9.2: Tracking with Feedforward Controller
- Page ID
- 24430
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Tracking Design with Feedforward Controller
A tracking system is designed to asymptotically reach and maintain zero steady-state error with respect to a reference input, \(r\left(t\right)\). This can be achieved by using \(r\left(t\right)\) to cancel \(e\left(t\right)\) in the steady state.
The reference signal can be used with a feed forward gain to cancel the error signal. Toward this end, let the state variable model be given as: \[\dot{x}(t)=Ax(t)+bu(t), \;\;y(t)=c^{T} x(t) \nonumber \]
The asymptotic tracking control law is defined as: \[u(t)=-k^{T} x(t)+k_{r} r(t) \nonumber \] where \(k_{r}\) is a scalar gain. The closed-loop system is formed as: \[\dot{x}(t)=\left(A-bk^{T} \right)x(t)+bk_{r} r(t), y(t)=c^{T} x(t) \nonumber \]
Assuming the closed-loop system is stable, let \(\dot{x}(t)=0\); the steady-state values of the state variables are obtained as: \[x_{ss} =-\left(A-bk^{T} \right)^{-1} bk_{r} r_{ss} . \nonumber \]
The steady-state output is given as: \[y_{ss} =-c^{T} \left(A-bk^{T} \right)^{-1} b\, k_{r} r_{ss} \nonumber \]
Define the closed-loop transfer function: \[T(s)=c^{T} \left(sI-A+bk^{T} \right)^{-1} b \nonumber \]
Then, the steady-state output is: \(y_{ss} =T(0)k_{r} r_{ss}\).
In order to ensure \(y_{ss} =r_{ss}\), we may choose, \(k_{r} =T(0)^{-1}\).
Further, if \(y=x_{1}\), then \(k_{r}\) may be selected as \(k_{r} =k_{1}\), to ensure \(r-y=0\) in the steady-state.
The state and output equations for a DC motor model are given as: \[\frac{\rm d}{\rm dt} \left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right]=\left[\begin{array}{cc} {-100} & {-5} \\ {5} & {-10} \end{array}\right]\left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right]+\left[\begin{array}{c} {100} \\ {0} \end{array}\right]V_a,\;\; \omega =\left[\begin{array}{cc} {0} & {1} \end{array}\right]\left[\begin{array}{c} {i_a} \\ {\omega } \end{array}\right]. \nonumber \]
A pole placement controller for the DC motor model was previously designed as: \(V_{a} =-0.4i_{a} -7.15\omega\). The control law is revised to include a feedforward gain: \(V_{a} =-0.4i_{a} -7.15\omega +k_{r} r\), where we want to choose \(k_r\) for zero steady-state error.
By including the controller, the closed-loop system model is given as: \[\frac{\rm d}{\rm dt} \left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right]=\left[\begin{array}{cc} {-140} & {-720} \\ {5} & {-10} \end{array}\right]\left[\begin{array}{c} {i_a } \\ {\omega } \end{array}\right]+\left[\begin{array}{c} {100} \\ {0} \end{array}\right]k_{r} r \nonumber \]
The state variables assume the following values in steady-state: \(\left[\begin{array}{c} {i_{a,ss} } \\ {\omega _{ss} } \end{array}\right]=\left[\begin{array}{c} {0.2} \\ {0.1} \end{array}\right]k_{r} r_{ss}\).
The steady-state value of the output is given as: \(y_{ss} =0.1k_{r} r_{ss}\). Then, we may choose \(k_{r} =10\) for zero steady-state error.
The resulting conrol law is given as: \(V_{a} =-0.4i_{a} -7.15\omega +10r\).
A plot of the motor response with feedforward compensation is shown in Fig. 9.2.1. As expected, the motor speed asymptotically approaches unity in the steady-state.
The state variable model of a mass–spring–damper is given as: \[\frac{\rm d}{\rm dt} \left[\begin{array}{c} {x} \\ {v} \end{array}\right]=\left[\begin{array}{cc} {0} & {1} \\ {-10} & {-1} \end{array}\right]\left[\begin{array}{c} {x} \\ {v} \end{array}\right]+\left[\begin{array}{c} {0} \\ {1} \end{array}\right]f, \;\;x=\left[\begin{array}{cc} {1} & {0} \end{array}\right]\left[\begin{array}{c} {x} \\ {v} \end{array}\right] \nonumber \]
A pole placement controller for the model was previously designed as: \(f=-3v\).
The control law is modified to include a feedforward gain as: \(f=-3v+k_{r} r.\)
The closed-loop system model is given as: \(\frac{\rm d}{\rm dt} \left[\begin{array}{c} {x} \\ {v} \end{array}\right]=\left[\begin{array}{cc} {0} & {1} \\ {-10} & {-4} \end{array}\right]\left[\begin{array}{c} {x} \\ {v} \end{array}\right]+k_{r} r\).
The steady-state output is: \(y_{ss} =0.1k_{r} r_{ss}\). Hence, we may choose \(k_{r} =10\) for error-free tracking. The resulting control law is given as: \(f=-3v+10\, r.\)