Consider the following auxiliary system:

$$ \dot{\tilde{x}}_{s} = \tilde{A}_{s}\tilde{x}_{s} + \tilde{B}_{s} \tilde{u}_{s}, \qquad \tilde{y}_{s} = \tilde{C}_{s}\tilde{x}_{s} + \tilde{D}_{s} \tilde{u} _{s}, $$

(20)

where

$$\begin{aligned}& \tilde{A}_{s} = A_{s} + B_{s}H(I - D_{s}H)^{ - 1}C_{s},\qquad \tilde{B}_{s} = B_{s} + B_{s}H(I - D_{s}H)^{ - 1}D_{s}, \\& \tilde{C}_{s} = (I - D_{s}H)^{ - 1}C_{s}, \qquad \tilde{D}_{s} = (I - D _{s}H)^{ - 1}D_{s}. \end{aligned}$$

It is easy to see that the design of the proper dynamic output feedback controller for reduced-order system (3)-(4) is equivalent to the strictly proper one for auxiliary system (19) with the following form:

$$ \dot{\tilde{\zeta }} (t) = M\tilde{\zeta } (t) + N \tilde{y}_{s}(t), \qquad \tilde{u}_{s}(t) = G\tilde{\zeta } (t), $$

(21)

where *M*, *N* and *G* are defined in (6).

Combining (19) and (20), we can see that the robustness problem considered here is to find some sufficient conditions such that the static output feedback gain matrix *H* and the strictly proper dynamic output feedback gain matrices *M*, *N* and *G* can assign the desired stable poles to the auxiliary systems and fast subsystem (5)-(6), respectively.

For the static output feedback of fast subsystem (5)-(6), using the pole-assignment technique, the gain matrix *H* can be designed to place \(\min (n_{ 1},p)\) eigenvalues arbitrarily close to their desired locations [10]. In fact, there are a lot of existing works addressing this problem, and various methods have been proposed, e.g., Riccati equation approach, rank-constrained condition, approach based on structural properties, bilinear matrix inequality (BMI) approaches, min-max optimization techniques, and linear matrix inequality approaches [28, 29]. Among them, the LMI approaches are much more efficient in dealing with synthesis problems [30–32], thus many results have also been obtained. In addition, the survey on the development of static output feedback can be found in [33].

For the strictly proper dynamic output feedback design of auxiliary system (20), a sufficient and necessary condition in terms of LMIs is given in the following theorem.

### Theorem 1

*Consider auxiliary system* (20). *There exists a strictly proper dynamic output feedback in the form of* (21) *such that the resulting closed system is asymptotically stable if and only if there exist matrices*
\(X > 0\), \(Y > 0\), Φ *and* Ψ *satisfying the following LMIs*:

\left(\begin{array}{cc}X& I\\ I& Y\end{array}\right)\ge 0,

(22)

$$\begin{aligned}& \tilde{A}_{s}^{T}X + X^{T}\tilde{A}_{s} + \Phi \tilde{C}_{s} + \tilde{C}_{s}^{T} \Phi^{T} < 0, \end{aligned}$$

(23)

$$\begin{aligned}& \tilde{A}_{s}Y + Y^{T}\tilde{A}_{s}^{T} - \tilde{B}_{s}\Psi - \Psi ^{T}\tilde{B}_{s}^{T} < 0. \end{aligned}$$

(24)

*If* (22)-(24) *hold*, *then the dynamic output feedback controller gain matrices in* (21) *can be chosen as*

$$ \begin{gathered} M = \bigl(X - Y^{ - 1}\bigr)^{ - T}\bigl(\tilde{A}_{s}^{T}Y^{ - 1} + X^{T}\tilde{A} _{s} - X^{T}\tilde{B}_{s} \Psi Y^{ - 1} + \Phi \tilde{C}_{s} - \Phi \tilde{D}_{s} \Psi Y^{ - 1}\bigr), \\ N = \bigl(Y^{ - 1} - X\bigr)^{ - T}\Phi , \qquad G = - \Psi Y^{ - 1}. \end{gathered} $$

