143 KiB
A brief and practical introduction to $\mathcal{H}_\infty$ Control
- Introduction
- Introduction to Model Based Control
- Classical Open Loop Shaping
- A first Step into the $\mathcal{H}_\infty$ world
- Modern Interpretation of Control Specifications
- $\mathcal{H}_\infty$ Shaping of closed-loop transfer functions
- Mixed-Sensitivity $\mathcal{H}_\infty$ Control - Example
- Conclusion
- Resources
Introduction ignore
The purpose of this document is to give a practical introduction to the wonderful world of $\mathcal{H}_\infty$ Control.
No attend is made to provide an exhaustive treatment of the subject. $\mathcal{H}_\infty$ Control is a very broad topic and entire books are written on it. Therefore, for more advanced discussion, please have a look at the recommended references at the bottom of this document.
When possible, Matlab scripts used for the example/exercises are provided such that you can easily test them on your computer.
The general structure of this document is as follows:
- A short introduction to model based control is given in Section sec:model_based_control
- Classical open loop shaping method is presented in Section sec:open_loop_shaping. It is also shown that $\mathcal{H}_\infty$ synthesis can be used for open loop shaping
- Important concepts indispensable for $\mathcal{H}_\infty$ control such as the $\mathcal{H}_\infty$ norm and the generalized plant are introduced in Section sec:h_infinity_introduction
- A very important step in $\mathcal{H}_\infty$ control is to express the control specifications (performances, robustness, etc.) as an $\mathcal{H}_\infty$ optimization problem. Such procedure is described in Section sec:modern_interpretation_specification
- One of the most useful use of the $\mathcal{H}_\infty$ control is the shaping of closed-loop transfer functions. Such technique is presented in Section sec:closed-loop-shaping
- Finally, complete examples of the use of $\mathcal{H}_\infty$ Control for practical problems are provided in Section sec:h_infinity_mixed_sensitivity.
Introduction to Model Based Control
<<sec:model_based_control>>
Introduction ignore
Model Based Control - Methodology
<<sec:model_based_control_methodology>>
The typical methodology for Model Based Control techniques is schematically shown in Figure fig:control-procedure.
It consists of three steps:
- Identification or modeling: a mathematical model $G(s)$ representing the plant dynamics is obtained
-
Translate the specifications into mathematical criteria:
- Specifications: Response Time, Noise Rejection, Maximum input amplitude, Robustness, …
- Mathematical Criteria: Cost Function, Shape of transfer function, Phase/Gain margin, Roll-Off, …
- Synthesis: research of a controller $K(s)$ that satisfies the specifications for the model of the system
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addsub) at (0, 0){};
\node[block, right=1.5 of addsub] (controller) {Controller};
\node[block, right=1.5 of controller] (plant) {Plant};
\node[block, above=1 of controller] (controller_design) {Synthesis};
\node[block, above=1 of plant] (model_plant) {Model};
\draw[<-] (addsub.west) -- ++(-1, 0) node[above right]{$r$};
\draw[->] (addsub) -- (controller.west) node[above left]{$\epsilon$};
\draw[->] (controller) -- (plant.west) node[above left]{$u$};
\draw[->] (plant.east) -- ++(1, 0) node[above left]{$y$};
\draw[] ($(plant.east) + (0.5, 0)$) -- ++(0, -1);
\draw[->] ($(plant.east) + (0.5, -1)$) -| (addsub.south);
\draw[->, dashed] (plant) -- node[midway, right, labelc, solid]{1} (model_plant);
\draw[->, dashed] (controller_design) --node[midway, right, labelc, solid]{3} (controller);
\draw[->, dashed] (model_plant) -- (controller_design);
\draw[<-, dashed] (controller_design.west) -- node[midway, above, labelc, solid]{2} ++(-1, 0) node[left, style={align=center}]{Specifications};
\end{tikzpicture}
In this document, we will suppose a model of the plant is available (step 1 already performed), and we will focus on steps 2 and 3.
In Section sec:open_loop_shaping, steps 2 and 3 will be described for a control techniques called classical (open-)loop shaping.
Then, steps 2 and 3 for the $\mathcal{H}_\infty$ Loop Shaping of closed-loop transfer functions will be discussed in Sections sec:modern_interpretation_specification, sec:closed-loop-shaping and sec:h_infinity_mixed_sensitivity.
From Classical Control to Robust Control
<<sec:comp_classical_modern_robust_control>>
Many different model based control techniques have been developed since the birth of classical control theory in the '30s.
Classical control methods were developed starting from 1930 based on tools such as the Laplace and Fourier transforms. It was then natural to study systems in the frequency domain using tools such as the Bode and Nyquist plots. Controllers were manually tuned to optimize criteria such as control bandwidth, gain and phase margins.
The '60s saw the development of control techniques based on a state-space. Linear algebra and matrices were used instead of the frequency domain tool of the class control theory. This allows multi-inputs multi-outputs systems to be easily treated. Kalman introduced the well known Kalman estimator as well the notion of optimality by minimizing quadratic cost functions. This set of developments is loosely termed Modern Control theory.
By the 1980's, modern control theory was shown to have some robustness issues and to lack the intuitive tools that the classical control methods were offering. This led to a new control theory called Robust control that blends the best features of classical and modern techniques. This robust control theory is the subject of this document.
The three presented control methods are compared in Table tab:comparison_control_methods.
Note that in parallel, there have been numerous other developments, including non-linear control, adaptive control, machine-learning control just to name a few.
| Classical Control | Modern Control | Robust Control | |
|---|---|---|---|
| Date | 1930- | 1960- | 1980- |
| Tools | Transfer Functions | State Space formulation | Systems and Signals Norms ($\mathcal{H}_\infty$, $\mathcal{H}_2$ Norms) |
| Nyquist Plots | Riccati Equations | Closed Loop Transfer Functions | |
| Bode Plots | Open/Closed Loop Shaping | ||
| Phase and Gain margins | Weighting Functions | ||
| Disk margin | |||
| Singular Value Decomposition | |||
| Control Architectures | Proportional, Integral, Derivative | Full State Feedback, LQR | General Control Configuration |
| Leads, Lags | Kalman Filters, LQG | Generalized Plant | |
| Advantages | Study Stability | Automatic Synthesis | Automatic Synthesis |
| Simple | MIMO | MIMO | |
| Natural | Optimization Problem | Optimization Problem | |
| Guaranteed Robustness | |||
| Easy specification of performances | |||
| Disadvantages | Manual Method | No Guaranteed Robustness | Required knowledge of specific tools |
| Only SISO | Difficult Rejection of Perturbations | Need a reasonably good model of the system | |
| No clear way to limit input usage |
\begin{tikzpicture}
% Scale
\def\yscale{0.8}
\def\xscale{1.0}
% Colors
\def\colorstart{blue}
\def\colorend{red}
% Axis
\draw [->] (-0.5,0) -- (10*\xscale,0) node[below left]{Robustness};
\draw [->] (0,-0.5) -- (0,10*\yscale) node[below left, rotate=90, anchor=south east]{Performance};
% Color Bar
\shade[draw, bottom color=\colorstart, top color=\colorend, fill opacity=0.5] (10*\xscale, 1*\yscale) rectangle (11*\xscale, 9*\yscale);
\node[rotate=90, above] at (10*\xscale, 5*\yscale) {Required information on plant};
\node[above] at (10.5*\xscale, 1*\yscale) {little};
\node[below] at (10.5*\xscale, 9*\yscale) {large};
% ===================================
% Classical Control
% ===================================
% Control Types
\node[align=center] (pid) at (7.0*\xscale, 1.2*\yscale) {PID\\Lead-Lag};
\begin{scope}[on background layer]
% Control Families
\node[ellipse, draw, dashed, minimum width=3.0*\xscale cm, minimum height=2.0*\yscale cm,
fill=\colorstart!90!\colorend, fill opacity=0.5, text opacity=1]
(classicalcontrol) at (pid) {};
\end{scope}
\node[above, align=center] at (classicalcontrol.north) {\textbf{Classical control} (1930)\\{\small SISO, Manual Method}};
% ===================================
% ===================================
% Modern Control
% ===================================
% Control Types
\node[align=center] (lqg) at (2.0*\xscale, 7.5*\yscale) {LQR\\LQG};
\begin{scope}[on background layer]
\node[ellipse, draw, dashed, minimum width=2.0*\xscale cm, minimum height=2.0*\yscale cm,
fill=\colorstart!20!\colorend, fill opacity=0.5, text opacity=1]
(moderncontrol) at (lqg) {};
\end{scope}
\node[above, align=center] at (moderncontrol.north) {\textbf{Modern control} (1960)\\{\small MIMO, Optimal}};
% ===================================
% ===================================
% Robust Control
% ===================================
% Control Types
\node[align=center] (hinf) at (4.5*\xscale, 4.8*\yscale) {$H_\infty$\\$H_2$};
\node[] (mu) at (5.5*\xscale, 4.8*\yscale) {$\mu$};
\begin{scope}[on background layer]
\node[ellipse, draw, dashed, minimum width=3.0*\xscale cm, minimum height=2.5*\yscale cm,
shade, left color=\colorstart!50!\colorend, right color=\colorstart!10!\colorend, fill opacity=0.5, text opacity=1]
(robustcontrol) at ($0.5*(hinf)+0.5*(mu)$) {};
\end{scope}
\node[above, align=center] at (robustcontrol.north) {\textbf{Robust control} (1990)\\{\small MIMO, Robust}};
% ===================================
\end{tikzpicture}Example System
<<sec:example_system>>
Throughout this document, multiple examples and practical application of presented control strategies will be provided. Most of them will be applied on a physical system presented in this section.
