Part II Control

Chapter 8 Control

We consider input-state-output systems with a state \(x:[0,\infty )\to \mR ^n\), external input \(w:[0,\infty )\to \mR ^{m_1}\), control input \(u:[0,\infty )\to \mR ^{m_2}\), performance output \(z:[0,\infty )\to \mR ^{p_1}\) and measured output \(y:[0,\infty )\to \mR ^{p_2}\) described by

\begin{equation} \label {eq:control:xzy} \dot {x}=Ax+B_1w+B_2u,\qquad z=C_1x+D_{11}w+D_{12}u,\qquad y=C_2x+D_{21}w+D_{22}u, \end{equation}

with the initial condition \(x(0)=x^0\) where \(x^0\in \mR ^n\) and

\begin{gather} \label {eq:control:ABCDmatrices} A\in \mR ^{n\times n},~ B_1\in \mR ^{n\times m_1},~ B_2\in \mR ^{n\times m_2},\notag \\ C_1\in \mR ^{p_1\times n},~ D_{11}\in \mR ^{p_1\times m_1},~ D_{12}\in \mR ^{p_1\times m_2},~ C_2\in \mR ^{p_2\times n},~ D_{21}\in \mR ^{p_2\times m_1}, D_{22}\in \mR ^{p_2\times m_2}. \end{gather}

We are free to choose the control input \(u\). There are three typical forms that this can take:

  • • Full Information Feedback: \(u=F_1w+F_2x\) where \(F_1\in \mR ^{m_2\times m_1}\) and \(F_2\in \mR ^{m_2\times n}\);

  • • State Feedback: \(u=Fx\) where \(F\in \mR ^{m_2\times n}\);

  • • Output Feedback:

    \[ \dot {x}_c=A_cx_c+B_cy,\qquad u=C_cx_c+D_cy, \]

    where \(x_c:[0,\infty )\to \mR ^{n_c}\) is the controller state with initial condition \(x_c(0)=x_c^0\in \mR ^{n_c}\) and

    \[ A_c\in \mR ^{n_c\times n_c},~B_c\in \mR ^{n_c\times p_2},~ C_c\in \mR ^{m_2\times n_c},~D_c\in \mR ^{m_2\times p_2}. \]

We obtain the closed-loop system by combining the equations (8.1) and the relevant feedback equation for \(u\). We can usually eliminate \(u\) from consideration entirely as follows.

  • • For Full Information Feedback the closed-loop system is

    \[ \dot {x}=(A+B_2F_2)x+(B_1+B_2F_1)w,\qquad z=(C_1+D_{12}F_2)x+\left (D_{11}+D_{12}F_1\right )w, \]

  • • for State Feedback the closed-loop system is

    \[ \dot {x}=(A+B_2F)x+B_1w,\qquad z=(C_1+D_{12}F)x+D_{11}w, \]

  • • for Output Feedback with \(D_c=0\) the closed-loop system is

    \[ \bbm {\dot {x}\\\dot {x}_c} =\bbm {A&B_2C_c\\B_cC_2&A_c+B_cD_{22}C_c}\bbm {x\\x_c} +\bbm {B_1\\B_cD_{21}}w,\qquad z=\bbm {C_1&D_{12}C_c}\bbm {x\\x_c}+D_{11}w. \]

    (when \(D_c\neq 0\) but with \(I-D_{22}D_c\) invertible a similar but more complicated formula holds).

The objective for choosing \(u\) will be (some combination of) the following.

  • • (internal) Closed-loop Stability: if \(w=0\), then for all initial conditions \(x^0\) and \(x_c^0\), we have \(\lim _{t\to \infty }x(t)=0\) and \(\lim _{t\to \infty }x_c(t)=0\).

  • • Regulation: assuming that \(w\) is given by

    \[ \dot {x}_e=A_ex_e,\qquad w=C_ex_e, \]

    with the initial condition \(x_e(0)=x^0_e\in \mR ^{n_e}\) where

    \[ A_e\in \mR ^{n_e\times n_e},~C_e\in \mR ^{m_1\times n_e}, \]

    for all \(x^0\), \(x_c^0\) and \(x_e^0\) we have

    \[ \lim _{t\to \infty }z(t)=0. \]

  • • \(H^2\) minimization: minimize

    \[ \int _{-\infty }^\infty \|F_{zw}(\omega )\|^2_\HS \,d\omega , \]

    where \(F_{zw}\) is the frequency response of the closed-loop system (from \(w\) to \(z\)) and \(\|\cdot \|_\HS \) denotes a certain norm on matrices (the Hilbert-Schmidt norm) or equivalently (by Plancherel’s Theorem) minimize

    \[ \int _0^\infty \|h_{zw}(t)\|^2_\HS \,dt, \]

    where \(h_{zw}\) is the impulse response of the closed-loop system (from \(w\) to \(z\)).

  • • \(H^\infty \) minimization: minimize

    \[ \sup _{\omega \in \mR }\|F_{zw}(\omega )\|, \]

    where \(F_{zw}\) is the frequency response of the closed-loop system (from \(w\) to \(z\)) and \(\|\cdot \|\) denotes a certain norm on matrices (the operator/spectral norm).

  • Remark 8.1.  For the \(H^\infty \) problem we will actually study only the suboptimal case: for a given \(\gamma >0\) find a control of the relevant form which achieves

    \[ \sup _{\omega \in \mR }\|F_{zw}(\omega )\|<\gamma . \]

  • Remark 8.2.  For regulation, Full Information Feedback can utilize the whole state \(x_e\) of the exosystem rather than just the output \(w\) of the exosystem.