Reduced-order observer-based robust leader-following control of heterogeneous discrete-time multi-agent systems with system uncertainties

Abstract

In this paper, the leader-following control of heterogeneous discrete-time multi-agent systems (HD_MASs) in the presence of system uncertainties under directed topology is addressed. It aims to achieve reference tracking, disturbance rejection and robust control while the references and disturbances are generated by an autonomous exosystem. In practice, these agents are often different types of devices, thus they have different internal dynamics. Moreover, it is difficult to measure all states of each aircraft due to high cost or technical limitation. In this case, a novel leader-following output consensus problem is formulated and solved in this paper. Firstly, an appropriate linear transformation is proposed to divide the state information of each agent into measurable and unmeasurable parts. Then the reduced-order observer is designed only for unmeasurable parts. Based on the designed observer, the distributed feedback controller is proposed such that the outputs of all followers reach the same trajectory with the leader. In light of the internal model principle and discrete-time algebraic Riccati equation, the robust leader-following consensus of HD_MASs is achieved. Furthermore, this paper extends the results to continuous-time multi-agent systems. Finally, several simulation experiments are presented to verify the effectiveness of the theoretical results.

Appendices

Appendix A: Proof of Lemma 3.1

Since the matrix pairs (A_i, B_i) are stabilizable, (A_i, C_i) are detectable, for any $\lambda _{i}\in \mathcal {C}$, the matrix pairs (A_i − λ_iI_n, B) with n × (n + q) dimensions are full row rank, and the matrix pairs $\left ({\begin {array}{*{20}{c}} A_{i}-\lambda _{i} I_{n} \\ C_{i} \end {array}} \right )$ with (n + p) × n dimensions are full column rank. For any invertible matrix $\mathcal {T}_{i}$, one has $Rank~ \mathcal {T}_{i}^{-1}(A_{i}-\lambda _{i} I_{n}, B_{i})\mathcal {T}_{i}=Rank~ (\hat {A}_{i}-\lambda _{i} I_{n}, \hat {B}_{i}\mathcal {T}_{i})=n,$ i.e., there exist matrices K_i such that $\hat {A}_{i}+\hat {B}_{i}\mathcal {T}_{i}K_{i}$ are Schur. Then, there exist matrices $\bar {K}_{i}=\mathcal {T}_{i}K_{i}$ such that $\hat {A}_{i}+\hat {B}_{i}\bar {K}_{i}$ are Schur. For any invertible matrix $\mathcal {T}_{i}$, one also has $Rank~ \left ({\begin {array}{*{20}{c}} A_{i}-\lambda _{i} I_{n} \\ C_{i} \end {array}} \right )=Rank~ \left ({\begin {array}{*{20}{c}} \mathcal {T}_{i}^{-1} & \mathbf {0}\\ \mathbf {0} & I_{p} \end {array}} \right ) \left ({\begin {array}{*{20}{c}} A_{i}-\lambda _{i} I_{n} \\ C_{i} \end {array}} \right )\mathcal {T}_{i}$. Since $C_{i}\mathcal {T}_{i}=(I_{p}, \mathbf {0})$, we have $\left ({\begin {array}{*{20}{c}} \mathcal {T}_{i}^{-1} & \mathbf {0}\\ \mathbf {0} & I_{p} \end {array}} \right )\left ({\begin {array}{*{20}{c}} A_{i}-\lambda _{i} I_{n} \\ C_{i} \end {array}} \right )\mathcal {T}_{i}=\left ({\begin {array}{*{20}{c}} A_{i}^{11}-\lambda _{i} I_{p} & A_{i}^{12}\\ A_{i}^{21} & A_{i}^{22}-\lambda _{i} I_{n-p}\\ I_{p} & \mathbf {0} \end {array}} \right )$. Therefore, the matrices $\left ({\begin {array}{*{20}{c}} A_{i}^{12} \\ A_{i}^{22}-\lambda _{i} I_{n-p} \end {array}} \right )$are full column rank, i.e., the matrix pairs $(A_{i}^{22}, A_{i}^{12})$ are detectable, for any $\lambda _{i}\in \mathcal {C}$. The proof has been completed.

