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< h1 class = "title" > Robust Control - \(\mathcal{H}_\infty\) Synthesis< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org983bccb" > 1. Introduction to the Control Methodology - Model Based Control< / a >
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< ul >
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< li > < a href = "#org18f2c71" > 1.1. Control Methodology< / a > < / li >
< li > < a href = "#org49ba16e" > 1.2. Some Background: From Classical Control to Robust Control< / a > < / li >
< li > < a href = "#org3998425" > 1.3. Example System< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org2f14f6e" > 2. Classical Open Loop Shaping< / a >
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< ul >
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< li > < a href = "#orgadf1fa2" > 2.1. Introduction to Open Loop Shaping< / a > < / li >
< li > < a href = "#org480ec5c" > 2.2. Example of Open Loop Shaping< / a > < / li >
< li > < a href = "#org81a2b10" > 2.3. \(\mathcal{H}_\infty\) Loop Shaping Synthesis< / a > < / li >
< li > < a href = "#org8f1b4c8" > 2.4. Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orga39391a" > 3. First Step in the \(\mathcal{H}_\infty\) world< / a >
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< ul >
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< li > < a href = "#orgafc7a12" > 3.1. The \(\mathcal{H}_\infty\) Norm< / a > < / li >
< li > < a href = "#org5ab38ae" > 3.2. \(\mathcal{H}_\infty\) Synthesis< / a > < / li >
< li > < a href = "#org3371822" > 3.3. The Generalized Plant< / a > < / li >
< li > < a href = "#orgbf413fb" > 3.4. From a Classical Feedback Architecture to a Generalized Plant< / a > < / li >
< li > < a href = "#org5b0f8f6" > 3.5. The General Synthesis Problem Formulation< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org0a60e2d" > 4. Modern Interpretation of the Control Specifications< / a >
< ul >
< li > < a href = "#orgeaf9df5" > 4.1. Introduction< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org73188d9" > 5. Resources< / a > < / li >
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< / ul >
< / div >
< / div >
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< div id = "outline-container-org983bccb" class = "outline-2" >
< h2 id = "org983bccb" > < span class = "section-number-2" > 1< / span > Introduction to the Control Methodology - Model Based Control< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< div id = "outline-container-org18f2c71" class = "outline-3" >
< h3 id = "org18f2c71" > < span class = "section-number-3" > 1.1< / span > Control Methodology< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
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< p >
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The typical methodology when applying Model Based Control to a plant is schematically shown in Figure < a href = "#orgc0a3638" > 1< / a > .
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It consists of three steps:
< / p >
< ol class = "org-ol" >
< li > < b > Identification or modeling< / b > : \(\Longrightarrow\) mathematical model< / li >
< li > < b > Translate the specifications into mathematical criteria< / b > :
< ul class = "org-ul" >
< li > < span class = "underline" > Specifications< / span > : Response Time, Noise Rejection, Maximum input amplitude, Robustness, … < / li >
< li > < span class = "underline" > Mathematical Criteria< / span > : Cost Function, Shape of TF< / li >
< / ul > < / li >
< li > < b > Synthesis< / b > : research of \(K\) that satisfies the specifications for the model of the system< / li >
< / ol >
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< div id = "orgc0a3638" class = "figure" >
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< p > < img src = "figs/control-procedure.png" alt = "control-procedure.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Typical Methodoly for Model Based Control< / p >
< / div >
< p >
In this document, we will mainly focus on steps 2 and 3.
