Output Feedback Controller Synthesis for Negative Imaginary Systems
James Dannatt Ian Petersen


High Performance Monte Carlo Simulation of Ising Model on TPU Clusters
    Kun Yang, Yi-Fan Chen, George Roumpos, Chris Colby, John Anderson
    {kuny, yifanchen, roumposg, ccolby, janders}@google.com
======================================================================

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This paper presents necessary and sufficient conditions for deriving a strictly proper dynamic controller which satisfies the negative imaginary output feedback control problem. Our synthesis method divides the output feedback control problem into a state feedback problem and a dual output injection problem. Thus, a controller is formed using the solutions to a pair of dual algebraic Riccati equations. Finally, an illustrative example is offered to show how this method may be applied.





§ INTRODUCTION



Negative imaginary (NI) systems theory is concerned with stable systems with a relative degree of zero, one and two that have a phase response in the interval [-π, 0] for non negative frequencies <cit.>. These systems were first introduced in <cit.>, motivated by the study of linear mechanical systems with collocated force inputs and position outputs. Since the introduction of NI systems theory, it has found use in a large number of applications. A summary of many of these applications can be found in <cit.>. Along with introducing a definition of NI systems, <cit.> showed that the positive feedback interconnection of an NI system with a strictly negative imaginary (SNI) system is internally stable as long as the closed-loop dc gain is less than unity. Thus, if the plant uncertainty of a system is known to be SNI, a feedback controller can be constructed such that the closed-loop system is NI. The resulting positive feedback interconnection can then guaranteed to be robustly stable under the appropriate DC gain condition <cit.>. The construction of such a controller has prompted the development of both state and output feedback results.



The state feedback control problem for negative imaginary systems is concerned with designing a controller K for the positive feedback control scheme u=Kx. K must be chosen such that the corresponding closed-loop system has the negative imaginary property when all of the systems state-variables are available for feedback. The earliest NI state feedback results were presented in <cit.> and <cit.> using the solution of an LMI as the basis for controller design. <cit.> then drew on the H_∞ literature[See <cit.> and <cit.>.], in order to propose a state feedback controller synthesis method that used the solution to an ARE to form a controller. Unfortunately, neither of these methods were ideal as they allowed the closed-loop system to have poles at the origin. In fact an origin pole was unavoidable when a controller was designed using the method in <cit.>. Recognizing this, <cit.> applied a perturbation of the plant matrix in the design process to guarantee asymptotic stability of the closed-loop system. This method ensured stability, however only sufficient conditions were offered and no proof was provided ensuring the perturbed system maintained the NI property. This was rectified in <cit.> were both necessary and sufficient conditions for state feedback controller synthesis were offered. The closed-loop system in <cit.> was shown to have both the NI property and a prescribed degree of stability.



In practice the full system state is not always available for use. The negative imaginary output feedback control problem is then finding a controller H for the control scheme u=Hx̂, where x̂ is an estimate of the systems state-variables. H is chosen such that the corresponding closed-loop system has the negative imaginary property. This problem was first addressed in <cit.> where both static and dynamic output controllers were offered. Alternatively, <cit.> presented a method of controller synthesis using the solution to dual AREs as in <cit.>. In each of these cases, only sufficient conditions were offered.



Here, we present necessary and sufficient conditions for deriving a dynamic controller that solves the output feedback control problem. Our approach uses the solution to two dual AREs like in <cit.>. In our method, the solutions of our AREs may be obtained through the Schur decomposition of a matrix and the solution of two Lyapunov equations, avoiding the common problem of singular Hamiltonians associated with the NI AREs. Our controller results in a closed-loop system with the NI property. Thus, if the plant uncertainty is SNI, the resulting positive feedback interconnection will be robustly stable when the DC gain condition of <cit.> is satisfied.



To assist with exposition, proofs may be found in the appendix.