Appendix B: Proof of Lemma 3.3

Since the gain matrix L_i is defined as $L_{i}=A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}$, we have

$$ \begin{array}{@{}rcl@{}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} &\quad&(A_{i}-s_{i}L_{i}C_{i})P_{i}(A_{i}-s_{i}L_{i}C_{i})^{*}-P_{i} \\ &= & A_{i}P_{i}{A_{i}^{T}}-s_{i}^{*}A_{i}P_{i}{C_{i}^{T}}{L_{i}^{T}}-s_{i}L_{i}C_{i}P_{i}{A_{i}^{T}}+s_{i}s_{i}^{*}L_{i}C_{i}P_{i}{C_{i}^{T}}{L_{i}^{T}}-P_{i}\\ & = & A_{i}P_{i}{A_{i}^{T}}-s_{i}^{*}A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}-s_{i}A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} &\quad& +s_{i}s_{i}^{*}A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}-P_{i}\\ & = &A_{i}P_{i}{A_{i}^{T}}-P_{i}-A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}+(1+s_{i}s_{i}^{*}-s_{i}^{*}-s_{i})A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}.\\ & = &-Q_{i}+(1+s_{i}s_{i}^{*}-s_{i}^{*}-s_{i})A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}. \end{array} $$

(27)

According to Lyapunov stability theory, $\forall P_{i}={P_{i}^{T}}>0$, the matrices A_i − s_iL_iC_i are stabilizable for $s_{i}\in \mathcal {C}$, if and only if (A_i − s_iL_iC_i)P_i(A_i − s_iL_iC_i)^∗− P_i < 0, i.e.

$$ -Q_{i}+(1+s_{i}s_{i}^{*}-s_{i}^{*}-s_{i})A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}<0. $$

(28)

Since $Q_{i}={Q_{i}^{T}}$, there exist matrices $Q_{si}=Q_{s_{i}}^{T}>0$ such that $Q_{s_{i}}=Q_{s_{i}}^{-1}Q_{i}$. Post-multiplying and pre-multiplying (28) by $Q_{s_{i}}^{-1}$, respectively, one has

$$ -I_{n}+|s_{i}-1|^{2}Q_{s_{i}}^{-1}A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}Q_{s_{i}}^{-1}<0, $$

(29)

where $1+s_{i}s_{i}^{*}-s_{i}^{*}-s_{i}=(1-s_{i})(1-s_{i})^{*}=|s_{i}-1|^{2}$. By the (29), we get

$$ -1+|s_{i}-1|^{2}\lambda_{k}[Q_{s_{i}}^{-1}A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}Q_{s_{i}}^{-1}]<0, $$

(30)

if the eigenvalues $\lambda _{k}[Q_{s_{i}}^{-1}A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}Q_{s_{i}}^{-1}]>0$ and satisfy $|s_{i}-1|^{2}<\frac {1}{max_{k=1,2,\ldots , n}\lambda _{k}[Q_{s_{i}}^{-1}A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}Q_{s_{i}}^{-1}]},$ the (30) holds. Moreover, if $\lambda _{k}[Q_{s_{i}}^{-1}A_{i}P_{i}{C_{i}^{T}}(C_{i}P_{i}{C_{i}^{T}})^{-1}C_{i}P_{i}{A_{i}^{T}}Q_{s_{i}}^{-1}]=0, k=1, 2,\ldots , n$, the (30) also holds for any $s_{i}\in \mathcal {C}$. Similarly, if $c_{i}\in \mathcal {C}$ is distributed in the stable domain ${\varPhi }_{c_{i}}=\{c_{i}\in \mathcal {C}: |c_{i}-1|^{2}<\delta _{c_{i}}\},$ where $\delta _{c_{i}}^{-1}=max_{k=1,2,\ldots , n}\lambda _{k}[Q_{c_{i}}^{-1}{A_{i}^{T}}P_{i}B_{i}({B_{i}^{T}}P_{i}B_{i})^{-1}{B_{i}^{T}}P_{i}A_{i}Q_{c_{i}}^{-1}], Q_{c_{i}}=Q_{c_{i}}^{-1}Q_{i}>0$, the matrices A_i + c_iB_iK_i are Schur. This completes the proof.