< / p >
< / div >
< / div >
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< div id = "outline-container-org49ba16e" class = "outline-3" >
< h3 id = "org49ba16e" > < span class = "section-number-3" > 1.2< / span > Some Background: From Classical Control to Robust Control< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
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< table id = "orgd1f0c28" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > Table summurazing the main differences between classical, modern and robust control< / caption >
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< colgroup >
< col class = "org-left" / >
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< col class = "org-center" / >
< col class = "org-center" / >
< col class = "org-center" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-center" > < b > Classical Control< / b > < / th >
< th scope = "col" class = "org-center" > < b > Modern Control< / b > < / th >
< th scope = "col" class = "org-center" > < b > Robust Control< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < b > Date< / b > < / td >
< td class = "org-center" > 1930-< / td >
< td class = "org-center" > 1960-< / td >
< td class = "org-center" > 1980-< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Tools< / b > < / td >
< td class = "org-center" > Transfer Functions< / td >
< td class = "org-center" > State Space formulation< / td >
< td class = "org-center" > Systems and Signals Norms (\(\mathcal{H}_\infty\), \(\mathcal{H}_2\) Norms)< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" > Nyquist Plots< / td >
< td class = "org-center" > Riccati Equations< / td >
< td class = "org-center" > Closed Loop Transfer Functions< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" > Bode Plots< / td >
< td class = "org-center" >   < / td >
< td class = "org-center" > Open/Closed Loop Shaping< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" > Phase and Gain margins< / td >
< td class = "org-center" >   < / td >
< td class = "org-center" > Weighting Functions< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" >   < / td >
< td class = "org-center" >   < / td >
< td class = "org-center" > Disk margin< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Control Architectures< / b > < / td >
< td class = "org-center" > Proportional, Integral, Derivative< / td >
< td class = "org-center" > Full State Feedback< / td >
< td class = "org-center" > General Control Configuration< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" > Leads, Lags< / td >
< td class = "org-center" > LQR, LQG< / td >
< td class = "org-center" >   < / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" >   < / td >
< td class = "org-center" > Kalman Filters< / td >
< td class = "org-center" >   < / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Advantages< / b > < / td >
< td class = "org-center" > Study Stability< / td >
< td class = "org-center" > Automatic Synthesis< / td >
< td class = "org-center" > Automatic Synthesis< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" > Simple< / td >
< td class = "org-center" > MIMO< / td >
< td class = "org-center" > MIMO< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" > Natural< / td >
< td class = "org-center" > Optimization Problem< / td >
< td class = "org-center" > Optimization Problem< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" >   < / td >
< td class = "org-center" >   < / td >
< td class = "org-center" > Guaranteed Robustness< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" >   < / td >
< td class = "org-center" >   < / td >
< td class = "org-center" > Easy specification of performances< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Disadvantages< / b > < / td >
< td class = "org-center" > Manual Method< / td >
< td class = "org-center" > No Guaranteed Robustness< / td >
< td class = "org-center" > Required knowledge of specific tools< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-center" > Only SISO< / td >
< td class = "org-center" > Difficult Rejection of Perturbations< / td >
< td class = "org-center" > Need a reasonably good model of the system< / td >
< / tr >
< / tbody >
< / table >
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< / div >
< / div >
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< div id = "outline-container-org3998425" class = "outline-3" >
< h3 id = "org3998425" > < span class = "section-number-3" > 1.3< / span > Example System< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
< p >
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Let’ s consider the model shown in Figure < a href = "#org6c29b14" > 2< / a > .
It could represent a suspension system with a payload to position or isolate using an force actuator and an inertial sensor.
The notations used are listed in Table < a href = "#org2590b80" > 2< / a > .
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< / p >
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< div id = "org6c29b14" class = "figure" >
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< p > < img src = "figs/mech_sys_1dof_inertial_contr.png" alt = "mech_sys_1dof_inertial_contr.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 2: < / span > Test System consisting of a payload with a mass \(m\) on top of an active system with a stiffness \(k\), damping \(c\) and an actuator. A feedback controller \(K(s)\) is added to position / isolate the payload.< / p >