§ NOTATION



Let ℝ and ℂ denote the fields of real and complex numbers respectively. The notation Im[G(jω)] refers to the imaginary component of the frequency response G(jω). Analogously Re[G(jω)] refers to the real component of G(jω). C^* refers to the complex conjugate transpose of a matrix or vector C and R^∼(s) represents R^T(-s). The notation ρ(A) denotes the spectral radius of A.




§ PRELIMINARIES



The following section provides a review of the relevant NI and SNI definitions and lemmas needed in proving our main result.



Consider the linear time-invariant (LTI) system described by
c 
	ẋ(t) = Ax(t) + Bu(t) 

	y(t) = Cx(t) + Du(t)

where A ∈ℝ^n × n, B ∈ℝ^n × m, C ∈ℝ^m × n and D ∈ℝ^m × m.
We denote the proper, real, rational transfer function matrix of this system as G(s) = C(sI-A)^-1B + D.


  [Dual System]
For the system (<ref>), the transpose of this system is defined as
c 
	ẋ(t) = A^Tx(t) + C^Tu(t) 

	y(t) = B^Tx(t) + D^Tu(t)

and is denoted by the proper, real-rational transfer function matrix G(s)^T = B^T(sI-A^T)^-1C^T + D^T.


We now provide a formal definition of both negative imaginary and strictly negative imaginary systems.


definition
  [Negative Imaginary] <cit.>
 A square transfer function matrix G(s) is NI if the following conditions are satisfied:
	
		
  * G(s) has no pole in Re[s]>0.

		
  * For all ω≥ 0 such that jw is not a pole of G(s), j(G(jω) - G(jω)^*) ≥ 0.

		
  * If s=jω_0, ω_0 > 0 is a pole of G(s) then it is a simple pole. Furthermore, if s=jω_0, ω_0 > 0 is a pole of G(s), then the residual matrix K = lim_s → jω_0 (s-jω_0)jG(s) is positive semidefinite Hermitian.

		
  * If s=0 is a pole of G(s), then it is either a simple pole or a double pole. If it is a double pole, then, lim_s → 0 s^2G(s) ≥ 0.
	



Also, an LTI system (<ref>) is said to be NI if the corresponding transfer function matrix G(s) = C(sI-A)^-1B + D is NI.


 
[Strictly Negative Imaginary] <cit.> 
 A square transfer function matrix G(s) is SNI if the following conditions are satisfied:
	
		
  * G(s) has no poles in Re[s] ≥ 0.

		
  * For all ω > 0 such that jω is not a pole of G(s), j(G(jω) - G(jω)^*) > 0.
	



Also, an LTI system (<ref>) is said to be SNI if the corresponding transfer function matrix G(s) = C(sI-A)^-1B + D is SNI.

  [G(s) is NI  s(G(s)-D) is PR] <cit.> 
 A square, real, rational and proper transfer function matrix G(s) is NI if and only if F(s) = s(G(s)-D) is positive real (PR).


The following lemmas are required in the proof of our main result.


  [G(s) is NI  G(s)^T is NI] <cit.> 
 A square, real, rational and proper transfer function matrix G(s) is NI (respectively SNI) if and only if G(s)^T is NI (respectively SNI).




 [ARE Negative Imaginary Lemma] <cit.> 

Let [ A   B
C   D ] be a minimal realization of a real, rational transfer function matrix G(s) and suppose CB + B^TC^T > 0. Then the following statements are equivalent:


  * G(s) is NI.

  * The ARE

    PA + A^TP + (CA-B^TP)^TR^-1(CA-B^TP) = 0
 has a positive semi-definite solution P ≥ 0.

  * The ARE

    ZA^T + AZ + (B-ZA^TC^T)R^-1(B^T-CAZ) = 0
 has a positive semi-definite solution Z ≥ 0.



  [ARE Strictly Negative Imaginary Lemma] <cit.>
Let [ A   B
C   D ] be a minimal realization of a real, rational transfer function matrix G(s) and suppose CB + B^TC^T > 0. Then the following statements are equivalent:


  * G(s) is SNI and A has no imaginary-axis eigenvalues.

