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\(\)
Chapter J Problem Sheet 10 (Lectures 20–21)
-
1. Consider the first order scalar differential equation
\[ \dot {x}=x+w_1+u, \]
the performance output
\[ z=\bbm {x\\u}, \]
and the measurement
\[ y=x+w_2, \]
i.e.
\[ A=2,~~B_1=\bbm {1&0},~~B_2=1,~~C_1=\bbm {1\\0},~~D_{12}=\bbm {0\\1},~~C_2=1,~~D_{21}=\bbm {0&1}. \]
-
(a) Determine whether or not \(D_{21}\) is surjective;
-
(b) Determine whether or not \((A,C_2)\) is detectable;
-
(c) Determine whether or not the Rosenbrock surjectivity condition holds;
-
(d) Determine the stabilizing solution of the algebraic Riccati equation (21.4);
-
(e) Determine the unique controller which solves the \(H^2\) measurement feedback problem (you may use the solution of the other Riccati equation from Problem Sheet 9).
-
2. Consider the undamped second order scalar differential equation
\[ \ddot {q}(t)+q(t)=w_1(t)+u(t), \]
with the state \(x=\sbm {q\\\dot {q}}\) and the performance output
\[ z=\bbm {q\\\varepsilon u}, \]
where \(\varepsilon >0\), and with \(\delta >0\) the measured output
\[ y=q+\delta w_2. \]
i.e.
\(\seteqnumber{0}{J.}{0}\)
\begin{gather*}
A=\bbm {0&1\\-1&0},\quad B_1=\bbm {0&0\\1&0},\quad B_2=\bbm {0\\1},\quad C_1=\bbm {1&0\\0&0},\quad D_{12}=\bbm {0\\\varepsilon }, \\ C_2=\bbm {1&0},\quad D_{21}=\bbm {0&\delta }.
\end{gather*}
-
(a) Determine whether or not \(D_{21}\) is surjective;
-
(b) Determine whether or not \((A,C_2)\) is detectable;
-
(c) Determine whether or not the Rosenbrock surjectivity condition holds;
-
(d) Determine the stabilizing solution of the algebraic Riccati equation (21.4);
-
(e) Determine the unique controller which solves the \(H^2\) measurement feedback problem (you may use the solution of the other Riccati equation from Problem Sheet 9).
Bibliography
The suspension system model is based on [Hrovat, 1997] (authored by an employee of Ford Research). The fixed structure optimization is based on [Scheibe and Smith, 2009]. The tape drive model is based on [Cherubini, 2022] (authored by an
employee of IBM Research) and other articles by that author.
Further background on material in the first seven chapters can for example be found in [Bolton, 2015] and on the material in the later chapters in [Trentelman et al., 2002] (for the material on Controllability and Observability also
[Logemann and Ryan, 2014]).
Bibliography
-
[Bolton, 2015] Bolton, W. (2015). Mechatonics. sixth edition.
-
[Cherubini, 2022] Cherubini, G. (2022). Advanced control systems for data storage on magnetic tape: A long-lasting success story. IEEE Control Systems Magazine, 42(4):8–11.
-
[Hrovat, 1997] Hrovat, D. (1997). Survey of advanced suspension developments and related optimal control applications. Automatica, 33:1781–1817.
-
[Logemann and Ryan, 2014] Logemann, H. and Ryan, E. P. (2014). Ordinary Differential Equations: Analysis, Qualitative Theory and Control.
-
[Scheibe and Smith, 2009] Scheibe, F. and Smith, M. C. (2009). Analytical solutions for optimal ride comfort and tyre grip for passive vehicle suspensions. Vehicle Systems Dynamics, 47:1229–1252.
-
[Trentelman et al., 2002] Trentelman, H. L., Stoorvogel, A. A., and Hautus, M. (2002). Control Theory for Linear Systems.