(25)

### Proof

(Sufficiency) When (22)-(24) is satisfied, we first show that there exist matrices *X*, *Y*, Φ and Ψ such that \(Y^{ - 1} - X\) is nonsingular. In fact, if \(Y^{ - 1} - X\) is singular, then we can choose any positive definite matrix *X̄* and let \(\theta \in (0,1)\) satisfying that *θ* is not an eigenvalue of \((Y^{ - 1} - X)\bar{X}^{ - 1}\) and is small enough such that *X* is replaced by \(X + \theta \bar{X}\), the inequality in (22)-(24) still holds. Then it is not difficult to see that \(X + \theta \bar{X} - Y ^{ - 1}\) is nonsingular. Applying the dynamic output feedback controller with the parameters given in (25), we obtain the following closed-loop system:

\left(\begin{array}{c}{\dot{x}}_{s}(t)\\ \dot{\zeta}(t)\end{array}\right)=\left(\begin{array}{cc}{\tilde{A}}_{s}& -{\tilde{B}}_{s}\mathrm{\Psi}{Y}^{-1}\\ {({Y}^{-1}-X)}^{-T}\mathrm{\Phi}{\tilde{C}}_{s}& \mathrm{\Xi}\end{array}\right)\left(\begin{array}{c}{x}_{s}(t)\\ \zeta (t)\end{array}\right),

(26)

where \(\Xi = (X - Y^{ - 1})^{ - T}(\tilde{A}_{s}^{T}Y^{ - 1} + X^{T} \tilde{A}_{s} - X^{T}\tilde{B}_{s}\Psi Y^{ - 1} + \Phi \tilde{C}_{s})\). Set

{P}_{s}=\left(\begin{array}{cc}X& {Y}^{-1}-X\\ {Y}^{-1}-X& X-{Y}^{-1}\end{array}\right).

Considering that \(X - (Y^{ - 1} - X)(X - Y^{ - 1})^{ - 1}(Y^{ - 1} - X) = X + Y^{ - 1} - X > 0\). According to the Schur complement lemma, one has \(P_{s} > 0\). Furthermore, after some algebraic operations, it can be verified

{\mathrm{\Lambda}}_{s}^{T}{P}_{s}+{P}_{s}^{T}{\mathrm{\Lambda}}_{s}=\left(\begin{array}{cc}{\mathrm{\Delta}}_{11}& -{\mathrm{\Delta}}_{11}\\ -{\mathrm{\Delta}}_{11}& {\mathrm{\Delta}}_{22}\end{array}\right),

where

$$\begin{aligned}& \Delta_{11} = \tilde{A}_{s}^{T}X + X^{T}\tilde{A}_{s} + \Phi \tilde{C}_{s} + \tilde{C}_{s}^{T}\Phi^{T}, \\& \Delta_{22} = \Delta_{11} + \tilde{A}_{s}^{T}Y^{ - 1} + Y^{ - T} \tilde{A}_{s} - Y^{ - T}\bigl( \tilde{B}_{s}\Psi + \Psi^{T}\tilde{B}_{s} ^{T}\bigr)Y^{ - 1}. \end{aligned}$$

Pre- and post-multiplying both sides of (24) with \(Y^{ - T}\) and \(Y^{ - 1}\), we get

$$ \tilde{A}_{s}^{T}Y^{ - 1} + Y^{ - T} \tilde{A}_{s} - Y^{ - T}\bigl( \tilde{B}_{s}\Psi + \Psi^{T}\tilde{B}_{s}^{T}\bigr)Y^{ - 1} < 0. $$

It follows from (23) that \(\Delta_{22} < 0\). Noticing \(\Delta_{11} < 0\), \(\Delta_{22} - \Delta_{11} < 0\), and using the Schur complement lemma again, we obtain

\left(\begin{array}{cc}{\mathrm{\Delta}}_{11}& -{\mathrm{\Delta}}_{11}\\ -{\mathrm{\Delta}}_{11}& {\mathrm{\Delta}}_{22}\end{array}\right)<0.