  * The ARE

    PA + A^TP + (CA-B^TP)^TR^-1(CA-B^TP) = 0
 has a positive definite solution P>0 and all the eigenvalues of the matrix A-BR^-1C(A-B^TP) lie in the open left half of the complex plane or at the origin.

  * The ARE

    ZA^T + AZ + (B-ZA^TC^T)R^-1(B^T-CAZ) = 0
 has a positive definite solution Z>0 all the eigenvalues of the matrix A-(B-YA^TC^T)R^-1CA lie in the open left half of the complex plane or at the origin.






 
If a matrix matrix P>0 solves the ARE (<ref>), then the matrix Z = P^-1 is a solution to the ARE

    ZA_0^T + A_0Z + ZQ̅Z + BR^-1B^T    = 0,

where
c
    A_0 = A-BR^-1CA, 

    R = CB + B^TC^T, 

    Q̅ = A^TC^TR^-1CA.
  


 <cit.>
Consider the real, rational, LTI system (<ref>) with transfer function G(s). The operation of the system by

    ẋ = Ax + Bu ⟼ẋ = Ax + Bu + Ly

is called output injection and can be written as

    [ A   B
    C   D ]↦[ A + LC   B + LD
    C   D ].


Output injection does not change the detectability of a system <cit.>.

Before the output feedback control problem can be addressed, the state feedback control problem for negative imaginary systems must be discussed.




§ STATIC STATE FEEDBACK



Consider the state space representation of a linear uncertain system described by

    ẋ   = Ax + B_1w + B_2u 
    
      z          = C_1x 
    
      y          = C_2x + D_21w

This system has uncertainty Δ(s) with  state space representation:

    ẋ_Δ   = A_Δx_Δ + B_Δz 
    
      w          = C_Δx_Δ + D_Δz

Also, assume that all of this system's state-variables are available for feedback.



The state feedback control problem for negative imaginary systems is then, under the control scheme u=Kx, can we design a controller K such that the corresponding closed-loop uncertain system

    ẋ   = (A + B_2K)x + B_1w 
    
      z          = C_1x        
    ẋ_Δ   = A_Δx_Δ + B_Δz 
    
      w          = C_Δx_Δ + D_Δz

has the NI property.



The following lemma is one solution to this problem.




 [NI State Feedback Control Lemma] <cit.> Consider the uncertain system (<ref>)-(<ref>) that satisfies C_1B_2 non-singular and C_1B_1 + B_1^TC_1^T > 0. The following statements are equivalent:


  * There exists a static state feedback matrix K such that the closed-loop system (<ref>)-(<ref>) is NI.

  * There exists matrices T ≥ 0 and S ≥ 0 such that

    -A_22T - TA_22^T + B_f2RB_f2^T    = 0, 
    
      -A_22S - SA_22^T + B_22R^-1B_22^T    = 0, 
    
      T-S    > 0,

where the matrices A_22, B_f2 and B_22 are obtained from the Schur decomposition outlined below in (<ref>)-(<ref>).

  * There exists a positive semi-definite matrix P_f≥0 that satisfies the ARE

    P_fA_f + A_f^TP_f +P_f(B̃_̃1̃R^-1B̃_̃1̃^T-B_fRB_f^T)P_f = 0,

where A_f,B_f and B̃_̃1̃ are defined below in (<ref>)-(<ref>).

If any of the above statements hold, then a corresponding state feedback controller matrix K is given by,

    K    = (C_1B_2)^-1(B_1^TP - C_1A - R(B_2^TC_1^T)^-1B_2^TP),

where P = UP_fU^T and P_f = [ 0   0
0   (T-S)^-1 ]≥ 0.
U is an orthogonal matrix obtained through the following real Schur transformation [See Section 5.4 of <cit.>].






 §.§ Schur decomposition



Consider the following real Schur transformation of the matrix A-B_2(C_1B_2)^-1C_1A which we can apply to the system (<ref>), (<ref>) to give


    A_f    = U^T(A-B_2(C_1B_2)^-1C_1A)U   =
      [ A_11   A_12
    
      0    A_22 ],
    
    
      B_f    = U^T(B_2(C_1B_2)^-1 - B_1R^-1) =
      [ B_f1
    B_f1 ],
    B̃_̃1̃   = U^TB_1 =
      [ B_11
    B_22 ].

