UDC 62-501.12

MATHEMATICS

E. D. YAKUBOVICH

EXPONENTIAL STABILIZATION OF LINEAR SYSTEMS

(Presented by Academician V. I. Smirnov on 4 X 1968)

Methods are well known for determining the parameters entering into the coefficients of a linear system of differential equations for which the solutions of the system satisfy the estimate \(|\mathbf{x}(t)| \leq C e^{-\alpha t} |\mathbf{x}(0)|\) with a prescribed \(\alpha \geq 0\). In this, however, the constant \(C\) remains, generally speaking, undetermined, which reduces the practical significance of these methods. In the present paper we consider the problem (under a number of assumptions) of choosing the parameters of the system for which the estimate \(|\mathbf{x}(t)| < e^{-\alpha t} |\mathbf{x}(0)|\) is valid, where \(\alpha \geq 0\) and the norm \(|\mathbf{x}|=\sqrt{\mathbf{x}^{*}H\mathbf{x}}\) \((H=H^{*}>0)\) are regarded as prescribed*.

\(1^\circ\). Consider the linear control system

\[ dx/dt=Ax+Ru,\qquad \mathbf{u}=-S^{*}\mathbf{x} \tag{1} \]

and the positive definite form

\[ V(\mathbf{x})=\mathbf{x}^{*}H\mathbf{x},\qquad H=H^{*}>0. \tag{2} \]

Here \(\mathbf{x}\) is an \(n\)-dimensional state vector; \(\mathbf{u}\) is a control vector of order \(m\), \(m \leq n\) (the number \(m\) is called below the number of controls); \(A,H,R,S\) are constant real matrices of orders \(n\times n\), \(n\times n\), \(n\times m\), \(n\times m\), respectively; the asterisk denotes transposition.

The problem is to determine, from given matrices \(A,H,R\) (or \(A,H,S\)) and a number \(\alpha \geq 0\), a matrix \(S\) (or \(R\)) such that, on every solution of system (1), the inequality

\[ \dot V(\mathbf{x})+2\alpha V(\mathbf{x})<0,\qquad \mathbf{x}\ne 0 \tag{3} \]

is satisfied.

If inequality (3) is satisfied, then we shall say that the matrices \(R,S\) \(\alpha\)-stabilize the system \(dx/dt=Ax\) with respect to the form (2), or, more briefly, that system (1), (2) is \(\alpha\)-stabilized. In this case we shall call the matrix \(S\) (or \(R\)) paired with the matrix \(R\) (or \(S\)), and the matrix \(R\) (or \(S\)) for which a paired one exists will be called admissible.

We note that if the control system is \(\alpha\)-stabilized, then it is asymptotically stable. The converse (for the given form (2)), obviously, is not true**.

Consider the matrix \(C_\alpha\) of the quadratic form \(-\dot V(x)-2\alpha V(x)\), where \(\dot V(\mathbf{x})\) is computed by virtue of the first equation (1) in the absence of control \((u=0)\):
\(C_\alpha=-(A^{*}H+HA+2\alpha H)\). Suppose that the form \(\mathbf{x}^{*}C_\alpha \mathbf{x}\) is indefinite and has \(\mu_\alpha \geq 1\) nonpositive squares***.

Theorem 1. In order that the system \(dx/dt=Ax\) be capable of being \(\alpha\)-stabilized with respect to the form (2), it is necessary that the number of controls \(m\) and the number \(\mu_\alpha\) be related by the inequality \(m \geq \mu_\alpha\).

The theorems 2–4 formulated below give conditions for admissibility of the matrix \(R\) and determine the form of the paired matrix \(S\).

Let \(L\) be an arbitrary \(n\times k\) matrix of rank \(k\). By \(\{L\}_k\) we shall denote the subspace spanned by its columns. For a given subspace—

* The notation \(K=K^{*}>0\) means that the form \(\mathbf{x}^{*}K\mathbf{x}\) is positive definite.
** Recall that \(\alpha \geq 0\). What follows is also valid for \(\alpha<0\).
*** For \(\mu_\alpha=0\) the problem admits the trivial solution \(S=0\) (or \(R=0\)).

the space \(\{L\}_k\) of dimension \(k\) will be denoted by \(L\), an arbitrary matrix whose columns form a basis in \(\{L\}_k\). Consider the subspace \(\{HR\}_m\), its orthogonal complement \(\{Z\}_{n-m}\), and the corresponding matrix \(Z\). Obviously, \(Z^*HR=0\).

Theorem 2. Let \(R\) be a given matrix of order \(n \times m\) and rank \(m\). In order that the matrix \(R\) be admissible, it is necessary and sufficient that the matrix \(Z^*C_\alpha Z\) be positive definite.