This system is shown in Figure fig:mech_sys_1dof_inertial_contr. It could represent an active suspension stage supporting a payload. The inertial motion of the payload is measured using an inertial sensor and this is feedback to a force actuator. Such system could be used to actively isolate the payload (disturbance rejection problem) or to make it follow a trajectory (tracking problem).
The notations used on Figure fig:mech_sys_1dof_inertial_contr are listed and described in Table tab:example_notations.
\begin{tikzpicture}
% Parameters
\def\massw{3}
\def\massh{1}
\def\spaceh{1.8}
% Ground
\draw[] (-0.5*\massw, 0) -- (0.5*\massw, 0);
% Mass
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5](m){$m$};
% Spring, Damper, and Actuator
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[damper] ( 0, 0) -- ( 0, \spaceh) node[midway, left=0.3]{$c$};
\draw[actuator] ( 0.3*\massw, 0) -- (0.3*\massw, \spaceh) node[midway](F){};
% Displacements
\draw[dashed] (0.5*\massw, 0) -- ++(0.5, 0);
\draw[->] (0.6*\massw, 0) -- ++(0, 0.5) node[below right]{$d$};
% Inertial Sensor
\node[inertialsensor] (inertials) at (0.5*\massw, \spaceh+\massh){};
\node[addb={+}{-}{}{}{}, right=0.8 of inertials] (subf) {};
\node[block, below=0.4 of subf] (K){$K(s)$};
\draw[->] (inertials.east) node[above right]{$y$} -- (subf.west);
\draw[->] (subf.south) -- (K.north) node[above right]{$\epsilon$};
\draw[<-] (subf.north) -- ++(0, 0.6) node[below right]{$r$};
\draw[->] (K.south) |- (F.east) node[above right]{$u$};
\end{tikzpicture}
| Notation | Description | Value | Unit |
|---|---|---|---|
| $m$ | Payload's mass to position / isolate | $10$ | [kg] |
| $k$ | Stiffness of the suspension system | $10^6$ | [N/m] |
| $c$ | Damping coefficient of the suspension system | $400$ | [N/(m/s)] |
| $y$ | Payload absolute displacement (measured by an inertial sensor) | [m] | |
| $d$ | Ground displacement, it acts as a disturbance | [m] | |
| $u$ | Actuator force | [N] | |
| $r$ | Wanted position of the mass (the reference) | [m] | |
| $\epsilon = r - y$ | Position error | [m] | |
| $K$ | Feedback controller | to be designed | [N/m] |
Derive the following open-loop transfer functions:
\begin{align} G(s) &= \frac{y}{u} \\ G_d(s) &= \frac{y}{d} \end{align}Hint
You can follow this generic procedure:
- List all applied forces ot the mass: Actuator force, Stiffness force (Hooke's law), …
- Apply the Newton's Second Law on the payload \[ m \ddot{y} = \Sigma F \]
- Transform the differential equations into the Laplace domain: \[ \frac{d\ \cdot}{dt} \Leftrightarrow \cdot \times s \]
- Write $y(s)$ as a function of $u(s)$ and $w(s)$
Results
\begin{align} G(s) &= \frac{1}{m s^2 + cs + k} \\ G_d(s) &= \frac{cs + k}{m s^2 + cs + k} \end{align}Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure fig:mech_sys_1dof_inertial_contr into a classical feedback architecture as shown in Figure fig:open_loop_shaping.
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
\node[block, right=0.8 of addfb] (K){$K(s)$};
\node[block, right=0.8 of K] (G){$G(s)$};
\node[addb={+}{}{}{}{}, right=0.8 of G] (addd){};
\node[block, above=0.5 of addd] (Gd){$G_d(s)$};
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- (addd.west);
\draw[<-] (Gd.north) -- ++(0, 0.8) node[below right]{$d$};
\draw[->] (Gd.south) -- (addd.north);
\draw[->] (addd.east) -- ++(1.2, 0);
\draw[->] ($(addd.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -1.0) -| (addfb.south);
\end{tikzpicture}
Let's define the system parameters on Matlab.
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]And now the system dynamics $G(s)$ and $G_d(s)$.
G = 1/(m*s^2 + c*s + k); % Plant
Gd = (c*s + k)/(m*s^2 + c*s + k); % DisturbanceThe Bode plots of $G(s)$ and $G_d(s)$ are shown in Figures fig:bode_plot_example_afm and fig:bode_plot_example_Gd.
Classical Open Loop Shaping
<<sec:open_loop_shaping>>
Introduction ignore
After an introduction to classical Loop Shaping in Section sec:open_loop_shaping_introduction, a practical example is given in Section sec:loop_shaping_example. Such Loop Shaping is usually performed manually with tools coming from the classical control theory.
However, the $\mathcal{H}_\infty$ synthesis can be used to automate the Loop Shaping process. This is presented in Section sec:h_infinity_open_loop_shaping and applied on the same example in Section sec:h_infinity_open_loop_shaping_example.
Introduction to Loop Shaping
<<sec:open_loop_shaping_introduction>>
Loop Shaping refers to a control design procedure that involves explicitly shaping the magnitude of the Loop Transfer Function $L(s)$.
The Loop Gain (or Loop transfer function) $L(s)$ usually refers to as the product of the controller and the plant (see Figure fig:open_loop_shaping):
\begin{equation} L(s) = G(s) \cdot K(s) \label{eq:loop_gain} \end{equation}Its name comes from the fact that this is actually the "gain around the loop".
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addsub) at (0, 0){};
\node[block, right=0.8 of addsub] (K) {$K(s)$};
\node[below] at (K.south) {Controller};
\node[block, right=0.8 of K] (G) {$G(s)$};
\node[below] at (G.south) {Plant};
\draw[<-] (addsub.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addsub) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- ++(0.8, 0) node[above left]{$y$};
\draw[] ($(G.east) + (0.5, 0)$) -- ++(0, -1.4);
\draw[->] ($(G.east) + (0.5, -1.4)$) -| (addsub.south);
\draw [decoration={brace, raise=5pt}, decorate] (K.north west) -- node[above=6pt]{$L(s)$} (G.north east);
\end{tikzpicture}
This synthesis method is one of main way controllers are design in the classical control theory. It is widely used and generally successful as many characteristics of the closed-loop system depend on the shape of the open loop gain $L(s)$ such as:
- Good Tracking: $L$ large
- Good disturbance rejection: $L$ large
- Attenuation of measurement noise on plant output: $L$ small
- Small magnitude of input signal: $L$ small
- Nominal stability: $L$ small (RHP zeros and time delays)
- Robust stability: $L$ small (neglected dynamics)
The shaping of the Loop Gain is done manually by combining several leads, lags, notches… This process is very much simplified by the fact that the loop gain $L(s)$ depends linearly on $K(s)$ eqref:eq:loop_gain. A typical wanted Loop Shape is shown in Figure fig:open_loop_shaping_shape. Another interesting Loop shape called "Bode Step" is described in cite:lurie02_system_archit_trades_using_bode.