This transformation is constructed such that all of the eigenvalues of the matrix A_11 are in the closed left half plane and A_22 is an anti-stable matrix.



The sub-matrix A_11 will always contain a zero eigenvalue. This follows directly from that fact that A-B_2(C_1B_2)^-1C_1A is singular.



If the state feedback controller K is applied to a system for which the plant uncertainty is known to be SNI and the DC gain condition of <cit.> is satisfied, then the resulting positive feedback interconnection of the plant uncertainty with the closed-loop transfer function is guaranteed to be robustly stable.


 [NI Dual Output Injection Lemma]
Consider the uncertain system (<ref>)-(<ref>) that satisfies D_21 non-singular and C_1B_1 + B_1^TC_1^T > 0. The following statements are equivalent:


  * The closed-loop system (<ref>)-(<ref>) is NI.

  * There exists an output injection matrix L such that the system

    ẋ   = (A + LC_2)^Tx + A^TC_1^Tw 
    z̃   = (B_1^T + D_21^TL^T)x + B_1^TC_1^T

is PR.

  * There exists a positive semi-definite matrix Z≥0 that satisfies the ARE

    Z(A + LC_2)^T + (A + LC_2)Z + Q^TR^-1Q = 0,

where Q = B_1^T+D_21^TL^T-C_1AZ.

If any of the above statements hold, a corresponding control matrix L that solves this systems dual output injection problem is given by

    L = (ZA^TC_1^T - B_1 - ZC_2^T(D_21)^-1R)(D_21^T)^-1.




We will now use both the state feedback and output injection lemmas in constructing a controller that satisfies the output feedback control problem.




§ NEGATIVE IMAGINARY DYNAMIC OUTPUT FEEDBACK
 


Consider the state space representation of a linear uncertain system

    ẋ   = Ax + B_1w + B_2u 
    
      z          = C_1x            
    
      y          = C_2x + D_21w

where A ∈ℝ^n × n, B_1 ∈ℝ^n × 1, B_2 ∈ℝ^n × r, C_1 ∈ℝ^1 × n, C_2 ∈ℝ^1 × n, D_21∈ℝ^n × n. This system is assumed to have SNI uncertainty Δ(s) with state space representation (<ref>)-(<ref>).



Suppose the system (<ref>)-(<ref>) satisfies the following assumptions:

 
  * (A,B_2) is stabilizable and (C_2,A) is detectable;
 
  * C_1B_2 is non-singular;
 
  * D_21 is non-singular;
 
  * R = C_1B_1 + B_1^TC_1^T > 0. 


If we apply the dynamic compensator

    ẋ_̇k̇   = A_kx_k + B_ky 
    
      u         = C_kx_k

to the system (<ref>)-(<ref>) the corresponding closed-loop system has the realization

    [ A   B_2C_k   B_1
    B_kC_2   A_k   B_kD_21
    C_1   0   0 ],

with closed-loop transfer function

    G_cl(s)    = C_c(sI - A_c)^-1B_c.


We now present both necessary and sufficient conditions for the existence of a dynamic output controller that will result in a closed-loop system with the NI property.



 
Consider the uncertain system (<ref>)-(<ref>) satisfying assumptions A1-A<ref>. Suppose there exist P≥0, Z≥0, F and L which satisfy:

 
  * c
  R(P) = PÃ + Ã^TP + Q̃^TR^-1Q̃ = 0,
  
where

    Ã   = A + B_2F ,    Q̃   = C_1(A+B_2F)-B_1^TP.

  
  * c
  S(Z) = ZA̅^T + A̅Z + Q̅R^-1Q̅^T = 0,
  
  where

    A̅   = A + LC_2,    Q̅   = B_1+LD_21-ZA^TC_1^T.


  * ρ (ZP) < 1.