Suppose that the matrix \(Z^*C_\alpha Z\) is not positive definite and has \(\nu\) \((0<\nu<n-m)\) nonpositive eigenvalues. By Theorem 2, the matrix \(R\) is not admissible. Theorem 3 answers the question of how one can “complete” the matrix \(R\) by adjoining to it a matrix \(R_2\) with the minimal number of columns (i.e., by minimally increasing the number of controls), so that the resulting matrix \(R_1=\|R,R_2\|\) becomes admissible. Consider a subspace \(\{Z_0\}_{n-m-\nu}\subset\{Z\}_{n-m}\) such that \(Z_0^*C_\alpha Z_0>0\), and construct its orthogonal complement \(\{Y\}_\nu\subset\{Z\}_{n-m}\), \(Y^*Z_0=0\).

Theorem 3. Let the matrix \(Z^*C_\alpha Z\) have \(\nu\) \((0<\nu<n-m)\) nonpositive eigenvalues. The system (1), (2) can be \(\alpha\)-stabilized by introducing, in addition, \(\nu\) controls. In this case the matrix \(R_1=\|R,H^{-1}Y\|\) is admissible.

Theorem 4. Let the matrix \(R\) of order \(n\times m\) be admissible. The general form of a matrix \(S\) paired with it is given by the formula \(S=\lambda HR+U\), where \(\lambda>\lambda_0>0\) is an arbitrary number for which the inequality \(C_\alpha+\lambda HRR^*H>0\) is satisfied, and \(U\) is an arbitrary matrix of order \(n\times m\) satisfying the condition \(UU^*<\lambda(C_\alpha+\lambda HRR^*H)\). Here and below, \(\lambda_0\) is the minimal number such that \(C_\alpha+\lambda_0HRR^*H\geq0\).

Since inequality (3) is symmetric with respect to the matrices \(HR\) and \(S\), there are theorems, analogous to Theorems 2–4, that solve the problem of choosing the matrix \(S\).

Consider a problem close to the one solved above. Let, in the system \(dx/dt=Ax+Ru\), \(\mathbf v=-S^*\mathbf x\), the matrices \(A,R,S\) be given and let the closed-loop system for \(\mathbf u=\mathbf v\) not be \(\alpha\)-stabilized. Consider the question of whether the system can be \(\alpha\)-stabilized by introducing the connection \(\mathbf u=\tau^*\mathbf v\), where \(\tau\) is some constant \(m\times m\) matrix.

Theorem 5. Let the matrices \(A,R,S\) be given and let \(R\) be an admissible matrix. Put \(K=\lambda(C_\alpha+\lambda HRR^*R)\), \(\sigma=\lambda(S^*K^{-1}S)^{-1}SK^{-1}HR\), \(U_1=S\sigma-\lambda HR\), where \(\lambda>\lambda_0>0\) is an arbitrary number for which the inequality \(K>0\) is satisfied. In order that the matrix \(R\) have a paired matrix \(S_1\) of the form \(S_1=S\tau\), where \(\tau\) is some \(m\times m\) matrix, it is necessary and sufficient that the inequality \(U_1U_1^*<K\) be satisfied.

When the last inequality is satisfied, one can take \(S_1=S\sigma\).

2°. In the following lemmas, the vectors \(x\) and \(y\) have orders \(n\) and \(m\), respectively; \(F\) is a matrix of order \(n\times m\) and rank \(m\).

Lemma 1. In order that the quadratic form \(O(x,y)\) be positive definite on the subspace \(F^*x=y\), it is necessary and sufficient that there exist \(\lambda_0>0\) such that the form \(Q(x,y)+\lambda(x^*F-y^*)(F^*x-y)\) is positive definite on the whole space \(\{x,y\}\) for \(\lambda>\lambda_0\).

The proof of this, apparently known, lemma, obtained by the author jointly with I. E. Zuber, is omitted here.

Lemma 2. Let \(x^*Cx\) be an indefinite form. For the existence of a number \(\lambda_0>0\) such that the inequality \(M=C+\lambda FF^*>0\) is satisfied for all \(\lambda>\lambda_0\), it is necessary and sufficient that, for an arbitrary matrix \(Y\) of order \(n\times(n-m)\) and rank \(n-m\) such that \(Y^*F=0\), the inequality \(Y^*CY>0\) hold.

Necessity is obvious.

Sufficiency. Let \(Y^*CY>0\). Consider the matrix \(T=\|T_F,T_Y\|\), the first \(m\) columns of which form an orthonormal basis in the sub-

space spanned by the column vectors of the matrix \(F\), and the last \(n-m\) columns form an orthonormal basis in the subspace that is the orthogonal complement of the first. Then, obviously, \(T^*=T^{-1}\) and

\[ F_0=T^*F=\left\|\begin{matrix}\Phi\\0\end{matrix}\right\|,\qquad Y_0=T^*Y=\left\|\begin{matrix}0\\\Psi\end{matrix}\right\|. \]

Here \(\Phi,\Psi\) are nonsingular matrices of orders \(m\times m\) and \(m\times(n-m)\).