\begin{tikzpicture}
% Phase Axis
\draw[->] (-0.3, -0.5) -- ++(8, 0) node[above]{$\omega$}; \draw[<-] (0, 0)
node[left]{$\angle L(j\omega)$} -- ++(0, -2.3);
% Gain Axis
\draw[->] (-0.3, 2) -- ++(8, 0) node[above]{$\omega$}; \draw[->] (0, 0.5) --
++(0, 3) node[left]{$\left|L(j\omega)\right|$};
% Gain Slopes
\draw[shift={(0,2)}] (0.5, 1.25) -- node[midway, above]{$-2$} (2, 0.5) --
node[midway, above]{$-1$} (6, -0.5) -- node[midway, below left]{$-2$} (7.5,
-1.25);
% Forbiden region
\path[shift={(0,1.8)}, fill=red!50!white] (0.5, 1.25) -- (2, 0.5) -| coordinate[near start](lfshaping) cycle;
\path[shift={(0,2.2)}, fill=red!50!white] (6, -0.5) -- (7.5, -1.25) |- coordinate[near end](hfshaping) cycle;
\draw[<-] (lfshaping) -- ++(0, -0.8) node[below, align=center]{{\scriptsize Ref. tracking}\\{\scriptsize Dist. rejection}};
\draw[<-] (hfshaping) -- ++(0, 0.8) node[above, align=center]{{\scriptsize Noise attenuation}};
% Crossover frequency
\node[below] (wc) at (4,2){$\omega_c$};
\draw[<-] (wc.south) -- ++(0, -0.4) node[below, align=center]{{\scriptsize Bandwidth}};
% Phase
\draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4, -1.25)[out=0, in=-180] to
(6, -2) -- (7.5, -2); \draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4,
-1.25)[out=0, in=-180] to (6, -2) -- (7.5, -2);
% Phase Margin
\draw[->, dashed] (4, -2) -- (4, -1.25) node[above]{{\scriptsize Phase Margin}};
\draw[dashed] (0, -2) node[left]{$-\pi$} -- (7.5, -2);
\end{tikzpicture}
The shaping of closed-loop transfer functions is obviously not as simple as they don't depend linearly on $K(s)$. But this is were the $\mathcal{H}_\infty$ Synthesis will be useful! More details on that in Sections sec:modern_interpretation_specification and sec:closed-loop-shaping.
Example of Manual Open Loop Shaping
<<sec:loop_shaping_example>>
Let's take our example system described in Section sec:example_system and design a controller using the Open-Loop shaping synthesis approach. The specifications are:
- Disturbance rejection: Highest possible rejection below 1Hz
- Positioning speed: Bandwidth of approximately 10Hz
- Noise attenuation: Roll-off of -40dB/decade past 30Hz
- Robustness: Gain margin > 3dB and Phase margin > 30 deg
Using SISOTOOL, design a controller that fulfills the specifications.
sisotool(G)Hint
You can follow this procedure:
- In order to have good disturbance rejection at low frequency, add a simple or double integrator
- In terms of the loop gain, the bandwidth can be defined at the frequency $\omega_c$ where $|l(j\omega_c)|$ first crosses 1 from above. Therefore, adjust the gain such that $L(j\omega)$ crosses 1 at around 10Hz
- The roll-off at high frequency for noise attenuation should already be good enough. If not, add a low pass filter
- Add a Lead centered around the crossover frequency (10 Hz) and tune it such that sufficient phase margin is added. Verify that the gain margin is good enough.
Let's say we came up with the following controller.
K = 14e8 * ... % Gain
1/(s^2) * ... % Double Integrator
1/(1 + s/2/pi/40) * ... % Low Pass Filter
(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
The bode plot of the Loop Gain is shown in Figure fig:loop_gain_manual_afm and we can verify that we have the wanted stability margins using the margin command:
[Gm, Pm, ~, Wc] = margin(G*K)| Requirements | Manual Method |
|---|---|
| Gain Margin $> 3$ [dB] | 3.1 |
| Phase Margin $> 30$ [deg] | 35.4 |
| Crossover $\approx 10$ [Hz] | 10.1 |
$\mathcal{H}_\infty$ Loop Shaping Synthesis
<<sec:h_infinity_open_loop_shaping>>
The synthesis of controllers based on the Loop Shaping method can be automated using the $\mathcal{H}_\infty$ Synthesis.
Using Matlab, it can be easily performed using the loopsyn command:
K = loopsyn(G, Lw);where:
Gis the (LTI) plantLwis the wanted loop shapeKis the synthesize controller
Matlab documentation of loopsyn (link).
Therefore, by just providing the wanted loop shape and the plant model, the $\mathcal{H}_\infty$ Loop Shaping synthesis generates a stabilizing controller such that the obtained loop gain $L(s)$ matches the specified one with an accuracy $\gamma$.
Even though we will not go into details and explain how such synthesis is working, an example is provided in the next section.
Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
<<sec:h_infinity_open_loop_shaping_example>>
To apply the $\mathcal{H}_\infty$ Loop Shaping Synthesis, the wanted shape of the loop gain should be determined from the specifications. This is summarized in Table tab:open_loop_shaping_specifications.
Such shape corresponds to the typical wanted Loop gain Shape shown in Figure fig:open_loop_shaping_shape.
| Specification | Corresponding Loop Shape | |
|---|---|---|
| Disturbance Rejection | Highest possible rejection below 1Hz | Slope of -40dB/decade at low frequency to have a high loop gain |
| Positioning Speed | Bandwidth of approximately 10Hz | $L$ crosses 1 at 10Hz: $\vert L_w(j2 \pi 10)\vert = 1$ |
| Noise Attenuation | Roll-off of -40dB/decade past 30Hz | Roll-off of -40dB/decade past 30Hz |
| Robustness | Gain margin > 3dB and Phase margin > 30 deg | Slope of -20dB/decade near the crossover |
Then, a (stable, minimum phase) transfer function $L_w(s)$ should be created that has the same gain as the wanted shape of the Loop gain. For this example, a double integrator and a lead centered on 10Hz are used. Then the gain is adjusted such that the $|L_w(j2 \pi 10)| = 1$.
Using Matlab, we have:
Lw = 2.3e3 * ...
1/(s^2) * ... % Double Integrator
(1 + s/(2*pi*10/sqrt(3)))/(1 + s/(2*pi*10*sqrt(3))); % Lead
The $\mathcal{H}_\infty$ open loop shaping synthesis is then performed using the loopsyn command:
[K, ~, GAM] = loopsyn(G, Lw);The obtained Loop Gain is shown in Figure fig:open_loop_shaping_hinf_L and matches the specified one by a factor $\gamma \approx 2$.
When using the $\mathcal{H}_\infty$ Synthesis, it is usually recommended to analyze the obtained controller.
This is usually done by breaking down the controller into simple elements such as low pass filters, high pass filters, notches, leads, etc.
Let's briefly analyze the obtained controller which bode plot is shown in Figure fig:open_loop_shaping_hinf_K:
- two integrators are used at low frequency to have the wanted low frequency high gain
- a lead is added centered with the crossover frequency to increase the phase margin
- 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)
Let's finally compare the obtained stability margins of the $\mathcal{H}_\infty$ controller and of the manually developed controller in Table tab:open_loop_shaping_compare.
| Specifications | Manual Method | $\mathcal{H}_\infty$ Method |
|---|---|---|
| Gain Margin $> 3$ [dB] | 3.1 | 31.7 |
| Phase Margin $> 30$ [deg] | 35.4 | 54.7 |
| Crossover $\approx 10$ [Hz] | 10.1 | 9.9 |
A first Step into the $\mathcal{H}_\infty$ world
<<sec:h_infinity_introduction>>
Introduction ignore
In this section, the $\mathcal{H}_\infty$ Synthesis method, which is based on the optimization of the $\mathcal{H}_\infty$ norm of transfer functions, is introduced.
After the $\mathcal{H}_\infty$ norm is defined in Section sec:h_infinity_norm, the $\mathcal{H}_\infty$ synthesis procedure is described in Section sec:h_infinity_synthesis .
The generalized plant, a very useful tool to describe a control problem, is presented in Section sec:generalized_plant. The $\mathcal{H}_\infty$ is then applied to this generalized plant in Section sec:h_infinity_general_synthesis.
Finally, an example showing how to convert a typical feedback control architecture into a generalized plant is given in Section sec:generalized_plant_derivation.