Then, a strictly proper controller which results in a closed-loop system with the negative imaginary property is given by (<ref>), (<ref>) where

    C_k    = F, 
    
        B_k    = -(I-ZP)^-1L, 
    
        A_k    = A+B_2C_k-B_kC_2-(B_1 + (I-ZP)^-1LD_21) 
       · R^-1(C_1(A+B_2C_k)-B_1^TP).


Conversely, suppose there exists a controller of the form (<ref>), (<ref>) such that the closed-loop system is SNI and minimal. Then there exists P>0, Z>0, F and L which satisfy

 
  * c
  R(P) = PÃ + Ã^TP + Q̃^TR^-1Q̃ = 0,
  
where

    Ã   = A + B_2F ,    Q̃   = C_1(A+B_2F)-B_1^TP.

  
  * c
  S(Z) = ZA̅^T + A̅Z + Q̅R^-1Q̅^T = 0,
  
  where

    A̅   = A + LC_2,    Q̅   = B_1+LD_21-ZA^TC_1^T.


  * ρ (ZP) ≤ 1.



 <cit.>
Consider the uncertain system (<ref>)-(<ref>) satisfying assumptions A1-A<ref> and suppose there exist P>0, Z>0, F and L which satisfy Theorem <ref>. If the matrix A_c is Hurwitz, then the resulting closed-loop system is SNI.



The controller (<ref>)-(<ref>) is applied to a system for which the plant uncertainty is assumed to be SNI. If the dc gain condition of <cit.> is satisfied, then the resulting positive feedback interconnection of the plant uncertainty with the closed-loop transfer function is guaranteed to be robustly stable.


The following lemma is needed in the proof of Theorem <ref>.



Consider the uncertain system (<ref>)-(<ref>) satisfying assumptions A1-A<ref>. If there exists P≥0, Z≥0, F and L which satisfy Theorem <ref>, then there exists a matrix V≥0 which satisfies the ARE

    V(A_e + L_eC_e2) + (A_e + L_eC_e2)^TV + Q_v^TR^-1Q_v = 0,

where

    A_e    = A - B_1R^-1C_1(A+B_2F) + B_1R^-1B_1^TP, 
    
      C_e1   = C_1B_2F, 
    
      C_e2   = C_2 - D_21R^-1C_1(A+B_2F) + D_21R^-1B_1^TP, 
    
      Q_v    = C_e1 -(B_1+L_eD_21)V.






§ ILLUSTRATIVE EXAMPLE



Consider the linear uncertain system (<ref>)-(<ref>), where

    A =  [ -1   0   0
    0   -1   1
    1   1   -1 ], B_1 = [ 1
    2
    1 ],   B_2 = [ 1
    1
    1 ], 
    
    C_1 = [ 1   0   0 ], C_2 = [ 1   2   0 ], D_21 = I.

We can easily verify this system satisfies A1-A<ref>. Also, the matrices P=0 and Z=0 satisfy conditions (a)-(c) of Theorem <ref>. Thus we can construct matrices F= [ 1   0   0 ] and L = [ -1   -2   -1 ]^T and our dynamic compensator with the form (<ref>), (<ref>) can be constructed as

    [ -1   -2   0   1
    -1   -5   1   2
    1   -1   -1   1
    1   0   0   0 ].


We can now verify our closed-loop system with transfer function G_cl(s) = C_cl(sI - A_cl)^-1B_cl, where

    A_cl   =  [ 0   0   0   -1   0   0
    1   -1   1   -1   0   0
    2   1   -1   -1   0   0
    0   0   0   -2   -2   0
    0   0   0   -2   -5   1
    0   0   0   0   -1   -1 ],
    B_cl = [ 1
    2
    1
    0
    0
    0 ], 
    
    C_cl   = [ 1   0   0   0   0   0 ],

satisfies the the ARE

    Σ A_cl + A_cl^TΣ
       + (C_clA_cl-B_cl^TΣ)^TR^-1(C_clA_cl - B_cl^TΣ) = 0,

with

    Σ   =  [ 0   0   0   0   0   0
    0   0   0   0   0   0
    0   0   0   0   0   0
    0   0   0   0.178   -0.053   -0.024
    0   0   0   -0.053   0.019   0.010
    0   0   0   -0.024   0.010   0.010 ]≥ 0.