Appendix C: Proof of Theorem 3.1

If the digraph $\mathcal {G}$ contains a directed spanning tree, according to Lemma 2.1, all eigenvalues of the matrix $\theta {\mathcal{H}}$ have positive real parts. By the Jordan canonical form theorem, there is a nonsingular matrix T_s ∈ R^N×N satisfying $\theta {\mathcal{H}}=T_{s}JT_{s}^{-1}$, where $J=block~ diag(J_{N_{1}}(\lambda _{1}^{\prime }), \ldots , J_{N_{m}}(\lambda _{m}^{\prime })),N_{1}+\ldots +N_{m}=N, \lambda _{1}^{\prime }<\ldots <\lambda _{m}^{\prime }, \lambda _{1}=\cdots =\lambda _{N_{1}}=\lambda _{1}^{\prime }, \lambda _{N_{1}+1}=\lambda _{N_{1}+2}=\cdots =\lambda _{N_{1}+N_{2}}=\lambda _{2}^{\prime }, \ldots \lambda _{N_{1}+N_{2}+\cdots +N_{m-1}+1}=\lambda _{N_{1}+N_{2}+\cdots +N_{m-1}+2}=\cdots =\lambda _{N_{1}+N_{2}+\cdots +N_{m-1}+N_{m}}=\lambda _{m}^{\prime }, J_{N_{l}}(\lambda _{l}^{\prime })=\left ({\begin {array}{*{20}{c}} \lambda _{l}^{\prime } & 1 & & \\ &\lambda _{l}^{\prime }& {\ddots } & \\ & & {\ddots } & 1\\ & & & \lambda _{l}^{\prime } \end {array}} \right ), l=1,\ldots , m,$ Let $T_{1}=\left ({\begin {array}{*{20}{c}} I_{Np} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & I_{N(n-p)}& \mathbf {0} & \mathbf {0}\\ \mathbf {0} & \mathbf {0} & \theta {\mathcal{H}}\otimes I_{s_{m}} & \mathbf {0}\\ \mathbf {0} & I_{N(n-p)} & \mathbf {0} & I_{N(n-p)} \end {array}} \right ),\\ T_{2}=\left ({\begin {array}{*{20}{c}} T_{s}\otimes I_{p} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & T_{s}\otimes I_{n-p} & \mathbf {0} & \mathbf {0}\\ \mathbf {0} & \mathbf {0} & T_{s}\otimes I_{s_{m}} & \mathbf {0}\\ \mathbf {0} & \mathbf {0} & \mathbf {0} & T_{s}\otimes I_{n-p} \end {array}} \right ),$$\bar {A}_{c} =T_{2}^{-1}T_{1}^{-1}A_{c}T_{1}T_{2}= \left ({\begin {array}{*{20}{c}} \bar {A}_{c}^{11} & \bar {A}_{c}^{12}\\ \mathbf {0} & A_{22}-LA_{12} \end {array}} \right ),$ where