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< / div >
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< table id = "org2590b80" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 2:< / span > Example system variables< / caption >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > < b > Notation< / b > < / th >
< th scope = "col" class = "org-left" > < b > Description< / b > < / th >
< th scope = "col" class = "org-left" > < b > Value< / b > < / th >
< th scope = "col" class = "org-left" > < b > Unit< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > \(m\)< / td >
< td class = "org-left" > Payload’ s mass to position / isolate< / td >
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< td class = "org-left" > \(10\)< / td >
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< td class = "org-left" > [kg]< / td >
< / tr >
< tr >
< td class = "org-left" > \(k\)< / td >
< td class = "org-left" > Stiffness of the suspension system< / td >
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< td class = "org-left" > \(10^6\)< / td >
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< td class = "org-left" > [N/m]< / td >
< / tr >
< tr >
< td class = "org-left" > \(c\)< / td >
< td class = "org-left" > Damping coefficient of the suspension system< / td >
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< td class = "org-left" > \(400\)< / td >
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< td class = "org-left" > [N/(m/s)]< / td >
< / tr >
< tr >
< td class = "org-left" > \(y\)< / td >
< td class = "org-left" > Payload absolute displacement (measured by an inertial sensor)< / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > [m]< / td >
< / tr >
< tr >
< td class = "org-left" > \(d\)< / td >
< td class = "org-left" > Ground displacement, it acts as a disturbance< / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > [m]< / td >
< / tr >
< tr >
< td class = "org-left" > \(u\)< / td >
< td class = "org-left" > Actuator force< / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > [N]< / td >
< / tr >
< tr >
< td class = "org-left" > \(r\)< / td >
< td class = "org-left" > Wanted position of the mass (the reference)< / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > [m]< / td >
< / tr >
< tr >
< td class = "org-left" > \(\epsilon = r - y\)< / td >
< td class = "org-left" > Position error< / td >
< td class = "org-left" >   < / td >
< td class = "org-left" > [m]< / td >
< / tr >
< tr >
< td class = "org-left" > \(K\)< / td >
< td class = "org-left" > Feedback controller< / td >
< td class = "org-left" > to be designed< / td >
< td class = "org-left" > [N/m]< / td >
< / tr >
< / tbody >
< / table >
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< div class = "exercice" id = "org60eec50" >
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< p >
Derive the following open-loop transfer functions:
< / p >
\begin{align}
G(s) & = \frac{y}{u} \\
G_d(s) & = \frac{y}{d}
\end{align}
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< details > < summary > Hint< / summary >
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< p >
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You can follow this generic procedure:
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< / p >
< ol class = "org-ol" >
< li > List all applied forces ot the mass: Actuator force, Stiffness force (Hooke’ s law), … < / li >
< li > Apply the Newton’ s Second Law on the payload
\[ m \ddot{y} = \Sigma F \]< / li >
< li > Transform the differential equations into the Laplace domain:
\[ \frac{d\ \cdot}{dt} \Leftrightarrow \cdot \times s \]< / li >
< li > Write \(y(s)\) as a function of \(u(s)\) and \(w(s)\)< / li >
< / ol >
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< / details >
< details > < summary > Results< / summary >
\begin{align}
G(s) & = \frac{1}{m s^2 + cs + k} \\
G_d(s) & = \frac{cs + k}{m s^2 + cs + k}
\end{align}
< / details >
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< / div >
< p >
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Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure < a href = "#org6c29b14" > 2< / a > into a classical feedback form as shown in Figure < a href = "#org1e99301" > 6< / a > .
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< / p >
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< div id = "org9295a3c" class = "figure" >
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< p > < img src = "figs/classical_feedback_test_system.png" alt = "classical_feedback_test_system.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Block diagram corresponding to the example system< / p >
< / div >
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< p >
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Let’ s define the system parameters on Matlab.
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "linenr" > 1: < / span > k = 1e6; < span class = "org-comment" > % Stiffness [N/m]< / span >
< span class = "linenr" > 2: < / span > c = 4e2; < span class = "org-comment" > % Damping [N/(m/s)]< / span >
< span class = "linenr" > 3: < / span > m = 10; < span class = "org-comment" > % Mass [kg]< / span >
< / pre >
< / div >
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< p >
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And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown in Figures < a href = "#orgc2b7d7d" > 4< / a > and < a href = "#orgf203cd9" > 5< / a > ).
