88oflncs, Springer-Verlag, NY, 987 Q-C Zhong, Robust Control of Time-Delay Systems, Springer-Verlag Limited, London, UK, 6, to appear 3 K Zhou and PP Khargonekar, On the weighted sensitivity minimization problem for delay systems, Syst Control Lett, vol8,pp 37 3, DS Flamm and SK Mitter, H sensitivity minimization for delay systems, Syst Control Lett, vol 9, pp 7 4, G Tadmor, Weighted sensitivity minimization in systems with a single input delay: A state space solution, SAM J Control Optim, vol 35, no 5, pp , Q-C Zhong, Frequency domain solution to delay-type Nehari problem, Automatica, vol 39, no 3, pp , 3, See Automatica vol 4, no 7, 4, p83 for minor corrections 7 Gohberg, S Goldberg, and MA Kaashoek, Classes of Linear Operators, vol, Birkhäuser, Basel, Q-C Zhong, On distributed delay in linear control laws Part : Discrete-delay implementations, EEE Trans Automat Control, vol 49, no, pp 74 8, (a) = (b) = (c) = (d) = 7 Fig 6 The Nyquist plot of (a< and = ) 39

6 (a) = (c) = (b) = (d) =5 Fig 7 The Nyquist plot of (a> and =) 39