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Stability robustness of a feedback interconnection of systems with negative imaginary frequency response

A necessary and sufficient condition, expressed simply as the DC loop gain (ie the loop gain at zero frequency) being less than unity, is given in this paper to guarantee the internal stability of a feedback interconnection of Linear Time-Invariant (LTI) Multiple-Input Multiple-Output (MIMO) systems with negative imaginary frequency response. Systems with negative imaginary frequency response arise for example when considering transfer functions from force actuators to co-located position sensors, and are commonly important in for example lightly damped structures. The key result presented here has similar application to the small-gain theorem, which refers to the stability of feedback interconnections of contractive gain systems, and the passivity theorem (or more precisely the positive real theorem in the LTI case), which refers to the stability of feedback interconnections of positive real systems. A complete state-space characterisation of systems with negative imaginary frequency response is also given in this paper and also an example that demonstrates the application of the key result is provided.


Notation
Let RH n×n ∞ denote the set of real-rational stable transfer function matrices of dimension (n × n). Let R and C denote fields of real and complex numbers respectively, and R n×n and C n×n denote real and complex matrices respectively of dimension (n × n). Let λ i (A) denote the i-th eigenvalue of a square complex matrix A and λ(A) denote the maximum eigenvalue for a square complex matrix A that has only real eigenvalues. Let ℜ(a) and ℑ(a) denote the real and imaginary parts respectively of a ∈ C. Let A T and A * denote the transpose and the complex conjugate transpose of a complex matrix and N (s) satisfy certain known properties (e.g. they are both bounded-real with product of gains less than unity, or they are both positive-real, etc), it is also possible to derive powerful theorems (such as the small-gain theorem [2], [3], or the passivity theorem [4], [5], etc) that use only limited information on M (s) and N (s) to establish the internal stability of this feedback interconnection. This is powerful and interesting because it provides a mechanism to derive robust stability results when systems are perturbed by uncertain dynamics that are quantified only in terms of restricted information (e.g. stable and contractive gain for the small gain theorem, or stable and positive real for the passivity theorem, etc).
In this paper, we derive a new result of a similar flavor. We assume that both M (s) and N (s) are LTI MIMO stable systems with "negative imaginary frequency response" 1 and use this information to derive a necessary and sufficient internal stability condition using only limited information on M (s) and N (s).
We now show why systems with negative imaginary frequency response are important in engineering applications. We will do this via a simple example. Consider a lightly damped structure with co-located position sensors and force actuators. Lightly damped structures with co-located position sensors and force actuators can typically be modeled by a (possibly infinite) sum of second order transfer functions as follows: For the purpose of control systems design, however, one typically tends to include only a small finite number of modes (h ≪ H) in the modeling of such systems, thereby giving rise to spillover unmodeled dynamics (i.e. unmodeled dynamics due to the lightly-damped modes not included in the plant model). That is, let P (s) be the truncated plant model used for control systems design and ∆(s) be the spillover dynamics, both given by: It is typically an important, though difficult, design specification to ensure that the closed-loop system retains stability in the presence of such spillover dynamics ∆(s). Since the relative degree of such spillover dynamics is more than unity, standard positive real analysis [4], [5] will not be very helpful in establishing robust stability and since these systems tend to be highly resonant, application of the small gain theorem [2], [3] would typically be very conservative. However, it is readily noticed that such spillover dynamics ∆(s) are stable and satisfy a negative imaginary frequency response property. The DC gain of the spillover dynamics is simply H i=(h+1) k i . Also, satisfaction of stability and the negative imaginary frequency response property is invariant to values of ζ i > 0 and ω n,i > 0 for all i ∈ [h + 1, H]. Consequently, provided a controller C(s) is designed so as to make the closed-loop transfer function C 1+P C from plant output disturbances to plant input satisfy a negative imaginary frequency response property with DC gain strictly less than 1/( H i=(h+1) k i ), then robust stability to all spillover dynamics ∆(s), and more, will hold for any value of ζ i > 0 and ω n,i > 0 for all i ∈ [h + 1, H], and hence C(s) will also robustly stabilize P ∆ (s).
Note that a similar condition specifically for a subclass of SISO systems has existed in the Positive Position Feedback control literature [6], [7] for some time. It is also not difficult to see how such a condition arises in SISO systems via a Nyquist diagram sketch. In fact, most controller synthesis and analysis in Positive Position Feedback control is based on graphical techniques using Nyquist plots, or non-convex parameter optimization [8]. In this paper, we do three things: (a) we formalize the robustness qualities of feedback interconnections of systems with negative imaginary frequency response via a mathematical theorem and corollary (as opposed to graphical Nyquist sketches); (b) we extend the ideas to MIMO LTI systems and show that even in that case, a necessary and sufficient condition for the internal stability of such systems is that the DC loop-gain (measured in a precise sense) is less than unity; and (c) we give a complete state-space characterization of MIMO LTI systems with negative imaginary frequency response which may, in future work, underpin controller synthesis.
It is worth also pointing out that while a transfer function from force actuators to co-located position sensors has typically negative imaginary frequency response, the corresponding transfer function from force actuators to co-located velocity sensors has typically a positive real response. Consequently, it is legitimate to wonder whether simply replacing M (s) with sM (s) and N (s) with − 1 s N (s) in Figure 1 to obtain a negative feedback interconnection and using standard positive real analysis [4], [5] would do the trick? The an-swer to this question is "no, it does not" as 1 s N (s) is not stable, sM (s) is not always guaranteed to be proper, and most importantly positive real analysis yields an unconditional stability result whereas the interconnection of two systems with negative imaginary frequency response will always be conditionally stable (see Theorem 5). Of course, there are some connections between positive real systems and systems with negative imaginary frequency response which in fact will be exploited in Lemma 1 where we give a complete state-space characterization of systems with negative imaginary frequency response, but the differences are also important and should not be discounted (e.g. arising from frequencies ω = 0 and ω = ∞).