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "linenr" > 4: < / span > G = 1< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > c< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k); < span class = "org-comment" > % Plant< / span >
< span class = "linenr" > 5: < / span > Gd = (c< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k)< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > c< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k); < span class = "org-comment" > % Disturbance< / span >
< / pre >
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< / div >
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< div id = "orgc2b7d7d" class = "figure" >
< p > < img src = "figs/bode_plot_example_afm.png" alt = "bode_plot_example_afm.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Bode plot of the plant \(G(s)\)< / p >
< / div >
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< div id = "orgf203cd9" class = "figure" >
< p > < img src = "figs/bode_plot_example_Gd.png" alt = "bode_plot_example_Gd.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > Magnitude of the disturbance transfer function \(G_d(s)\)< / p >
< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org2f14f6e" class = "outline-2" >
< h2 id = "org2f14f6e" > < span class = "section-number-2" > 2< / span > Classical Open Loop Shaping< / h2 >
< div class = "outline-text-2" id = "text-2" >
< / div >
< div id = "outline-container-orgadf1fa2" class = "outline-3" >
< h3 id = "orgadf1fa2" > < span class = "section-number-3" > 2.1< / span > Introduction to Open Loop Shaping< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
< div class = "definition" id = "orgba735c5" >
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< p >
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The < b > Loop Gain< / b > \(L(s)\) usually refers to as the product of the controller and the plant (Figure < a href = "#org1e99301" > 6< / a > ):
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< / p >
\begin{equation}
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L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
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\end{equation}
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< div id = "org1e99301" class = "figure" >
< p > < img src = "figs/open_loop_shaping.png" alt = "open_loop_shaping.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Classical Feedback Architecture< / p >
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< / div >
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< / div >
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< div class = "definition" id = "org53d42d3" >
< p >
< b > Open Loop Shaping< / b > refers to a control design technique where the controller \(K(s)\) is designed such that the < b > Open Loop Gain< / b > \(L(s)\) has desirable shape.
< / p >
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< / div >
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< p >
This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain \(L(s)\):
< / p >
< ul class = "org-ul" >
< li > < b > Performance< / b > : \(L\) large< / li >
< li > < b > Good disturbance rejection< / b > : \(L\) large< / li >
< li > < b > Limitation of measurement noise on plant output< / b > : \(L\) small< / li >
< li > < b > Small magnitude of input signal< / b > : \(K\) and \(L\) small< / li >
< li > < b > Nominal stability< / b > : \(L\) small (RHP zeros and time delays)< / li >
< li > < b > Robust stability< / b > : \(L\) small (neglected dynamics)< / li >
< / ul >
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< p >
The Open Loop shape is usually done manually has the loop gain \(L(s)\) depends linearly on \(K(s)\) \eqref{eq:loop_gain}.
< / p >
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< p >
\(K(s)\) then consists of a combination of leads, lags, notches, etc. such that \(L(s)\) has the wanted shape.
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< / p >
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< / div >
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< / div >
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< div id = "outline-container-org480ec5c" class = "outline-3" >
< h3 id = "org480ec5c" > < span class = "section-number-3" > 2.2< / span > Example of Open Loop Shaping< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
< div class = "exampl" id = "org9e97650" >
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< p >
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Let’ s take our example system and try to apply the Open-Loop shaping strategy to design a controller that fulfils the following specifications:
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< / p >
< ul class = "org-ul" >
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< li > < b > Performance< / b > : Bandwidth of approximately 10Hz< / li >
< li > < b > Noise Attenuation< / b > : Roll-off of -40dB/decade past 30Hz< / li >
< li > < b > Robustness< / b > : Gain margin > 3dB and Phase margin > 30 deg< / li >
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< / ul >
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< / div >
< div class = "exercice" id = "orgdf10f9e" >
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< p >
Using < code > SISOTOOL< / code > , design a controller that fulfill the specifications.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > sisotool(G)
< / pre >
< / div >
< / div >
< p >
In order to have the wanted Roll-off, two integrators are used, a lead is also added to have sufficient phase margin.
< / p >
< p >
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The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure < a href = "#org2f8dcdc" > 7< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K = 14e8 < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Gain< / span >
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1< span class = "org-type" > /< / span > (s< span class = "org-type" > ^< / span > 2) < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Double Integrator< / span >
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1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 40) < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Low Pass Filter< / span >
(1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 10< span class = "org-type" > /< / span > sqrt(8)))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 10< span class = "org-type" > *< / span > sqrt(8))); < span class = "org-comment" > % Lead< / span >
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< / pre >
< / div >
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< div id = "org2f8dcdc" class = "figure" >
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< p > < img src = "figs/loop_gain_manual_afm.png" alt = "loop_gain_manual_afm.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Bode Plot of the obtained Loop Gain \(L(s) = G(s) K(s)\)< / p >
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< / div >
< p >
And we can verify that we have the wanted stability margins:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > [Gm, Pm, < span class = "org-type" > ~< / span > , Wc] = margin(G< span class = "org-type" > *< / span > K)
< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-left" / >
< col class = "org-right" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
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< th scope = "col" class = "org-right" > Manual Method< / th >
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< / tr >
< / thead >
< tbody >
< tr >
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< td class = "org-left" > Gain Margin \(> 3\) [dB]< / td >
< td class = "org-right" > 3.1< / td >
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< / tr >
< tr >
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< td class = "org-left" > Phase Margin \(> 30\) [deg]< / td >
< td class = "org-right" > 35.4< / td >
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< / tr >
< tr >
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< td class = "org-left" > Crossover \(\approx 10\) [Hz]< / td >
< td class = "org-right" > 10.1< / td >
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< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-org81a2b10" class = "outline-3" >
< h3 id = "org81a2b10" > < span class = "section-number-3" > 2.3< / span > \(\mathcal{H}_\infty\) Loop Shaping Synthesis< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
< p >
The Open Loop Shaping synthesis can be performed using the \(\mathcal{H}_\infty\) Synthesis.