Thus, \(\Lambda_{s}^{T}P_{s} + P_{s}^{T}\Lambda_{s} < 0\). That is, closed-loop system (20) is asymptotically stable.

(Necessity) Suppose that there exists a dynamic output feedback controller in (21) such that closed-loop system (20) is asymptotically stable. Then there exists a positive definite matrix \(Q_{s} > 0\) such that

$$ \Lambda_{s}^{T}Q_{s} + Q_{s}^{T} \Lambda_{s} < 0. $$

(27)

Denote

{Q}_{s}=\left(\begin{array}{cc}{Q}_{s1}& {Q}_{s2}\\ {Q}_{s3}& {Q}_{s4}\end{array}\right).

Then it is easy to see that \(Q_{s1} > 0\), \(Q_{s2} > 0\) and \(\Pi = Q _{s1} - Q_{s2}Q_{s4}^{ - 1}Q_{s3} > 0\). By the 1-1 block of (27), we have

$$ \tilde{A}_{s}^{T}Q_{s1} + Q_{s1}^{T} \tilde{A}_{s} + Q_{s3}^{T}N \tilde{C}_{s} + \tilde{C}_{s}^{T}N^{T}Q_{s3} < 0. $$

Set \(X = Q_{s1}\), \(\Phi = Q_{s3}^{T}N\). Then it can be seen that (23) holds. Now, let

\mathrm{\Upsilon}=\left(\begin{array}{cc}I& O\\ -{Q}_{s4}^{-1}{Q}_{s3}& I\end{array}\right).

Pre- and post-multiplying (27) by \(\Upsilon^{T}\) and ϒ, respectively, we have

$$ \Pi^{T}\tilde{A}_{s} + \tilde{A}_{s}^{T} \Pi - \Pi^{T}\tilde{B}_{s}GQ _{s4}^{ - 1}Q_{s3} - Q_{s3}^{T}Q_{s4}^{ - T}G^{T} \tilde{B}_{s}^{T} \Pi < 0. $$

Set \(Y = \Pi^{ - 1}\), \(\Psi = GQ_{s4}^{ - 1}Q_{s3}Y\), and pre- and post-multiplying the above inequality by \(Y^{T}\) and *Y*, respectively, one has that (24) holds.

Observe that \(X > 0\) and \(X - Y^{ - 1} = Q_{s2}Q_{s4}^{ - 1}Q_{s3} \ge 0\). By the Schur complement, we have that (22) is satisfied. This completes the proof. □

### Remark 2

Theorem 1 presents a sufficient and necessary condition for the existence of the dynamic output feedback controller of the reduced-order subsystem. Furthermore, a workable way for solving the gain matrices *M*, *N* and *G* is also given. It is worth mentioning that, under this condition, there is no other restriction for the dynamic output feedback controller design, except the system itself.

### Remark 3

The proposed method in this paper guarantees that the proper dynamic output feedback controller designed for the reduced-order subsystem is a stabilizing one for the full-order system. In particular, it is not difficult to see that the stability of the limit case (i.e., slow subsystems and fast subsystems) of the resulting closed-loop systems is still preserved as \(\varepsilon \to 0\). It is worth pointing out that the singular perturbation approach is not yet fully adopted in many literature works. Instead, the singular system approach is adopted, in which the singular perturbation parameter \(\varepsilon > 0\) is viewed as a static scalar. This brings some simplification to system analysis and synthesis. However, the performance of the limit case of the closed-loop systems may not be guaranteed as \(\varepsilon \to 0\), and even more, the basic requirement for poles placement cannot be satisfied.