II. SOME TECHNICAL RESULTS
In this section, we generate the technical machinery that will enable us to concisely prove the main result in the next section. First, for the sake of brevity, let us define the following two sets of "stable systems with negative imaginary frequency response" as follows: The first lemma gives a complete state-space characterisation of elements in C . It hence also provides a test to easily check whether a transfer function matrix belongs to set C or not. Testing whether a transfer function belongs to set C s or not requires an additional check on transmission zeros of R(s) − R ∼ (s) in the open frequency region (0, ∞).

Lemma 1: Let
A B C D be a minimal state-space realisation of a transfer matrix R(s). Then R(s) ∈ C if and only if A is Hurwitz, D = D * and there exists a real matrix Y > 0 such that Proof: The two statements are connected via a sequence of equivalent reformulations: This equivalence follows on nothing that R(s) This equivalence is via the Positive Real Lemma (see for example [4] together with the fact that (CA, A) is observable or [2,Thms 13.25,13.26] together with the fact that (L * L, A) is observable). (e) A is Hurwitz, D = D * and ∃X > 0, L, W such that The second lemma relates the gain at zero frequency and the gain at infinite frequency for systems with negative imaginary frequency response.

Proof: Given a minimal realisation R(s) =
A B C D ∈ C and applying Lemma 1, we get which concludes the proof for the non-strict inequality. Now, we focus on R(s) ∈ C s ⇔R(s) := (R(s) − D) ∈ C s (since D = D * by Lemma 1) and suppose there exists an x ∈ R n×n such thatR(0)x = 0. Then, it follows that CY C * x = 0 which implies that C * x = 0 as Y > 0. This then also gives that Bx = 0 via B = −AY C * which yieldŝ is nonsingular for all ω ∈ (0, ∞) and hence the only possible x ∈ R n×n such thatR(0)x = 0 is x = 0. This shows that R(0) is also nonsingular and thusR(0) > 0. This concludes the proof.
The following lemma gathers some straightforward computations which help us understand properties of systems with negative imaginary frequency response.
Lemma 3: Given R(s) ∈ C , ∆(s) ∈ C and R s (s) ∈ C s . Then R(s) + ∆(s) ∈ C and R s (s) + ∆(s) ∈ C s . Proof: Trivial. The final technical lemma provides a matrix result that states that unity is not in the spectrum of matrix AB when matrices A and B satisfy certain negative imaginary properties.