The $\mathcal{H}_\infty$ Norm
<<sec:h_infinity_norm>>
The $\mathcal{H}_\infty$ norm of a multi-input multi-output system $G(s)$ is defined as the peak of the maximum singular value of its frequency response
\begin{equation} \|G(s)\|_\infty = \max_\omega \bar{\sigma}\big( G(j\omega) \big) \end{equation}For a single-input single-output system $G(s)$, it is simply the peak value of $|G(j\omega)|$ as a function of frequency:
\begin{equation} \|G(s)\|_\infty = \max_{\omega} |G(j\omega)| \label{eq:hinf_norm_siso} \end{equation}Let's compute the $\mathcal{H}_\infty$ norm of our test plant $G(s)$ using the hinfnorm function:
hinfnorm(G)7.9216e-06
We can see in Figure fig:hinfinity_norm_siso_bode that indeed, the $\mathcal{H}_\infty$ norm of $G(s)$ does corresponds to the peak value of $|G(j\omega)|$.
$\mathcal{H}_\infty$ Synthesis
<<sec:h_infinity_synthesis>>
The $\mathcal{H}_\infty$ synthesis is a method that uses an algorithm (LMI optimization, Riccati equation) to find a controller that stabilizes the system and that minimizes the $\mathcal{H}_\infty$ norms of defined transfer functions.
Why optimizing the $\mathcal{H}_\infty$ norm of transfer functions is a pertinent choice will become clear when we will translate the typical control specifications into the $\mathcal{H}_\infty$ norm of transfer functions in Section sec:modern_interpretation_specification.
Then applying the $\mathcal{H}_\infty$ synthesis to a plant, the engineer work usually consists of the following steps:
- Write the problem as standard $\mathcal{H}_\infty$ problem using the generalized plant (described in the next section)
- Translate the specifications as $\mathcal{H}_\infty$ norms of transfer functions (Section sec:modern_interpretation_specification)
- Make the synthesis and analyze the obtained controller
As the $\mathcal{H}_\infty$ synthesis usually gives very high order controllers, an additional step that reduces the controller order is sometimes required for practical implementation.
Note that there are many ways to use the $\mathcal{H}_\infty$ Synthesis:
The Generalized Plant
<<sec:generalized_plant>>
The first step when applying the $\mathcal{H}_\infty$ synthesis is usually to write the problem as a standard $\mathcal{H}_\infty$ problem. This consist of deriving the Generalized Plant for the current problem.
The generalized plant, usually noted $P(s)$, is shown in Figure fig:general_plant. It has two sets of inputs $[w,\,u]$ and two sets of outputs $[z\,v]$ such that:
\begin{equation} \begin{bmatrix} z \\ v \end{bmatrix} = P \begin{bmatrix} w \\ u \end{bmatrix} \end{equation}The meaning of these inputs and outputs are summarized in Table tab:notation_general.
A practical example about how to derive the generalized plant for a classical control problem is given in Section sec:generalized_plant_derivation.
\begin{tikzpicture}
\node[block={2.0cm}{2.0cm}] (P) {$P$};
\node[above] at (P.north) {Generalized Plant};
% Input and outputs coordinates
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
% Connections and labels
\draw[<-] (inputw) -- ++(-0.8, 0) node[above right]{$w$};
\draw[<-] (inputu) -- ++(-0.8, 0) node[above right]{$u$};
\draw[->] (outputz) -- ++(0.8, 0) node[above left]{$z$};
\draw[->] (outputv) -- ++(0.8, 0) node[above left]{$v$};
\end{tikzpicture}
| Notation | Meaning |
|---|---|
| $P$ | Generalized plant model |
| $w$ | Exogenous inputs: commands, disturbances, noise |
| $z$ | Exogenous outputs: signals to be minimized |
| $v$ | Controller inputs: measurements |
| $u$ | Control signals |
The $\mathcal{H}_\infty$ Synthesis applied on the Generalized plant
<<sec:h_infinity_general_synthesis>>
Once the generalized plant is obtained, the $\mathcal{H}_\infty$ synthesis problem can be stated as follows:
- $\mathcal{H}_\infty$ Synthesis applied on the generalized plant
Find a stabilizing controller $K$ that, using the sensed outputs $v$, generates control signals $u$ such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer function from $w$ to $z$ is minimized.
After $K$ is found, the system is robustified by adjusting the response around the unity gain frequency to increase stability margins.
The obtained controller $K$ and the generalized plant are connected as shown in Figure fig:general_control_names.
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{2.0cm}] (P) {$P$};
\node[block={1.5cm}{1.5cm}, below=0.7 of P] (K) {$K$};
% Input and outputs coordinates
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
% Connections and labels
\draw[<-] (inputw) node[above left, align=right]{(weighted)\\exogenous inputs\\$w$} -- ++(-1.5, 0);
\draw[<-] (inputu) -- ++(-0.8, 0) |- node[left, near start, align=right]{control signals\\$u$} (K.west);
\draw[->] (outputz) node[above right, align=left]{(weighted)\\exogenous outputs\\$z$} -- ++(1.5, 0);
\draw[->] (outputv) -- ++(0.8, 0) |- node[right, near start, align=left]{sensed output\\$v$} (K.east);
\end{tikzpicture}
Using Matlab, the $\mathcal{H}_\infty$ Synthesis applied on a Generalized plant can be applied using the hinfsyn command (documentation):
K = hinfsyn(P, nmeas, ncont);where:
Pis the generalized plant transfer function matrixnmeasis the number of sensed output (size of $v$)ncontis the number of control signals (size of $u$)Kobtained controller (of sizencont x nmeas) that minimizes the $\mathcal{H}_\infty$ norm from $w$ to $z$.
Note that the general control configure of Figure fig:general_control_names, as its name implies, is quite general and can represent feedback control as well as feedforward control architectures.
From a Classical Feedback Architecture to a Generalized Plant
<<sec:generalized_plant_derivation>>
The procedure to convert a typical control architecture as the one shown in Figure fig:classical_feedback_tracking to a generalized Plant is as follows:
- Define signals of the generalized plant: $w$, $z$, $u$ and $v$
- Remove $K$ and rearrange the inputs and outputs to match the generalized configuration shown in Figure fig:general_plant
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
\node[block, right=0.8 of addfb] (K){$K(s)$};
\node[block, right=0.8 of K] (G){$G(s)$};
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- ++(1.2, 0);
\draw[->] ($(G.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -0.8) -| (addfb.south);
\end{tikzpicture} \begin{tikzpicture}
\node[block] (G) {$G(s)$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
\coordinate[above right=0.6 and 1.4 of addw] (u);
\coordinate[above=0.6 of u] (epsilon);
\coordinate[] (w) at ($(epsilon-|G.west)+(-1.4, 0)$);
\node[block, below left=0.8 and 0 of addw] (K) {$K(s)$};
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (epsilon) node[above left](z1){$\epsilon$};
\draw[->] ($(G.west)+(-0.4, 0)$)node[branch](start){} |- (u) node[above left](z2){$u$};
\draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.4, 0)$) -- (G.west);
\draw[->] (w) node[above]{$w = r$} -| (addw.north);
\draw [decoration={brace, raise=5pt}, decorate] (z1.north east) -- node[right=6pt]{$z$} (z2.south east);
\begin{scope}[on background layer]
\node[fit={(G.south-|start.west) ($(z1.north west)+(-0.4, 0)$)}, inner sep=6pt, draw, dashed, fill=black!20!white] (P) {};
\node[below] at (P.north) {Generalized Plant $P(s)$};
\end{scope}
\end{tikzpicture}Consider the feedback control architecture shown in Figure fig:classical_feedback_tracking. Suppose we want to design $K$ using the general $\mathcal{H}_\infty$ synthesis, and suppose the signals to be minimized are the control input $u$ and the tracking error $\epsilon$.
- Convert the control architecture to a generalized configuration
- Compute the transfer function matrix of the generalized plant $P$ using Matlab as a function or $K$ and $G$
Hint
First, define the signals of the generalized plant:
- Exogenous inputs: $w = r$
- Signals to be minimized: Usually, we want to minimize the tracking errors $\epsilon$ and the control signal $u$: $z = [\epsilon,\ u]$
- Controller inputs: this is the signal at the input of the controller: $v = \epsilon$
- Control inputs: signal generated by the controller: $u$
Then, Remove $K$ and rearrange the inputs and outputs as in Figure fig:general_plant.
Answer
The obtained generalized plant shown in Figure fig:mixed_sensitivity_ref_tracking.
Using Matlab, the generalized plant can be defined as follows:
P = [1 -G;
0 1;
1 -G]
P.InputName = {'w', 'u'};
P.OutputName = {'e', 'u', 'v'};Modern Interpretation of Control Specifications
<<sec:modern_interpretation_specification>>
Introduction ignore
- Section sec:closed_loop_tf
- Section sec:sensitivity_transfer_functions
- Section sec:module_margin
- Section sec:other_requirements
As shown in Section sec:open_loop_shaping, the loop gain $L(s) = G(s) K(s)$ is a useful and easy tool when manually designing controllers. This is mainly due to the fact that $L(s)$ is very easy to shape as it depends linearly on $K(s)$. Moreover, important quantities such as the stability margins and the control bandwidth can be estimated from the shape/phase of $L(s)$.