$\bar {A}_{c}^{11}=\left ({\begin {array}{*{20}{c}} A_{11}+B_{1}K_{2}(J\otimes I_{p}) & A_{12}+B_{1}K_{1}(J\otimes I_{n-p}) & B_{1}K_{3}(J\otimes I_{s_{m}}) \\ A_{21}+B_{2}K_{2}(J\otimes I_{p}) & A_{22}+B_{2}K_{1}(J\otimes I_{n-p})& B_{2}K_{3}(J\otimes I_{s_{m}}) \\ I_{N}\otimes G_{2} & \mathbf {0} & I_{N}\otimes G_{1} \end {array}} \right ), \bar {A}_{c}^{12}=\left ({\begin {array}{*{20}{c}} B_{1}K_{1}(J\otimes I_{n-p}) \\ B_{2}K_{1}(J\otimes I_{n-p})\\ \mathbf {0} \end {array}} \right )$. It is obvious that A_c is stabilizable if and only if $\bar {A}_{c}$ is stabilizable. According to Theorem 3 in [27], A_c is Schur if and only if $\left ({\begin {array}{*{20}{c}} \hat {A}_{i}+\lambda _{i}\hat {B}_{i}(K_{2i}, K_{1i}) &\lambda _{i}\hat {B}_{i}K_{3i}\\ (G_{2}, ~\mathbf {0}) & G_{1} \end {array}} \right )$ and $A_{i}^{22}-L_{i}A_{i}^{12}, i=1, 2, \ldots , N$ are Schur. This completes the proof.

Appendix D: Proof of Theorem 3.2

According to Theorem 3.1, A_c is stabilizable if and only if $A_{i}^{22}-L_{i}A_{i}^{12}$ and $\left ({\begin {array}{*{20}{c}} \hat {A}_{i}+\lambda _{i}\hat {B}_{i}(K_{2i}, K_{1i}) &\lambda _{i}\hat {B}_{i}K_{3i}\\ (G_{2}, ~\mathbf {0}) & G_{1} \end {array}} \right )$are Schur. It thus follows from Lemma 3.1 that $(A_{i}^{22}, A_{i}^{12})$ is completely detectable. The gain matrix $L_{i}=A_{i}^{22}P_{i}(A_{i}^{12})^{T}(A_{i}^{12}P_{i}(A_{i}^{12})^{T})^{-1}$ can be obtained by the following discrete-time algebraic Riccati equation

$$ \small A_{i}^{22}P_{i}(A_{i}^{22})^{T}-P_{i}-A_{i}^{22}P_{i}(A_{i}^{12})^{T}(A_{i}^{12}P_{i}(A_{i}^{12})^{T})^{-1}A_{i}^{12}P_{i}(A_{i}^{22})^{T}+Q_{i}=0. $$

(31)

Let $\mathcal {K}_{i}=(K_{2i}, K_{1i}, K_{3i})$, we have $\left ({\begin {array}{*{20}{c}} \hat {A}_{i}+\lambda _{i}\hat {B}_{i}(K_{2i}, K_{1i}) &\lambda _{i}\hat {B}_{i}K_{3i}\\ (G_{2},~ \mathbf {0}) & G_{1} \end {array}} \right )=\mathcal {A}_{i}+\lambda _{i}{\mathcal{B}}_{i}\mathcal {K}_{i}$. By Lemmas 3.1 and 3.2, the matrix pairs $(\mathcal {A}_{i}, {\mathcal{B}}_{i})$ are stabilizable. Then, according to Lemma 3.3, if λ_i is distributed in the stable domain Φ_i, $\mathcal {A}_{i}+\lambda _{i}{\mathcal{B}}_{i}\mathcal {K}_{i}$ is Schur, where $\mathcal {K}_{i}=-({\mathcal{B}}_{i}^{T}\mathcal {P}_{i}{\mathcal{B}}_{i})^{-1}{\mathcal{B}}_{i}^{T}\mathcal {P}_{i}\mathcal {A}_{i}$, and $\mathcal {P}_{i}$ can be solved by (12). Therefore, the system A_c is Schur. This completes the proof.