Therefore it follows from Lemma <ref> that G_cl(s) is NI.







§ CONCLUSIONS



This paper has presented both necessary and sufficient conditions for synthesizing a dynamic controller which solves the negative imaginary output feedback control problem. Our method divides the output feedback control problem into a state feedback problem and a dual output injection problem. These problems are then solved by finding the solutions to a pair of algebraic Riccati equations. The solutions to these equations may also be obtained through Schur decomposition which avoids the common problem of singular Hamiltonians in NI controller synthesis. Relaxing the necessary conditions for NI controller synthesis to include non strict solutions to the NI ARE remains an open problem.




§ APPENDIX: PROOFS




The proof of equivalence between 1) and 2) is shown in <cit.>.
1)  3) follows using the same proof applied to the transfer function matrix G(s)^T which is NI if and only if G(s) is NI by Lemma <ref>.



The proof of equivalence between 1) and 2) is shown in <cit.>.
1)  3) follows using the same proof applied to the transfer function matrix G(s)^T which is SNI if and only if G(s) is SNI by Lemma <ref>. Also, note that G(s) is SNI implies A^T has no imaginary-axis eigenvalues from Definition <ref>.




This is proved by straight forward algebraic manipulation. Noting that Z=P^-1 we simply pre-multiply (<ref>) by Z and post-multiply by Z as follows

    0    = PA + A^TP + (CA-B^TP)^TR^-1(CA-B^TP), 
    
      0    = Z(PA + A^TP + (CA-B^TP)^TR^-1
       ·(CA-B^TP))Z, 
    
      0    = ZA^T + AZ +(B^T-CAZ)^TR^-1(B^T-CAZ).




The equivalence between 2)  3) follows from Lemma 2.3 in <cit.>.
In order to show 1)  2) note that from Lemma <ref>, condition 1) holds if and only if s(G(s)-D) is PR. If we then consider the Dual Positive Real Lemma in <cit.>, 1)  2) follows directly.



This proof is analogous to finding the W matrix in the sufficiency proof of Theorem 3.1 in <cit.>.


It follows from condition (b) of Theorem <ref> that the ARE (<ref>) has solution Z≥0. Let A_w = A - B_1D_21^-1C_2 and R_w = PA_w + A_w^TP + R_z,
where

    R_z = A^TC_1^TD_21^-1C_2 + (A^TC_1^TD_21^-1C_2)^T 
    
     - (D_21^-1C_2)^TR(D_21^-1C_2).

We can manipulate (<ref>) as follows

    ZA̅^T + A̅Z + Q̅R^-1Q̅^T    = 0,  
    
      (A - B_1D_21^-1C_2)Z + Z(A - B_1D_21^-1C_2) + R_z    = 0, 
    
      A_wZ + ZA_w^T + Z(R_w - PA_w - A_w^TP)Z    = 0,  
    
      A_wZ - ZPA_wZ + ZA_w^T - ZA_w^TPZ + ZR_wZ    = 0,  
    
      (I - ZP)A_wZ + ZA_w^T(I-PZ) + ZR_wZ    = 0, 
    
      A_wW + WA_w^T + WR_w    = 0,

  where

    A̅   = A + LC_2,    Q̅   = B_1+LD_21-ZA^TC_1^T.

It follows from condition (c) of Theorem <ref> that W=Z(I-PZ)^-1 is positive semi-definite and well defined. After algebraic manipulation we can rewrite A_w and R_w as

    A_w    = A_e + B_1R^-1C_e1 - B_1D_21^-1(C_e2+D_21R^-1C_e1), 
    
        R_w    = C_e1^TR^-1C_e1
       - (D_21^-1C_e2+RC_e1)^TR^-1(D_21^-1C_e2+RC_e1),

where

    A_e    = A - B_1R^-1C_1(A+B_2F) + B_1R^-1B_1^TP, 
    
        C_e1   = C_1B_2F, 
    
       C_e2   = C_2 - D_21R^-1C_1(A+B_2F) + D_21R^-1B_1^TP.