However, the loop gain $L(s)$ does not directly give the performances of the closed-loop system, which are determined by the closed-loop transfer functions.
If we consider the feedback system shown in Figure fig:gang_of_four_feedback, we can link to the following specifications to closed-loop transfer functions. This is summarized in Table tab:spec_closed_loop_tf.
| Specification | Closed-Loop Transfer Function |
|---|---|
| Reference Tracking | From $r$ to $\epsilon$ |
| Disturbance Rejection | From $d$ to $y$ |
| Measurement Noise Filtering | From $n$ to $y$ |
| Small Command Amplitude | From $n,r,d$ to $u$ |
| Stability | All closed-loop transfer function |
| Robustness (stability margins) | Module margin (see Section sec:module_margin) |
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
\node[block, right=0.8 of addfb] (K){$K(s)$};
\node[addb, right=0.8 of K] (addd){};
\node[block, right=0.8 of addd] (G){$G(s)$};
\node[addb, below right=0.4 and 0.2 of G] (addn){};
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (addd.west);
\draw[<-] (addd.north) -- ++(0, 0.6) node[below right]{$d$};
\draw[->] (addd.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- ++(1.6, 0) node[above left]{$y$};
\draw[->] (G-|addn) node[branch]{} -- (addn.north);
\draw[<-] (addn.east) -- ++(0.8, 0) node[above left]{$n$};
\draw[->] (addn.west) -| (addfb.south);
\end{tikzpicture}
Closed Loop Transfer Functions
<<sec:closed_loop_tf>>
As the performances of a controlled system depend on the closed loop transfer functions, it is very important to derive these closed-loop transfer functions as a function of the plant $G(s)$ and controller $K(s)$.
Write the output signals $[\epsilon, u, y]$ as a function of the systems $K(s), G(s)$ and of the input signals $[r, d, n]$ as shown in Figure fig:gang_of_four_feedback.
Hint
Take one of the output (e.g. $y$), and write it as a function of the inputs $[d, r, n]$ going step by step around the loop:
\begin{aligned} y &= G u \\ &= G (d + K \epsilon) \\ &= G \big(d + K (r - n - y) \big) \\ &= G d + GK r - GK n - GK y \end{aligned}Isolate $y$ at the right hand side, and finally obtain: \[ y = \frac{GK}{1+ GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \]
Do the same procedure for $u$ and $\epsilon$
Answer
The following equations should be obtained:
\begin{align} y &= \frac{GK}{1 + GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \\ \epsilon &= \frac{1 }{1 + GK} r - \frac{G}{1 + GK} d - \frac{G }{1 + GK} n \\ u &= \frac{K }{1 + GK} r - \frac{1}{1 + GK} d - \frac{K }{1 + GK} n \end{align}We can see that they are 4 different transfer functions describing the behavior of the system in Figure fig:gang_of_four_feedback. These called the Gang of Four:
\begin{align} S &= \frac{1 }{1 + GK}, \quad \text{the sensitivity function} \\ T &= \frac{GK}{1 + GK}, \quad \text{the complementary sensitivity function} \\ GS &= \frac{G }{1 + GK}, \quad \text{the load disturbance sensitivity function} \\ KS &= \frac{K }{1 + GK}, \quad \text{the noise sensitivity function} \end{align}If a feedforward controller is included, a Gang of Six transfer functions can be defined. More on that in this short video.
And we have:
\begin{align} \epsilon &= S r - GS d - GS n \\ y &= T r + GS d - T n \\ u &= KS r - S d - KS n \end{align}Thus, for reference tracking, we have to shape the closed-loop transfer function from $r$ to $\epsilon$, that is the sensitivity function $S(s)$. Similarly, to reduce the effect of measurement noise $n$ on the output $y$, we have to act on the complementary sensitivity function $T(s)$.
Sensitivity Function
<<sec:sensitivity_transfer_functions>>
The sensitivity function is indisputably the most important closed-loop transfer function of a feedback system. In this section, we will see how the shape of the sensitivity function will impact the performances of the closed-loop system.
Suppose we have developed a "reference" controller $K_r(s)$ and made three small changes to obtained three controllers $K_1(s)$, $K_2(s)$ and $K_3(s)$. The obtained sensitivity functions are shown in Figure fig:sensitivity_shape_effect and the corresponding step responses are shown in Figure fig:sensitivity_shape_effect_step.
The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table tab:compare_sensitivity_shapes.
| Controller | Sensitivity Function Shape | Change of the Step Response |
|---|---|---|
| $K_1(s)$ | Larger bandwidth $\omega_b$ | Faster rise time |
| $K_2(s)$ | Larger peak value $\Vert S\Vert_\infty$ | Large overshoot and oscillations |
| $K_3(s)$ | Larger low frequency gain $\vert S(j\cdot 0)\vert$ | Larger static error |
\begin{tikzpicture}
\begin{axis}[%
width=8cm,
height=4cm,
at={(0,0)},
xmode=log,
xmin=0.01,
xmax=10000,
ymin=-80,
ymax=40,
ylabel={Magnitude [dB]},
xlabel={Frequency [Hz]},
ytick={40, 20, 0, -20, -40, -60, -80},
xminorgrids,
yminorgrids
]
\addplot [thick, color=black, forget plot]
table[row sep=crcr]{%
0.01 -60\\
0.1 -60\\
190 6\\
10000 6\\
};
\draw[<-] (0.05, -60) -- (0.1, -70);
\draw (0.1, -70) -- (2, -70) node[right, fill=white, draw]{\footnotesize{Small static error}};
\draw[<-] (70, -3) -- (3, -3) node[left, fill=white, draw]{\footnotesize{Speed}};
\draw[<-] (300, 6) -- (200, 20);
\draw (200, 20) -- (10, 20) node[left, fill=white, draw]{\footnotesize{Robustness}};
\end{axis}
\end{tikzpicture}- Closed-Loop Bandwidth
- The closed-loop bandwidth $\omega_b$ is the frequency where $|S(j\omega)|$ first crosses $1/\sqrt{2} = -3dB$ from below. In general, a large bandwidth corresponds to a faster rise time.
From the simple analysis above, we can draw a first estimation of the wanted shape for the sensitivity function in Figure fig:h-infinity-spec-S.
The wanted characteristics on the magnitude of the sensitivity function are then:
- A small magnitude at low frequency to make the static errors small
- A wanted minimum closed-loop bandwidth in order to have fast rise time and good rejection of perturbations
- A small peak value in order to limit large overshoot and oscillations. This generally means higher robustness. This will become clear in the next section about the module margin.
Robustness: Module Margin
<<sec:module_margin>>
Let's start by an example demonstrating why the phase and gain margins might not be good indicators of robustness.
Let's consider the following plant $G_t(s)$:
w0 = 2*pi*100;
xi = 0.1;
k = 1e7;
Gt = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1);Let's say we have designed a controller $K_t(s)$ that gives the loop gain shown in Figure fig:phase_gain_margin_model_plant.
Kt = 1.2e6*(s + w0)/s;The following characteristics can be determined from Figure fig:phase_gain_margin_model_plant:
- bandwidth of $\approx 10\, \text{Hz}$
- infinite gain margin (the phase of the loop-gain never reaches -180 degrees
- more than 90 degrees of phase margin
This might indicate very good robustness properties of the closed-loop system.
Now let's suppose the "real" plant $G_r(s)$ as a slightly lower damping factor:
xi = 0.03;The obtained "real" loop gain is shown in Figure fig:phase_gain_margin_real_plant. At a frequency little bit above 100Hz, the phase of the loop gain reaches -180 degrees while its magnitude is more than one which indicated instability.
It is confirmed by checking the stability of the closed loop system:
isstable(feedback(Gr,K))0
Therefore, even a small change of the plant parameter makes the system unstable even though both the gain margin and the phase margin for the nominal plant are excellent.
This is due to the fact that the gain and phase margin are robustness indicators for a pure change or gain or a pure change of phase but not a combination of both.
Let's now determine a new robustness indicator based on the Nyquist Stability Criteria.
- Nyquist Stability Criteria (for stable systems)
- If the open-loop transfer function $L(s)$ is stable, then the closed-loop system is unstable for any encirclement of the point $−1$ on the Nyquist plot.