Appendix E: Proof of Theorem 3.3

According to the proof in Theorem 3.2, it is easy to find suitable feedback gain matrix such that the nominal form A_c of system matrix $\bar {A}_{c}$ is Schur. To solve the robust leader-following consensus problem by distributed feedback controller (9), the following Sylvester equation is considered

$$ \bar{X}_{c}A_{0}=\bar{A}_{c}\bar{X}_{c}+\bar{W}_{c}, $$

(32)

with $\bar {X}_{c}\in R^{N(2n-p+s_{m})\times s}$. For each sufficiently small Δ, $\bar {A}_{c}$ is stable. By the Assumption 3.2, (32) has an unique solution $\bar {X}_{c}$. Let $\bar {X}_{c}=(\bar {X}_{c1}^{T}, \bar {X}_{c2}^{T}, \bar {X}_{c3}^{T}, \bar {X}_{c4}^{T})^{T}$, where $\bar {X}_{c1}, \bar {X}_{c2}, \bar {X}_{c3}$ and $\bar {X}_{c4}$ have appropriate dimensions, we have

$$ \begin{array}{lll} \bar{X}_{c3}A_{0} &= (\theta \mathcal{H}\otimes G_{2})\bar{X}_{c1}+(I_{N}\otimes G_{1})\bar{X}_{c3}-(\theta \mathcal{A}_{0}\otimes G_{2}F_{0})(1_{N}\otimes I_{s})\\ &= (\theta \mathcal{H}\otimes G_{2})\bar{X}_{c1}+(I_{N}\otimes G_{1})\bar{X}_{c3}-(\theta \mathcal{H}\otimes G_{2}F_{0})(1_{N}\otimes I_{s})\\ &= (I_{N}\otimes G_{1})\bar{X}_{c3}+(I_{N}\otimes G_{2})[(\theta \mathcal{H}\otimes I_{p})\bar{X}_{c1}-(\theta \mathcal{H}\otimes F_{0})(1_{N} \otimes I_{s})]. \end{array} $$

(33)

Since (I_N ⊗ G₁, I_N ⊗ G₂) incorporates a pN-copy internal model of A₀, on the basis of Lemma 3.2, we obtain

$$ (\theta \mathcal{H}\otimes I_{p})\bar{X}_{c1}-(\theta \mathcal{H}\otimes F_{0})(1_{N} \otimes I_{s})=(\theta \mathcal{H}\otimes I_{p})[\bar{X}_{c1}-(I_{N}\otimes F_{0})(1_{N} \otimes I_{s})]=0. $$

(34)

Moreover, the matrix ${\mathcal{H}}$ is reversible, we have $\bar {X}_{c1}-(I_{N}\otimes F_{0})(1_{N} \otimes I_{s})=0$. Let $\hat {\zeta }(k)=\zeta (k)-\bar {X}_{c}\omega (k)$, and consider the following equation

$$ \small \hat{\zeta}(k+1)= \zeta(k+1)-\bar{X}_{c}\omega(k+1)= \bar{A}_{c}\zeta(k)+(\bar{W}_{c}-\bar{A}_{c}\bar{X}_{c}-\bar{W}_{c})\omega(k)= \bar{A}_{c}\hat{\zeta}(k). $$

(35)

Since $\bar {A}_{c}$ is Schur for each sufficiently small Δ, thus $\hat {\zeta }(k)\rightarrow 0 ~(k\rightarrow \infty )$, then $x_{m}(k)-\bar {X}_{c1}\omega (k)\rightarrow 0 ~(k\rightarrow \infty )$. The purpose of this paper is to solve the robust leader-follower consensus of HD_MASs (1) and (2), i.e. $e_{i}(k)=y_{i}(k)-y_{r}(k)\rightarrow 0, k\rightarrow \infty$. After the linear transformation (5), the error e_i(k) can be expressed as e_i(k) = x_mi(k) − y_r(k) = x_mi(k) − F₀ω(k). The global form of e_i(k) could be denoted as $e(k)=x_{m}(k)-(I_{N}\otimes F_{0})(1_{N}\otimes I_{s})\omega (k)=x_{m}(k)-\bar {X}_{c1}\omega (k)\rightarrow 0, k\rightarrow \infty$. Therefore, the robust leader-follower consensus of HD_MASs (1) and (2) under directed topology is solved.