< / p >
< p >
Even though we will not go into details, we will provide one example.
< / p >
< p >
Using Matlab, the \(\mathcal{H}_\infty\) synthesis of a controller based on the wanted open loop shape can be performed using the < code > loopsyn< / code > command:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = loopsyn(G, Gd);
< / pre >
< / div >
< p >
where:
< / p >
< ul class = "org-ul" >
< li > < code > G< / code > is the (LTI) plant< / li >
< li > < code > Gd< / code > is the wanted loop shape< / li >
< li > < code > K< / code > is the synthesize controller< / li >
< / ul >
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< div class = "seealso" id = "orgb2606d2" >
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< p >
Matlab documentation of < code > loopsyn< / code > (< a href = "https://www.mathworks.com/help/robust/ref/loopsyn.html" > link< / a > ).
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8f1b4c8" class = "outline-3" >
< h3 id = "org8f1b4c8" > < span class = "section-number-3" > 2.4< / span > Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
< p >
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Let’ s reuse the previous plant.
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< / p >
< p >
Translate the specification into the wanted shape of the open loop gain.
< / p >
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< ul class = "org-ul" >
< li > < b > Performance< / b > : Bandwidth of approximately 10Hz: \(|L_w(j2 \pi 10)| = 1\)< / li >
< li > < b > Noise Attenuation< / b > : Roll-off of -40dB/decade past 30Hz< / li >
< li > < b > Robustness< / b > : Gain margin > 3dB and Phase margin > 30 deg< / li >
< / ul >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Lw = 2.3e3 < span class = "org-type" > *< / span > ...
1< span class = "org-type" > /< / span > (s< span class = "org-type" > ^< / span > 2) < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Double Integrator< / span >
(1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 10< span class = "org-type" > /< / span > sqrt(3)))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 10< span class = "org-type" > *< / span > sqrt(3))); < span class = "org-comment" > % Lead< / span >
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< / pre >
< / div >
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< p >
The \(\mathcal{H}_\infty\) optimal open loop shaping is performed using the < code > loopsyn< / code > command:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > [K, < span class = "org-type" > ~< / span > , GAM] = loopsyn(G, Lw);
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< / pre >
< / div >
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< p >
The Bode plot of the obtained controller is shown in Figure < a href = "#org65199e4" > 8< / a > .
< / p >
< div class = "important" id = "orgede0579" >
< p >
It is always important to analyze the controller after the synthesis is performed.
< / p >
< p >
In the end, a synthesize controller is just a combination of low pass filters, high pass filters, notches, leads, etc.
< / p >
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< / div >
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< p >
Let’ s briefly analyze this controller:
< / p >
< ul class = "org-ul" >
< li > two integrators are used at low frequency to have the wanted low frequency high gain< / li >
< li > a lead is added centered with the crossover frequency to increase the phase margin< / li >
< li > a notch is added at the resonance of the plant to increase the gain margin (this is very typical of \(\mathcal{H}_\infty\) controllers, and can be an issue, more info on that latter)< / li >
< / ul >
< div id = "org65199e4" class = "figure" >
< p > < img src = "figs/open_loop_shaping_hinf_K.png" alt = "open_loop_shaping_hinf_K.png" / >
< / p >
< p > < span class = "figure-number" > Figure 8: < / span > Obtained controller \(K\) using the open-loop \(\mathcal{H}_\infty\) shaping< / p >
< / div >
< p >
The obtained Loop Gain is shown in Figure < a href = "#orga695393" > 9< / a > .