- Nyquist Plot
- The Nyquist plot shows the evolution of $L(j\omega)$ in the complex plane from $\omega = 0 \to \infty$.
For more information about the general Nyquist Stability Criteria, you may want to look at this video.
From the Nyquist stability criteria, it is clear that we want $L(j\omega)$ to be as far as possible from the $-1$ point (called the unstable point) in the complex plane. This minimum distance is called the module margin.
- Module Margin
- The Module Margin $\Delta M$ is defined as the minimum distance between the point $-1$ and the loop gain $L(j\omega)$ in the complex plane.
A typical Nyquist plot is shown in Figure fig:module_margin_example. The gain, phase and module margins are graphically shown to have an idea of what they represent.
As expected from Figure fig:module_margin_example, there is a close relationship between the module margin and the gain and phase margins. We can indeed show that for a given value of the module margin $\Delta M$, we have:
\begin{equation} \Delta G \ge \frac{\Delta M}{\Delta M - 1}; \quad \Delta \phi \ge \frac{1}{\Delta M} \end{equation}Let's now try to express the Module margin $\Delta M$ as an $\mathcal{H}_\infty$ norm of a closed-loop transfer function:
\begin{align*} \Delta M &= \text{minimum distance between } L(j\omega) \text{ and point } (-1) \\ &= \min_\omega |L(j\omega) - (-1)| \\ &= \min_\omega |1 + L(j\omega)| \\ &= \frac{1}{\max_\omega \frac{1}{|1 + L(j\omega)|}} \\ &= \frac{1}{\|S\|_\infty} \end{align*}The $\mathcal{H}_\infty$ norm of the sensitivity function $\|S\|_\infty$ is a measure of the Module margin $\Delta M$ and therefore an indicator of the system robustness.
\begin{equation} \Delta M = \frac{1}{\|S\|_\infty} \label{eq:module_margin_S} \end{equation}The wanted robustness of the closed-loop system can be specified by setting a maximum value on $\|S\|_\infty$.
Note that this is why large peak value of $|S(j\omega)|$ usually indicate robustness problems. And we know understand why setting an upper bound on the magnitude of $S$ is generally a good idea.
Typical, we require $\|S\|_\infty < 2 (6dB)$ which implies $\Delta G \ge 2$ and $\Delta \phi \ge 29^o$
To learn more about module/disk margin, you can check out this video.
TODO Other Requirements
<<sec:other_requirements>>
Interpretation of the $\mathcal{H}_\infty$ norm of systems:
- frequency by frequency attenuation / amplification
Let's note $G_t(s)$ the closed-loop transfer function from $w$ to $z$.
Consider an input sinusoidal signal $w(t) = \sin\left( \omega_0 t \right)$, then the output signal $z(t)$ will be equal to: \[ z(t) = A \sin\left( \omega_0 t + \phi \right) \] with:
- $A = |G_t(j\omega_0)|$ is the magnitude of $G_t(s)$ at $\omega_0$
- $\phi = \angle G_t(j\omega_0)$ is the phase of $G_t(s)$ at $\omega_0$
Noise Attenuation: typical wanted shape for $T$
| Open-Loop Shaping | Closed-Loop Shaping | |
|---|---|---|
| Reference Tracking | $L$ large | $S$ small |
| Disturbance Rejection | $L$ large | $GS$ small |
| Measurement Noise Filtering | $L$ small | $T$ small |
| Small Command Amplitude | $K$ and $L$ small | $KS$ small |
| Robustness | Phase/Gain margins | Module margin: $\Vert S\Vert_\infty$ small |
$\mathcal{H}_\infty$ Shaping of closed-loop transfer functions
<<sec:closed-loop-shaping>>
Introduction ignore
In the previous sections, we have seen that the performances of the system depends on the shape of the closed-loop transfer function.
Therefore, the synthesis problem is to design $K(s)$ such that closed-loop system is stable and such that various closed-loop transfer functions such as $S$, $KS$ and $T$ are shaped as wanted. This is clearly not simple as these closed-loop transfer functions does not depend linearly on $K$.
But don't worry, the $\mathcal{H}_\infty$ synthesis will do this job for us!
This Section sec:weighting_functions Section sec:weighting_functions_design Section sec:sensitivity_shaping_example Section sec:shaping_multiple_tf
How to Shape closed-loop transfer function? Using Weighting Functions!
<<sec:weighting_functions>>
If the $\mathcal{H}_\infty$ synthesis is applied on the generalized plant $P(s)$ shown in Figure fig:loop_shaping_S_without_W, it will generate a controller $K(s)$ such that the $\mathcal{H}_\infty$ norm of closed-loop transfer function from $r$ to $\epsilon$ is minimized. This closed-loop transfer function actually correspond to the sensitivity function. Therefore, it will minimize the the $\mathcal{H}_\infty$ norm of the sensitivity function: $\|S\|_\infty$.
However, as the $\mathcal{H}_\infty$ norm is the maximum peak value of the transfer function's magnitude, this synthesis is quite useless and clearly does not allow to shape the norm of $S(j\omega)$ over all frequencies.
\begin{tikzpicture}
\node[block] (G) {$G(s)$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
\coordinate[above right=1.0 and 1.4 of addw] (epsilon);
\coordinate[] (w) at ($(epsilon-|G.west)+(-1.0, 0)$);
\node[block, below left=0.8 and 0 of addw] (K) {$K(s)$};
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (epsilon) node[above](z1){$z = \epsilon$};
\draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.1, 0)$) -- (G.west);
\draw[->] (w) node[above]{$w = r$} -| (addw.north);
\begin{scope}[on background layer]
\node[fit={(G.south west) ($(z1.north west)+(-0.4, 0)$)}, inner sep=12pt, draw, dashed, fill=black!20!white] (P) {};
\node[below] at (P.north) {Generalized Plant $P(s)$};
\end{scope}
\end{tikzpicture}
The trick is to include a weighting function $W_S(s)$ in the generalized plant as shown in Figure fig:loop_shaping_S_with_W.
Now, the closed-loop transfer function from $w$ to $z$ is equal to $W_s(s)S(s)$ and applying the $\mathcal{H}_\infty$ synthesis to the weighted generalized plant $\tilde{P}(s)$ will generate a controller $K(s)$ such that $\|W_s(s)S(s)\|_\infty$ is minimized.
Let's now show how this is equivalent as shaping the sensitivity function:
\begin{align} & \left\| W_s(s) S(s) \right\|_\infty < 1\nonumber \\ \Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega\nonumber \\ \Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega \label{eq:sensitivity_shaping} \end{align}As shown in Equation eqref:eq:sensitivity_shaping, the $\mathcal{H}_\infty$ synthesis applying on the weighted generalized plant allows to shape the magnitude of the sensitivity transfer function.
Therefore, the choice of the weighting function $W_s(s)$ is very important: its inverse magnitude will define the wanted upper bound of the sensitivity function magnitude.
\begin{tikzpicture}
\node[block] (G) {$G(s)$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
\node[block, above right=1.0 and 1.0 of addw] (Ws) {$W_s(s)$};
\coordinate[right=0.8 of Ws] (epsilon);
\coordinate[] (w) at ($(epsilon-|G.west)+(-1.0, 0)$);
\begin{scope}[on background layer]
\node[fit={(G.south west) (Ws.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $\tilde{P}(s)$};
\end{scope}
\node[block, below=0.4 of P] (K) {$K(s)$};
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (Ws.west)node[above left]{$\epsilon$};
\draw[->] (Ws.east) -- (epsilon) node[above](z1){$z = \tilde{\epsilon}$};
\draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.2, 0)$) -- (G.west);
\draw[->] (w) node[above]{$w = r$} -| (addw.north);
\end{tikzpicture}
Using matlab, compute the weighted generalized plant shown in Figure fig:first_order_weight as a function of $G(s)$ and $W_S(s)$.
Hint
The weighted generalized plant can be defined in Matlab using two techniques:
- by writing manually the 4 transfer functions from $[w, u]$ to $[\tilde{\epsilon}, v]$
- by pre-multiplying the (non-weighted) generalized plant by a block-diagonal transfer function matrix containing the weights for the outputs $z$ and
1for the outputs $v$
Answer
The two solutions below can be used.
Pw = [Ws -Ws*G;
1 -G]; Pw = blkdiag(Ws, 1)*P;The second solution is however more general, and can also be used when weights are added at the inputs by post-multiplying instead of pre-multiplying.