If we then substitute these into (<ref>), we see that W≥0 is a solution to the ARE

    W(A_e + L_eC_e2)^T + (A_e + L_eC_e2)W + Q_eR^-1Q_e^T = 0,

where

    L_e    = -( B_1R^-1 + W(D_21^-1C_e2-R^-1C_e1)^T )(R^-1D_21^T)^-1, 
    
     Q_e    = B_1+L_eD_21-WC_e1^T.

It then follows from Lemma <ref> that the system

    [ (A_e + L_eC_e2)^T   C_e1^T
    (B_1+L_eD_21)^T   B_1^TC_1^T ]

is PR. Therefore it follows from the Dual Positive Real Lemma in <cit.> that the dual system

    [ A_e + L_eC_e2   B_1+L_eD_21
    C_e1   C_1B_1 ]

has a solution V≥0 that satisfies the ARE

    V(A_e + L_eC_e2) + (A_e + L_eC_e2)^TV + Q_v^TR^-1Q_v = 0,

where

    Q_v    = C_e1 -(B_1+L_eD_21)V.



We now offer a proof for our main result.


Consider the closed-loop system with realization (<ref>). If we choose (x^T,x^T - x_k^T)^T as our state vector then our new realization can be represented by the transfer function G_cl(s) = [ A_cl   B_cl
C_cl   0 ], where

    A_cl   = [ A+B_2C_k   -B_2C_k
    A-A_k+B_2C_k-B_kC_2   A_k-B_2C_k ], 
    
    B_cl   =
    [ B_1
    B_1-B_kD_21 ],       
    C_cl =
    [ C_1   0 ].


We will show sufficiency by showing that G_cl(s) with the controller given by (<ref>)-(<ref>) satisfies the NI ARE

    X(Σ)    = [ X_11   X_21^T
    X_21   X_22 ]
    = Σ A_cl + A_cl^TΣ
       + (C_clA_cl-B_cl^TΣ)^TR^-1(C_clA_cl - B_cl^TΣ) = 0,

with a suitable choice of Σ. Thus by Lemma <ref> the closed-loop system is NI. It follows from condition (c) that the controller(<ref>)-(<ref>) is well defined. Using the definition of P given in the conditions of the theorem and the definition of V given in Corollary <ref>, we may define Σ as

    Σ   =  [ P   0
    0   V ]≥ 0.


We now decompose (<ref>) and calculate

    X_11   = P(A+B_2C_K) + (A+B_2C_K)^TP 
       + (C_1(A+B_2C_K)-B_1^TP)^TR^-1
       ·(C_1(A+B_2C_K)-B_1^TP),
    
     X_21   = V(A-A_k+B_2C_K-B_kC_2) - (B_2C_K)^TP
       - ((C_1B_2C_K)^T + V(B_1-B_kD_21))R^-1
       ·(C_1(A+B_2C_K)-B_1^TP),
    
     X_22   = V(A_k-B_2C_K) + (A_k-B_2C_K)^TV 
       + ((C_1B_2C_K)^T + V(B_1-B_kD_21))R^-1
       ·((C_1B_2C_K)^T + V(B_1-B_kD_21))^T.


Our choice of C_k = F results in X_11 = R(P) and X_11 = 0 follows from condition (a). Similarly, A_k and B_k result in algebraic cancellation leading to X_21 = 0. Finally, after substitution and appropriate algebraic manipulation we are left with X_22 = (<ref>) = 0. Thus, (<ref>) is satisfied and G_cl(s) is NI.