III. THE MAIN RESULT
The key result in this paper is Theorem 5 below. It is an analysis theorem that states that provided one system belongs to class C and the other system belongs to class C s , then a necessary and sufficient condition 2 for internal stability of a positive feedback interconnection of these two systems is to check that the DC loop gain (i.e. the loop gain at zero frequency) is less than unity.
But T > 0 and Φ nonsingular also imply A has no eigenvalue at the origin. We now invoke Lemma 4 and use the fact that M (s) ∈ C and N (s) ∈ C s to conclude that det(I − M (jω)N (jω)) = 0 ∀ω ∈ (0, ∞), which in turn is equivalent to A having no eigenvalue at jω for all ω ∈ (0, ∞). ] One may wonder whether Integral Quadratic Constraint (IQC) theory [9] captures the sufficiency part of Theorem 5 or not. This question is subtle and its answer is non-trivial. However, in short the answer is "no, it does not". The subtlety of the question arises from the fact that IQC theory deals with the full frequency range ω ∈ R whereas systems N (s) ∈ C s satisfy the frequency domain inequality j[N (jω)−N (jω) * ] > 0 only on an open frequency interval ω ∈ (0, ∞). This strict inequality cannot be satisfied at ω = 0 because via Lemma 2 we know that N (0) = N (0) * . This fact causes the main theorem (Theorem 1) in [9], which underpins IQC theory, to be inapplicable by violation of its suppositions.

IV. ILLUSTRATIVE EXAMPLE
Consider the lightly damped mechanical plant depicted in Figure 2, which consists of two unit masses constrained to 1  slide rectilinearly on a frictionless table. Each mass is attached to a fixed wall via a spring of known unit stiffness and via a damper of known unit viscous resistance. Furthermore, the two unit masses are coupled together via a spring of uncertain stiffness k N/m and via a damper of uncertain viscous resistance α Ns/m. A force is applied to each mass (denoted by u 1 and u 2 respectively) and the displacement of each mass is measured (denoted by y 1 and y 2 respectively).
Although this is not an extremely difficult design problem, it does illustrate a number of important points arising from the results in this paper, as it contains key features such as an uncertain MIMO system with uncertainty that has negative imaginary frequency response. Similar examples have been considered in the literature as benchmark problems by a number of authors, including [1], [10] to mention a few.
It is clear that P ∆ (s) is uncertain because k and α are unknown.
For the purpose of control system design, we now choose to split the uncertain plant P ∆ (s) as P ∆ (s) = P (s) + ∆(s) where P (s) is the nominal completely known plant model and ∆(s) is the uncertain remainder. Via partial fraction expansion, we see that P (s) = Ψ diag( 1 2 p(s), 0)Ψ * and ∆(s) = Ψ −1 diag( 1 2 δ(s), 0)Ψ − * . It is then a simple computation to check that ∆(s) ∈ C for all α > 0 and k > 0. Now let us consider the controlled closed-loop system given in Figure 3, and let C(s) be chosen as C(s) := Ψ − * diag( −2(s 2 +s+1) 2s 3 +4s 2 +4s+3 , −1 s+1 )Ψ −1 . Then, define M (s) := −C(s)(I + P (s)C(s)) −1 to be the transfer function matrix mapping w to z so that the closed-loop system in Figure 3 can be rearranged into Figure 4 for robust stability analysis. Since P (s) ∈ RH 2×2 ∞ , internal stability of the nominal feedback loop (i.e. pretending ∆(s) = 0) is equivalent to M (s) ∈ RH 2×2 ∞ . Furthermore, since ∆(s) ∈ C , we additionally require M (s) ∈ C s to be able to apply the proposed results Theorem 5 and Corollary 6 to conclude robust stability.
V. CONCLUSIONS SISO LTI systems with negative imaginary frequency response have been studied in the context of positive position feedback control of lightly damped structures and the analysis/synthesis methods there depended on graphically Nyquist plots. This paper generalises the key stability result to MIMO LTI systems with negative imaginary frequency response showing that even in this case, a necessary and sufficient condition for the internal stability of such systems is that the DC loop gain (measured in a particular precise sense) is less than unity. We also gave in this paper a complete statespace characterisation of MIMO LTI systems with negative imaginary frequency response. This could possibly be used in future work to assist with synthesising systems with negative imaginary frequency response.
The next steps to extend applicability of this research are: (a) devise a controller synthesis procedure that generates systems that belong to either class C or C s ; and (b) generalise the analysis result given in this article to allow one (or possibly both) systems to be nonlinear and/or time-varying. Focusing on the latter, we point out that [11] has derived a theory for SISO nonlinear systems with counter-clockwise input-output dynamics that is closely related to this work. It is consequently interesting to see whether the ideas in [11] generalise to MIMO systems or not, or whether use of dissipativity theory may lead to the required MIMO nonlinear generalisations.