< / p >
< div id = "orga695393" class = "figure" >
< p > < img src = "figs/open_loop_shaping_hinf_L.png" alt = "open_loop_shaping_hinf_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Obtained Open Loop Gain \(L(s) = G(s) K(s)\) and comparison with the wanted Loop gain \(L_w\)< / p >
< / div >
< p >
Let’ s now compare the obtained stability margins of the \(\mathcal{H}_\infty\) controller and of the manually developed controller in Table < a href = "#orgcdf47bb" > 3< / a > .
< / p >
< table id = "orgcdf47bb" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 3:< / span > Comparison of the characteristics obtained with the two methods< / caption >
< colgroup >
< col class = "org-left" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Specifications< / th >
< th scope = "col" class = "org-right" > Manual Method< / th >
< th scope = "col" class = "org-right" > \(\mathcal{H}_\infty\) Method< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > Gain Margin \(> 3\) [dB]< / td >
< td class = "org-right" > 3.1< / td >
< td class = "org-right" > 31.7< / td >
< / tr >
< tr >
< td class = "org-left" > Phase Margin \(> 30\) [deg]< / td >
< td class = "org-right" > 35.4< / td >
< td class = "org-right" > 54.7< / td >
< / tr >
< tr >
< td class = "org-left" > Crossover \(\approx 10\) [Hz]< / td >
< td class = "org-right" > 10.1< / td >
< td class = "org-right" > 9.9< / td >
< / tr >
< / tbody >
< / table >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orga39391a" class = "outline-2" >
< h2 id = "orga39391a" > < span class = "section-number-2" > 3< / span > First Step in the \(\mathcal{H}_\infty\) world< / h2 >
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< div class = "outline-text-2" id = "text-3" >
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< / div >
< div id = "outline-container-orgafc7a12" class = "outline-3" >
< h3 id = "orgafc7a12" > < span class = "section-number-3" > 3.1< / span > The \(\mathcal{H}_\infty\) Norm< / h3 >
< div class = "outline-text-3" id = "text-3-1" >
< div class = "definition" id = "orgb2ccce0" >
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< p >
The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response
< / p >
\begin{equation}
\|G(s)\|_\infty = \max_\omega \bar{\sigma}\big( G(j\omega) \big)
\end{equation}
< p >
For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a function of frequency:
< / p >
\begin{equation}
\|G(s)\|_\infty = \max_{\omega} |G(j\omega)| \label{eq:hinf_norm_siso}
\end{equation}
< / div >
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< div class = "exampl" id = "orgda64a0c" >
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< p >
And compute its \(\mathcal{H}_\infty\) norm using the < code > hinfnorm< / code > function:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > hinfnorm(G)
< / pre >
< / div >
< pre class = "example" >
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7.9216e-06
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< / pre >
< p >
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The magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure < a href = "#orge580adb" > 10< / a > .
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The maximum value of the magnitude over all frequencies does correspond to the \(\mathcal{H}_\infty\) norm of \(G(s)\) as Equation \eqref{eq:hinf_norm_siso} implies.
< / p >
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< div id = "orge580adb" class = "figure" >
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< p > < img src = "figs/hinfinity_norm_siso_bode.png" alt = "hinfinity_norm_siso_bode.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Example of the \(\mathcal{H}_\infty\) norm of a SISO system< / p >
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< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org5ab38ae" class = "outline-3" >
< h3 id = "org5ab38ae" > < span class = "section-number-3" > 3.2< / span > \(\mathcal{H}_\infty\) Synthesis< / h3 >
< div class = "outline-text-3" id = "text-3-2" >
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< p >
< b > Optimization problem< / b > :
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\(\mathcal{H}_\infty\) synthesis is a method that uses an < b > algorithm< / b > (LMI optimization, Riccati equation) to find a controller of the same order as the system so that the \(\mathcal{H}_\infty\) norms of defined transfer functions are minimized.