Appendix F: Proof of Theorem 4.1

By Theorem 3.1, A_c is Hurwitz if and only if the matrices $A_{i}^{22}-L_{i}A_{i}^{12}$ and $\left ({\begin {array}{*{20}{c}} \hat {A}_{i}+\lambda _{i}\hat {B}_{i}(K_{2i}, K_{1i}) &\lambda _{i}\hat {B}_{i}K_{3i}\\ (G_{2}, ~\mathbf {0}) & G_{1} \end {array}} \right )$are Hurwitz, where λ_i(λ₁ ≤ λ₂ ≤⋯ ≤ λ_N), i = 1, 2,…, N are the eigenvalues of the matrix ${\mathcal{H}}$. Let $\mathcal {A}_{i}=\left ({\begin {array}{*{20}{c}} \hat {A}_{i} & \mathbf {0} \\ (G_{2},~ \mathbf {0}) & G_{1} \end {array}} \right ),$${\mathcal{B}}_{i}=\left ({\begin {array}{*{20}{c}} \hat {B}_{i}\\ \mathbf {0} \end {array}} \right ),$and $\mathcal {K}_{i}=(K_{2i}, K_{1i}, K_{3i})=-(min~Re(\lambda _{i}))^{-1}({\mathcal{B}}_{i})^{T}\mathcal {P}_{i}$, where $\mathcal {P}_{i}$ is the solution of the following Riccati equation

$$ \mathcal{A}_{i}^{T}\mathcal{P}_{i}+\mathcal{P}_{i}\mathcal{A}_{i}+I_{n+p}-\mathcal{P}_{i}\mathcal{B}_{i}\mathcal{B}_{i}^{T}\mathcal{P}_{i}=\mathbf{0}, $$

(36)

we have $\left ({\begin {array}{*{20}{c}} \hat {A}_{i}+\lambda _{i}\hat {B}_{i}(K_{2i}, K_{1i}) &\lambda _{i}\hat {B}_{i}K_{3i}\\ (G_{2},~ \mathbf {0}) & G_{1} \end {array}} \right )=\mathcal {A}_{i}+\lambda _{i}{\mathcal{B}}_{i}\mathcal {K}_{i}, i=1,2,\ldots , N$. By Lemma 3.2, the matrix pair $(\mathcal {A}_{i}, {\mathcal{B}}_{i}), i=1,2,\ldots , N$ is Hurwitz. By Lemma 4.1, $\mathcal {A}_{i}+\lambda _{i}{\mathcal{B}}_{i}\mathcal {K}_{i}$ is Hurwitz, i.e. the matrix $\left ({\begin {array}{*{20}{c}} \hat {A}_{i}+\lambda _{i}\hat {B}_{i}(K_{2i}, K_{1i}) &\lambda _{i}\hat {B}_{i}K_{3i}\\ (G_{2}, ~\mathbf {0}) & G_{1} \end {array}} \right )$is Hurwitz. Besides, on the basis of Lemma 3.1, the matrix $(A_{i}^{22}, A_{i}^{12})$ is completely detectable, and the gain matrix $L_{i}=P_{i}(A_{i}^{12})^{T}R_{i}^{-1}$ can be obtained by the following continuous-time algebraic Riccati equation

$$ A_{i}^{22}P_{i}+P_{i}(A_{i}^{22})^{T}+Q_{i}-P_{i}(A_{i}^{12})^{T}R_{i}^{-1}A_{i}^{12}P_{i}=0, $$

(37)

where $Q_{i}={Q_{i}^{T}}>0$ and $R_{i}={R_{i}^{T}}>0$ are arbitrary positive definite matrices. Based on the above analysis, the closed loop system A_c is Hurwitz. This completes the proof.