Let us partition the matrix \(C_0=T^*CT\) into blocks \(C_{11}, C_{12}=C_{21}^*, C_{22}\) of orders, respectively, \(m\times m\), \(m\times(n-m)\), \((n-m)\times(n-m)\). It is clear that \(Y^*CY=Y_0^*C_0Y_0\) and that the inequality \(Y^*CY>0\) has the form \(\Psi^*C_{22}\Psi>0\). Hence \(C_{22}>0\). Write the matrix \(M_0=T^*MT\) in block form:

\[ M_0=\left\|\begin{matrix} C_{11}+\lambda\Phi\Phi^* & C_{12}\\ C_{12}^* & C_{22} \end{matrix}\right\|. \]

Obviously, the condition \(C_{22}>0\) is sufficient (and necessary) for there to exist \(\lambda_0>0\) such that for \(\lambda>\lambda_0\) the inequality \(M_0>0\), and hence also the inequality \(M>0\), is satisfied.

Proof of Theorem 1. Put \(R^*Hx=y\). From \(V(x)+\alpha\dot V(x)=-Q(x,y)\) we obtain \(Q(x,y)=x^*C_\alpha x+y^*S^*x+x^*Sy\). Inequality (3) is equivalent to positive definiteness of the form \(Q(x,y)\) on the subspace \(R^*Hx=y\). By Lemma 1, for inequality (3) to hold it is necessary and sufficient that there exist \(\lambda_0>0\) such that for \(\lambda>\lambda_0\) the form

\[ Q(x,y)+\lambda(x^*HR-y^*)(R^*Hx-y) \]

is positive definite for all \(\{x,y\}\). It is easily shown that the latter condition is equivalent to the fact that, for \(\lambda>\lambda_0\),

\[ C_\alpha+\lambda HRR^*H>0. \]

Hence, by virtue of ([1], Ch. X, Theorem 16 \((A(x,x)=x^*C_\alpha x,\ \widetilde A(x,x)=A(x,x)+\lambda x^*HRR^*Hx,\ B(x,x)=x^*x,\ m=r)\)), the conclusion of the theorem follows.

Proof of Theorem 2. As shown above, the condition

\[ C_\alpha+\lambda HRR^*H>0 \]

is necessary and sufficient for the admissibility of the matrix \(R\). By Lemma 2 this inequality is equivalent to the inequality \(Z^*C_\alpha Z>0\), which proves the theorem.

Proof of Theorem 4. Since \(R\) is an admissible matrix, there exists \(\lambda_0>0\) such that for arbitrary \(\lambda>\lambda_0\) and some matrix \(S^*\) the inequality

\[ Q(x,y)+\lambda(x^*HR-y^*)(R^*Hx-y)>0 \]

is satisfied, or, equivalently to it, the inequalities

\[ C_\alpha+\lambda HRR^*H>0,\qquad \lambda(C_\alpha+\lambda HRR^*H)-(S-\lambda HR)(S^*-\lambda R^*H)>0. \]

Putting \(U=S-\lambda HR\), we obtain the assertion of the theorem.

Proof of Theorem 3. By assumption the matrix \(Z^*C_\alpha Z\) has exactly \(n-m-\nu\) positive eigenvalues. Therefore, by Theorem 2, the number of controls by which one can \(\alpha\)-stabilize the system (1), (2) is no less than \(m+\nu\). It is easy to see that it can be equal to \(m+\nu\). Indeed, the matrix \(R_1=\|R,H^{-1}Y\|\) has rank \(m+\nu\) and is admissible: \(Z_0^*HR_1=\|Z_0^*HR,\ Z_0^*Y\|\) and \(Z_0^*C_\alpha Z_0>0\).

Proof of Theorem 5. Necessity. Let the matrices \(R\) and \(S\) \(\alpha\)-stabilize the system (1), (2); let \(\lambda>\lambda_0\) be such that \(K>0\) and \(U=S\tau-\lambda HR\). By Theorem 4, \(UU^*<K\). Since \(K>0\), the inequality \(UU^*<K\) is equivalent to the inequality

\[ U^*K^{-1}U<I_m. \]

Put \(\tau=\sigma+\tau_0\); then \(U=U_1+S\tau_0\). Here the matrices \(U_1,\sigma,K\) have the form indicated in Theorem 5. Since \(S^*K^{-1}U_1=0\), it follows that

\[ U_1^*K^{-1}U_1=U^*K^{-1}U-\tau_0^*S^*K^{-1}S\tau_0<I_m. \]

Therefore \(U_1U_1^*<K\).

Sufficiency. From Theorem 4, for \(U=U_1\) we obtain that \(S_1=S\sigma\) is a pairing matrix.

Leningrad Electrotechnical Institute
named after V. I. Ulyanov (Lenin)

Received
30 IX 1968

REFERENCES

1. F. R. Gantmakher, Matrix Theory, Nauka, 1967.

* The matrix \(S\) may be both given and sought.