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< / p >
< p >
< b > Engineer work< / b > :
< / p >
< ol class = "org-ol" >
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< li > Write the problem as standard \(\mathcal{H}_\infty\) problem< / li >
< li > Translate the specifications as \(\mathcal{H}_\infty\) norms< / li >
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< li > Make the synthesis and analyze the obtain controller< / li >
< li > Reduce the order of the controller for implementation< / li >
< / ol >
< p >
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< b > Many ways to use the \(\mathcal{H}_\infty\) Synthesis< / b > :
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< / p >
< ul class = "org-ul" >
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< li > Traditional \(\mathcal{H}_\infty\) Synthesis< / li >
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< li > Mixed Sensitivity Loop Shaping< / li >
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< li > Fixed-Structure \(\mathcal{H}_\infty\) Synthesis< / li >
< li > Signal Based \(\mathcal{H}_\infty\) Synthesis< / li >
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< / ul >
< / div >
< / div >
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< div id = "outline-container-org3371822" class = "outline-3" >
< h3 id = "org3371822" > < span class = "section-number-3" > 3.3< / span > The Generalized Plant< / h3 >
< div class = "outline-text-3" id = "text-3-3" >
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< div id = "org7392644" class = "figure" >
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< p > < img src = "figs/general_plant.png" alt = "general_plant.png" / >
< / p >
< / div >
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< table id = "orgf7566b6" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 4:< / span > Notations for the general configuration< / caption >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" > Notation< / th >
< th scope = "col" class = "org-left" > Meaning< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > \(P\)< / td >
< td class = "org-left" > Generalized plant model< / td >
< / tr >
< tr >
< td class = "org-left" > \(w\)< / td >
< td class = "org-left" > Exogenous inputs: commands, disturbances, noise< / td >
< / tr >
< tr >
< td class = "org-left" > \(z\)< / td >
< td class = "org-left" > Exogenous outputs: signals to be minimized< / td >
< / tr >
< tr >
< td class = "org-left" > \(v\)< / td >
< td class = "org-left" > Controller inputs: measurements< / td >
< / tr >
< tr >
< td class = "org-left" > \(u\)< / td >
< td class = "org-left" > Control signals< / td >
< / tr >
< / tbody >
< / table >
\begin{equation}
\begin{bmatrix} z \\ v \end{bmatrix} = P \begin{bmatrix} w \\ u \end{bmatrix} = \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w \\ u \end{bmatrix}
\end{equation}
< / div >
< / div >
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< div id = "outline-container-orgbf413fb" class = "outline-3" >
< h3 id = "orgbf413fb" > < span class = "section-number-3" > 3.4< / span > From a Classical Feedback Architecture to a Generalized Plant< / h3 >
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< div id = "org6c156a7" class = "figure" >
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< p > < img src = "figs/classical_feedback.png" alt = "classical_feedback.png" / >
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< p > < span class = "figure-number" > Figure 12: < / span > Classical Feedback Architecture< / p >
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< table id = "org2b76c2f" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 5:< / span > Notations for the Classical Feedback Architecture< / caption >
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< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
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< thead >
< tr >
< th scope = "col" class = "org-left" > Notation< / th >
< th scope = "col" class = "org-left" > Meaning< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > \(G\)< / td >
< td class = "org-left" > Plant model< / td >
< / tr >
< tr >
< td class = "org-left" > \(K\)< / td >
< td class = "org-left" > Controller< / td >
< / tr >
< tr >
< td class = "org-left" > \(r\)< / td >
< td class = "org-left" > Reference inputs< / td >
< / tr >
< tr >
< td class = "org-left" > \(y\)< / td >
< td class = "org-left" > Plant outputs< / td >
< / tr >
< tr >
< td class = "org-left" > \(u\)< / td >
< td class = "org-left" > Control signals< / td >
< / tr >
< tr >
< td class = "org-left" > \(d\)< / td >
< td class = "org-left" > Input Disturbance< / td >
< / tr >
< tr >
< td class = "org-left" > \(\epsilon\)< / td >
< td class = "org-left" > Tracking Error< / td >
< / tr >
< / tbody >
< / table >
< p >
The procedure is:
< / p >
< ol class = "org-ol" >
< li > define signals of the generalized plant< / li >
< li > Remove \(K\) and rearrange the inputs and outputs< / li >
< / ol >
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< div class = "exercice" id = "org205bea9" >
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< p >
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Let’ s find the Generalized plant of corresponding to the tracking control architecture shown in Figure < a href = "#org506dd07" > 13< / a >
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< / p >
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< div id = "org506dd07" class = "figure" >
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< p > < img src = "figs/classical_feedback_tracking.png" alt = "classical_feedback_tracking.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Classical Feedback Control Architecture (Tracking)< / p >
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< p >
First, define the signals of the generalized plant:
< / p >
< ul class = "org-ul" >
< li > Exogenous inputs: \(w = r\)< / li >
< li > Signals to be minimized: \(z_1 = \epsilon\), \(z_2 = u\)< / li >
< li > Control signals: \(v = y\)< / li >
< li > Control inputs: \(u\)< / li >
< / ul >
< p >
Then, Remove \(K\) and rearrange the inputs and outputs.
