Chapter 12 Controllability I
We consider the input-state system
\(\seteqnumber{0}{12.}{0}\)\begin{equation} \label {eq:is} \dot {x}(t)=Ax(t)+Bu(t). \end{equation}
Here \(A\in \mR ^{n\times n}\) and \(B\in \mR ^{n\times m}\).
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Definition 12.1. The system (12.1) (or equivalently the pair of matrices \((A,B)\)) is said to be controllable if there exists a time \(T>0\) such that for all \(x^0,x^1\in \mR ^n\), there exists a control \(u:[0,T]\to \mR ^m\) such that \(x(0)=x^0\) and \(x(T)=x^1\).
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Definition 12.2. Let \(x^0,x^1\in \mR ^n\). If there exists a \(T>0\) such that there exists a control \(u:[0,T]\to \mR ^m\) such that \(x(0)=x^0\) and \(x(T)=x^1\), then \(x^1\) is said to be reachable from \(x^0\).
The reachable subspace consists of all states \(x^1\) that are reachable from \(x^0=0\).
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Definition 12.3. The Kalman controllability matrix is the \(n\times mn\) matrix
\[ \bbm {B&AB&A^2B&\ldots &A^{n-1}B}. \]
The controllability Gramian, indexed by \(T>0\), is the \(n\times n\) matrix
\[ Q_T=\int _0^T \e ^{At}BB^*\e ^{A^*t}\,dt. \]
The Hautus controllability matrix, indexed by \(s\in \mC \), is the \(n\times (n+m)\) matrix
\[ \bbm {sI-A&B}. \]
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Theorem 12.6. The following are equivalent:
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1. The input-state system is controllable;
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2. The Kalman controllability matrix is surjective;
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3. The Hautus controllability matrix is surjective for all \(s\in \mathbb {C}\);
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4. The Hautus controllability matrix is surjective for all eigenvalues \(s\) of \(A\);
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5. The controllability Gramian \(Q_T\) is invertible for some \(T>0\);
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6. The controllability Gramian \(Q_T\) is invertible for all \(T>0\);
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7. Every monic polynomial of degree \(n\) is assignable to the pair \((A,B)\);
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8. The reachable subspace equals \(\mR ^n\).
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Remark 12.7. In the single-input case (i.e. \(m=1\)) an explicit formula for an \(F\) such that the characteristic polynomial of \(A+BF\) equals the given monic polynomial \(p\) of degree \(n\) is:
\[ F=-\bbm {0&\ldots &0&1}\bbm {B&AB&A^2B&\ldots &A^{n-1}B}^{-1}p(A). \]
This is called Ackermann’s formula.
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Remark 12.8. A control which steers the system from state \(x^0\) to state \(x^1\) in time \(T\) is given in terms of the controllability Gramian as:
\[ u(t)=B^*\e ^{A^*(T-t)}Q_T^{-1}(x^1-\e ^{AT}x^0). \]
This control is special in the sense that it is the unique control \(u\) which achieves this transition from \(x^0\) to \(x^1\) that minimizes
\[ \int _0^T \|u(t)\|^2\,dt. \]
The minimum value of this control cost is given in terms of the controllability Gramian by
\[ \ipd {Q_T^{-1}(x^1-\e ^{AT}x^0)}{x^1-\e ^{AT}x^0}. \]
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Proposition 12.11. Let \(A\) be asymptotically stable.
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• The input-state system is controllable if and only if the infinite-time controllability gramian \(Q\) is invertible.
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• The reachable subspace equals the image of the infinite-time controllability gramian \(Q\).
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• The infinite-time controllability gramian \(Q\) is the unique solution of the control Lyapunov equation
\[ AQ+QA^*+BB^*=0. \]
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12.1 Examples
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Example 12.12. Consider the first order scalar differential equation
\[ \dot {x}+x=u. \]
We have
\[ A=-1,\qquad B=1. \]
Since \(n=1\), the Kalman controllability matrix is simply \(B\). Since a \(1\times 1\) matrix is surjective if and only if it is nonzero, we have that the Kalman controllability matrix is surjective and therefore the system is controllable.
The Hautus controllability matrix is
\[ \bbm {sI-A&B}=\bbm {s+1&1}. \]
This matrix is surjective for all \(s\in \mC \): the second column gives an invertible 1-by-1 matrix.
The controllability Gramian is
\[ Q_T=\int _0^T \e ^{-2t}\,dt =\left [-\frac {1}{2}\e ^{-2t}\right ]_{t=0}^T =\frac {1}{2}\left (1-\e ^{-2T}\right ). \]
Since \(Q_T\) is invertible (for any \(T>0\)), we again see that the system is controllable. From Remark 12.8 we then see that a control which steers the system from state \(x^0\) to state \(x^1\) in time \(T\) is given by
\(\seteqnumber{0}{12.}{1}\)\begin{align*} u(t)&=\e ^{t-T}~\frac {2}{1-\e ^{-2T}}(x^1-\e ^{-T}x^0). \end{align*} and that the control cost to achieve this is
\[ \frac {2}{1-\e ^{-2T}}\left (x^1-\e ^{-T}x^0\right )^2. \]
To understand this better, consider \(x^0=0\) and \(x^1\) with \(|x^1|=1\). Then the control cost is
\[ \frac {2}{1-\e ^{-2T}}, \]
which is a decreasing function in \(T\) with limit as \(T\downarrow 0\) equal to \(\infty \) and limit as \(T\to \infty \) equal to \(2\). We therefore see that although the system is controllable in any positive time \(T\), the cost becomes larger the smaller the given time \(T\) is (this is generally the case).
The control Lyapunov equation is
\[ -2Q+1=0, \]
which gives \(Q=\frac {1}{2}\) as the infinite-time controllability Gramian. Since this is invertible, we once again see that the system is controllable.