Design of Weighting Functions
<<sec:weighting_functions_design>>
Weighting function included in the generalized plant must be proper, stable and minimum phase transfer functions.
- proper
- more poles than zeros, this implies $\lim_{\omega \to \infty} |W(j\omega)| < \infty$
- stable
- no poles in the right half plane
- minimum phase
- no zeros in the right half plane
Matlab is providing the makeweight function that allows to design first-order weights by specifying the low frequency gain, high frequency gain, and the gain at a specific frequency:
W = makeweight(dcgain,[freq,mag],hfgain)with:
dcgain: low frequency gain[freq,mag]: frequencyfreqat which the gain ismaghfgain: high frequency gain
The Matlab code below produces a weighting function with the following characteristics (Figure fig:first_order_weight):
- Low frequency gain of 100
- Gain of 1 at 10Hz
- High frequency gain of 0.5
Ws = makeweight(1e2, [2*pi*10, 1], 1/2);
Quite often, higher orders weights are required.
In such case, the following formula can be used:
\begin{equation} W(s) = \left( \frac{ \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}} }{ \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}} }\right)^n \label{eq:weight_formula_advanced} \end{equation}The parameters permit to specify:
- the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$
- the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$
- the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
- the absolute slope between high and low frequency: $n$
A Matlab function implementing Equation eqref:eq:weight_formula_advanced is shown below:
function [W] = generateWeight(args)
arguments
args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
args.wc (1,1) double {mustBeNumeric, mustBePositive} = 2*pi
args.n (1,1) double {mustBeInteger, mustBePositive} = 1
end
if (args.Gc <= args.G0 && args.Gc <= args.G1) || (args.Gc >= args.G0 && args.Gc >= args.G1)
eid = 'value:range';
msg = 'Gc must be between G0 and G1';
throwAsCaller(MException(eid,msg))
end
s = zpk('s');
W = (((1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (args.G0/args.Gc)^(1/args.n))/((1/args.G1)^(1/args.n)*(1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (1/args.Gc)^(1/args.n)))^args.n;
endLet's use this function to generate three weights with the same high and low frequency gains, but but different slopes.
W1 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 1);
W2 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 2);
W3 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 3);The obtained shapes are shown in Figure fig:high_order_weight.
Shaping the Sensitivity Function
<<sec:sensitivity_shaping_example>>
Let's design a controller using the $\mathcal{H}_\infty$ synthesis that fulfils the following requirements:
- Bandwidth of at least 10Hz
- Small static errors for step responses
- Robustness: Large module margin $\Delta M > 0.5$ ($\Rightarrow \Delta G > 2$ and $\Delta \phi > 29^o$)
As usual, the plant used is the one presented in Section sec:example_system.
Translate the requirements as upper bounds on the Sensitivity function and design the corresponding Weight using Matlab.
Hint
The typical wanted upper bound of the sensitivity function is shown in Figure fig:h-infinity-spec-S-bis.
More precisely:
- Recall that the closed-loop bandwidth is defined as the frequency $|S(j\omega)|$ first crosses $1/\sqrt{2} = -3dB$ from below
- For the small static error, -60dB is usually enough as other factors (measurement noise, disturbances) will anyhow limit the performances
- Recall that the module margin is equal to the inverse of the $\mathcal{H}_\infty$ norm of the sensitivity function: \[ \Delta M = \frac{1}{\|S\|_\infty} \]
Remember that the wanted upper bound of the sensitivity function is defined by the inverse magnitude of the weight.
Answer
- $|W_s(j \cdot 2 \pi 10)| = \sqrt{2}$
- $|W_s(j \cdot 0)| = 10^3$
- $\|W_s\|_\infty = 0.5$
Using Matlab, such weighting function can be generated using the makeweight function as shown below:
Ws = makeweight(1e3, [2*pi*10, sqrt(2)], 1/2);
Or using the generateWeight function:
Ws = generateWeight('G0', 1e3, ...
'G1', 1/2, ...
'Gc', sqrt(2), 'wc', 2*pi*10, ...
'n', 2);Let's say we came up with the following weighting function:
Ws = generateWeight('G0', 1e3, ...
'G1', 1/2, ...
'Gc', sqrt(2), 'wc', 2*pi*10, ...
'n', 2);The weighting function is then added to the generalized plant.
P = [1 -G;
1 -G];
Pw = blkdiag(Ws, 1)*P;And the $\mathcal{H}_\infty$ synthesis is performed on the weighted generalized plant.
K = hinfsyn(Pw, 1, 1, 'Display', 'on');Test bounds: 0.5 <= gamma <= 0.51 gamma X>=0 Y>=0 rho(XY)<1 p/f 5.05e-01 0.0e+00 0.0e+00 3.000e-16 p Limiting gains... 5.05e-01 0.0e+00 0.0e+00 3.461e-16 p 5.05e-01 -3.5e+01 # -4.9e-14 1.732e-26 f Best performance (actual): 0.503
$\gamma \approx 0.5$ means that the $\mathcal{H}_\infty$ synthesis generated a controller $K(s)$ that stabilizes the closed-loop system, and such that:
\begin{aligned} & \| W_s(s) S(s) \|_\infty \approx 0.5 \\ & \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega \end{aligned}This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure fig:results_sensitivity_hinf.
Having $\gamma < 1$ means that the $\mathcal{H}_\infty$ synthesis found a controller such that the specified closed-loop transfer functions are bellow the specified upper bounds.
Having $\gamma$ slightly above one does not necessary means the obtained controller is not "good". It just means that at some frequency, one of the closed-loop transfer functions is above the specified upper bound by a factor $\gamma$.
Shaping multiple closed-loop transfer functions
<<sec:shaping_multiple_tf>>
As was shown in Section sec:modern_interpretation_specification, depending on the specifications, up to four closed-loop transfer function may be shaped (the Gang of four). This was summarized in Table tab:specification_modern.
For instance to limit the control input $u$, $KS$ should be shaped while to filter measurement noise, $T$ should be shaped.
When multiple closed-loop transfer function are shaped at the same time, it is refereed to as "Mixed-Sensitivity $\mathcal{H}_\infty$ Control" and is the subject of Section sec:h_infinity_mixed_sensitivity.
Depending on the closed-loop transfer function being shaped, different general control configuration are used and are described below.