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We obtain the generalized plant shown in Figure < a href = "#org08dc9a7" > 14< / a > .
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< / p >
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< div id = "org08dc9a7" class = "figure" >
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< p > < img src = "figs/mixed_sensitivity_ref_tracking.png" alt = "mixed_sensitivity_ref_tracking.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 14: < / span > Generalized plant of the Classical Feedback Control Architecture (Tracking)< / p >
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< p >
Using Matlab, the generalized plant can be defined as follows:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > P = [1 < span class = "org-type" > -< / span > G;
0 1;
1 < span class = "org-type" > -< / span > G]
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< h3 id = "org5b0f8f6" > < span class = "section-number-3" > 3.5< / span > The General Synthesis Problem Formulation< / h3 >
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< div class = "important" id = "org9695476" >
< p >
The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing controllers \(K\) which minimize
< / p >
\begin{equation}
\| F_l(P, K) \|_\infty = \max_{\omega} \overline{\sigma} \big( F_l(P, K)(j\omega) \big)
\end{equation}
< / div >
< div id = "org8ea68ac" class = "figure" >
< p > < img src = "figs/general_control_names.png" alt = "general_control_names.png" / >
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< p > < span class = "figure-number" > Figure 15: < / span > General Control Configuration< / p >
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< h2 id = "org0a60e2d" > < span class = "section-number-2" > 4< / span > Modern Interpretation of the Control Specifications< / h2 >
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< div id = "outline-container-orgeaf9df5" class = "outline-3" >
< h3 id = "orgeaf9df5" > < span class = "section-number-3" > 4.1< / span > Introduction< / h3 >
< div class = "outline-text-3" id = "text-4-1" >
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< ul class = "org-ul" >
< li > < b > Reference tracking< / b > Overshoot, Static error, Setling time
< ul class = "org-ul" >
< li > \(S(s) = T_{r \rightarrow \epsilon}\)< / li >
< / ul > < / li >
< li > < b > Disturbances rejection< / b >
< ul class = "org-ul" >
< li > \(G(s) S(s) = T_{d \rightarrow \epsilon}\)< / li >
< / ul > < / li >
< li > < b > Measurement noise filtering< / b >
< ul class = "org-ul" >
< li > \(T(s) = T_{n \rightarrow \epsilon}\)< / li >
< / ul > < / li >
< li > < b > Small command amplitude< / b >
< ul class = "org-ul" >
< li > \(K(s) S(s) = T_{r \rightarrow u}\)< / li >
< / ul > < / li >
< li > < b > Stability< / b >
< ul class = "org-ul" >
< li > \(S(s)\), \(T(s)\), \(K(s)S(s)\), \(G(s)S(s)\)< / li >
< / ul > < / li >
< li > < b > Robustness to plant uncertainty< / b > (stability margins)< / li >
< li > < b > Controller implementation< / b > < / li >
< / ul >
< p >
**
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< h2 id = "org73188d9" > < span class = "section-number-2" > 5< / span > Resources< / h2 >
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< div class = "yt" > < iframe width = "100%" height = "100%" src = "https://www.youtube.com/embed/?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin" frameborder = "0" allowfullscreen > < / iframe > < / div >
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< p >
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< div class = "yt" > < iframe width = "100%" height = "100%" src = "https://www.youtube.com/embed/?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o" frameborder = "0" allowfullscreen > < / iframe > < / div >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-11-27 ven. 23:12< / p >
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