The necessity of conditions (a),(b),(c) can be proven as follows.
Assume a strictly proper controller exists such that the closed-loop transfer function (<ref>) is strictly negative imaginary and minimal. Then by Lemma <ref>, there exists a matrix

    Σ   =  [ Σ_11   Σ_12
    Σ_12^T   Σ_22 ] > 0,

such that
c 
  ΣA_c + A_c^T Σ+ (B_c^TΣ-C_cA_c)^TR^-1(B_c^TΣ-C_cA_c) = 0, 


where

    A_c    =  [ A   B_2C_k
    B_kC_2   A_k ],    
     B_c    = [ B_1
    B_kD_21 ], 
    
     C_c    = [ C_1   0 ],    
     R    = C_cB_c + B_c^TC_c^T >0.


Define the following transformation matrix T̃ which will be used to diagonalize our system:

    T̃   =  [ I   0
    Ẽ   I ],    Ẽ = -Σ_22^-1Σ_12^T.


We can use this transformation matrix to define the following matrices:

    Σ̃   =  T̃^TΣT̃ = [ P   0
    0   Σ_22 ], 
    Ã   = T̃^-1A_cT̃  = [ A+B_2C_KẼ   *
    *   * ], 
    B̃   =  T̃^-1B_c = [ B_1
    * ], 
    C̃   =  C_cA_cT̃ = [ C_1A + C_1B_2C_kẼ   * ],

where

    P    = Σ_11 - Σ_12Σ_22^-1Σ_12^T > 0.


If we pre-multiply (<ref>) by T̃^T and post-multiply by T̃, we are left with the following equality:
c
  Σ̃Ã + Ã^TΣ̃ + (B̃Σ̃-C̃)^TR^-1(B̃Σ̃-C̃) = 0.

The (1,1) block matrix of this equality satisfies
c 
  P(A+B_2C_KẼ) + (A+B_2C_KẼ)^TP + Q̅^TR^-1Q̅ = 0, 


 where

    Q̅   = B_1P-C_1A-C_1B_2C_kẼ.

By choosing F = C_kẼ we can conclude that P> 0 and F satisfy condition (a).



Now in order to show condition (b) is satisfied, define the following transformation matrix

    T̅   =  [ I   -E̅
    0   I ],    E̅ = Σ_11^-1Σ_12.


We use this transformation matrix to define the following matrices:

    Σ̅   =  T̅^TΣT̅ = [ Σ̅_11   0
    0   * ], 
    A̅   = T̅^-1A_cT̅  = [ A+E̅B_KC_2   *
    *   * ], 
    B̅   =  T̅^-1B_c = [ B_1+E̅B_kD_21
    0 ], 
    C̅   =  C_cA_cT̅ = [ C_1A   * ].


If we pre-multiply (<ref>) by T̅^T and post-multiply by T̅, we are left with the following equality:

    Σ̅_11A̅ + A̅^TΣ̅_11 + (B̅^TΣ̅_11-C̅)^TR^-1(B̅^TΣ̅_11-C̅) = 0.

We now define the matrix

    Z̅ = Σ̅^-1   =  [ Σ̅_11^-1   0
    0   * ]

and by Lemma <ref> the following equality is also satisfied:

    Z̅A̅^T+A̅Z̅+(B̅^T-C̅Z̅)^TR^-1(B̅^T-C̅Z̅) = 0.

Thus, the (1,1) block matrix of this equality satisfies

    Z(A+E̅B_KC_2)^T + (A+E̅B_KC_2)Z + Q̅R^-1Q̅^T= 0,

 where

    Z    = Σ̅_11^-1 > 0, 
    Q̅   = B_1+E̅B_kD_21 - ZA^TC_1^T.

By choosing L = E̅B_k we can conclude that Z>0 and L satisfy condition (b).



Finally, note that with P and Z defined as in (<ref>) and (<ref>), we have

    Z^-1 - P =      Σ_11 - (Σ_11 - Σ_12Σ_22^-1Σ_12^T), 
    
      =     Σ_12Σ_22^-1Σ_12^T ≥ 0, 
    
      =      Z^-1(Z - ZPZ)Z^-1≥ 0.

Thus, condition (c) is satisfied.




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