S KS ignore
Shaping of S and KS
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
\node[block, above right=0.4 and 0.8 of addw] (W2) {$W_2$};
\node[block, above=0.5 of W2] (W1) {$W_1$};
\coordinate (Gin) at ($(G.west)+(-0.5, 0)$);
\begin{scope}[on background layer]
\node[fit={(Gin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=0.8 of W1] (z);
\coordinate[above left=1.8 and 1.4 of G] (w);
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.2, 0)$)node[branch]{} |- (W1.west);
\draw[->] (Gin)node[branch]{} |- (W2.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west);
\draw[->] (w) node[above right]{$w$} -| (addw.north);
\end{tikzpicture}
P = [W1 -G*W1
0 W2
1 -G];- $W_1(s)$ is used to shape $S$
- $W_2(s)$ is used to shape $KS$
S T ignore
Shaping of S and T
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, right=0.8 of G] (addw) {};
\node[block, above right=0.4 and 0.8 of addw] (W2) {$W_2$};
\node[block, above=0.5 of W2] (W1) {$W_1$};
\begin{scope}[on background layer]
\node[fit={(G.south west) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=0.8 of W1] (z);
\coordinate[above left=1.8 and 0.8 of G] (w);
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west);
\draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W2.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west);
\draw[->] (w) node[above right]{$w$} -| (addw.north);
\end{tikzpicture}
P = [W1 -G*W1
0 G*W2
1 -G];- $W_1$ is used to shape $S$
- $W_2$ is used to shape $T$
S GS ignore
Shaping of S and GS
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, left=0.8 of G] (addw) {};
\node[block, above right=0.4 and 0.8 of G] (W2) {$W_2$};
\node[block, above=0.5 of W2] (W1) {$W_1$};
\begin{scope}[on background layer]
\node[fit={(addw.west |- G.south) (W1.north east)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=1.0 of W1] (z);
\coordinate[above left=2.4 and 1.0 of addw] (w);
% Connections
\draw[->] (addw.east) -- (G.west);
\draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west);
\draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W2.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (G.east) -- (G-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (addw-|w) -- (addw.west);
\draw[->] (w) node[above right]{$w$} -| (addw.north);
\end{tikzpicture}
P = [W1 -W1
G*W2 -G*W2
G -G];- $W_1$ is used to shape $S$
- $W_2$ is used to shape $GS$
S T KS ignore
Shaping of S, T and KS
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, right=0.8 of G] (addw) {};
\node[block, above right=0.4 and 0.8 of addw] (W3) {$W_3$};
\node[block, above=0.2 of W3] (W2) {$W_2$};
\node[block, above=0.2 of W2] (W1) {$W_1$};
\coordinate (Gin) at ($(G.west)+(-0.5, 0)$);
\begin{scope}[on background layer]
\node[fit={(Gin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=0.8 of W1] (z);
\coordinate[above left=1.4 and 1.3 of G] (w);
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west);
\draw[->] (Gin)node[branch]{} |- (W2.west);
\draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W3.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (W3.east) -- (W3-|z) node[above left](z3){$z_3$};
\draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west);
\draw[->] (w) node[above right]{$w$} -| (addw.north);
\end{tikzpicture}
P = [W1 -G*W1
0 W2
0 G*W3
1 -G];- $W_1$ is used to shape $S$
- $W_2$ is used to shape $KS$
- $W_3$ is used to shape $T$
S T GS ignore
Shaping of S, T and GS
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, left=0.8 of G] (addw) {};
\node[block, above right=0.4 and 0.8 of G] (W3) {$W_3$};
\node[block, above=0.2 of W3] (W2) {$W_2$};
\node[block, above=0.2 of W2] (W1) {$W_1$};
\coordinate (addwin) at ($(addw.west)+(-0.5, 0)$);
\begin{scope}[on background layer]
\node[fit={(addwin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=0.8 of W1] (z);
\coordinate[above left=2.4 and 0.8 of addwin] (w);
% Connections
\draw[->] (addw.east) -- (G.west);
\draw[->] ($(addw.east)+( 0.3, 0)$)node[branch]{} |- (W1.west);
\draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W2.west);
\draw[->] (addwin)node[branch]{} |- (W3.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (W3.east) -- (W3-|z) node[above left](z3){$z_3$};
\draw[->] (G.east) -- (G-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (addw-|w) -- (addw.west);
\draw[->] (w) node[above right]{$w$} -| (addw.north);
\end{tikzpicture}
P = [W1 -W1
G*W2 -G*W2
0 W3
G -G];- $W_1$ is used to shape $S$
- $W_2$ is used to shape $GS$
- $W_3$ is used to shape $T$
S T KS GS ignore
Shaping of S, T, KS and GS
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addr) {};
\node[addb, left=0.6 of G] (addd) {};
\node[block, above right=0.4 and 0.8 of addr] (W2) {$W_2$};
\node[block, above=0.5 of W2] (W1) {$W_1$};
\node[block, above left=0.7 and 0.8 of addd] (W3) {$W_3$};
\node[block, above=0.5 of W3] (W4) {$1$};
\begin{scope}[on background layer]
\node[fit={(W3.west|-G.south) (W4.north -| W2.east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=1.0 of W1] (z);
\coordinate[left=1.0 of W3] (w);
% Connections
\draw[->] (G.east) -- (addr.west);
\draw[->] ($(addr.east)+( 0.2, 0)$)node[branch]{} |- (W1.west);
\draw[->] ($(addd.west)+(-0.4, 0)$)node[branch]{} |- (W2.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (addr.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (addd-|w) -- (addd.west);
\draw[->] (addd.east) -- (G.west);
\draw[->] (W3.east) -| (addd.north);
\draw[->] (W4.east) -| (addr.north);
\draw[->] (w) node[above right]{$w_1$} -- (W3.west);
\draw[->] (w|-W4) node[above right]{$w_2$} -- (W4.west);
\end{tikzpicture}
P = [ W1 -W1*G*W3 -G*W1
0 0 W2
1 -G*W3 -G];- $W_1$ is used to shape $S$
- $W_2$ is used to shape $KS$
- $W_1W_3$ is used to shape $GS$
- $W_2W_3$ is used to shape $T$
Limitation ignore
When shaping multiple closed-loop transfer functions, one should be verify careful about the three following points that are further discussed:
- The shaped closed-loop transfer functions are linked by mathematical relations and cannot be shaped
- Closed-loop transfer function can only be shaped in certain frequency range.
- The size of the obtained controller may be very large and not implementable in practice
Mathematical relations are linking the closed-loop transfer functions. For instance, the sensitivity function $S(s)$ and the complementary sensitivity function $T(s)$ as link by the following well known relation:
\begin{equation} S(s) + T(s) = 1 \end{equation}This means that $|S(j\omega)|$ and $|T(j\omega)|$ cannot be made small at the same time!
It is therefore not possible to shape the four closed-loop transfer functions independently. The weighting function should be carefully design such as these fundamental relations are not violated.
The control bandwidth is clearly limited by physical constrains such as sampling frequency, electronics bandwidth,
\begin{align*} &|G(j\omega) K(j\omega)| \ll 1 \Longrightarrow |S(j\omega)| = \frac{1}{1 + |G(j\omega)K(j\omega)|} \approx 1 \\ &|G(j\omega) K(j\omega)| \gg 1 \Longrightarrow |S(j\omega)| = \frac{1}{1 + |G(j\omega)K(j\omega)|} \approx \frac{1}{|G(j\omega)K(j\omega)|} \end{align*}Similar relationship can be found for $T$, $KS$ and $GS$.
Determine the approximate norms of $T$, $KS$ and $GS$ for large loop gains ($|G(j\omega) K(j\omega)| \gg 1$) and small loop gains ($|G(j\omega) K(j\omega)| \ll 1$).
Hint
You can follows this procedure for $T$, $KS$ and $GS$:
- Write the closed-loop transfer function $T(s)$ as a function of $K(s)$ and $G(s)$
- Take $|K(j\omega)G(j\omega)| \gg 1$ and conclude on $|T(j\omega)|$
- Take $|K(j\omega)G(j\omega)| \ll 1$ and conclude on $|T(j\omega)|$
Answer
The obtained constrains are shown in Figure fig:h-infinity-4-blocs-constrains.
Depending on the frequency band, the norms of the closed-loop transfer functions depend on the controller $K$ and therefore can be shaped. However, in some frequency bands, the norms do not depend on the controller and therefore cannot be shaped.
Therefore the weighting functions should only focus on certainty frequency range depending on the transfer function being shaped. These regions are summarized in Figure fig:h-infinity-4-blocs-constrains.
\begin{tikzpicture}
\begin{scope}[shift={(0, 0)}]
\draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim GK^{-1}$};
\draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim 1$};
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3) to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0);
\draw[dashed] rectangle ;
\begin{scope}[on background layer]
\node[fit={(-0.5, -2.7) (5.5, 1.4)}, inner sep=0pt, draw, dashed, fill=black!20!white] (S) {};
\node[below] at (S.north) {$S$};
\end{scope}
\end{scope}
\begin{scope}[shift={(6.4, 0)}]
\draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim K^{-1}$};
\draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim G$};
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-2) to[out=45,in=180+45] (1, -1) to[out=45, in=180] (2.5,-0.2) to[out=0,in=180-45] (4,-1) to[out=-45,in=180-45] (5, -2);
\begin{scope}[on background layer]
\node[fit={(-0.5, -2.7) (5.5, 1.4)}, inner sep=0pt, draw, dashed, fill=black!20!white] (GS) {};
\node[below] at (GS.north) {$GS$};
\end{scope}
\end{scope}
\begin{scope}[shift={(0, -4.4)}]
\draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -1.8) node[]{$\sim G^{-1}$};
\draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.3) node[]{$\sim K$};
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-1.5) to[out=45,in=180+45] (1, -0.5) to[out=45, in=180] (2.5,0.3) to[out=0,in=180-45] (4,-0.5) to[out=-45,in=180-45] (5, -1.5);
\begin{scope}[on background layer]
\node[fit={(-0.5, -2.7) (5.5, 1.4)}, inner sep=0pt, draw, dashed, fill=black!20!white] (KS) {};
\node[below] at (KS.north) {$KS$};
\end{scope}
\end{scope}
\begin{scope}[shift={(6.4, -4.4)}]
\draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim 1$};
\draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim GK$};
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,0) to[out=0,in=180] (1.5,0) to[out=0,in=180] (2.5,0.3) to[out=0,in=-45] (3,0) to[out=-45,in=180-45] (5, -2);
\begin{scope}[on background layer]
\node[fit={(-0.5, -2.7) (5.5, 1.4)}, inner sep=0pt, draw, dashed, fill=black!20!white] (T) {};
\node[below] at (T.north) {$T$};
\end{scope}
\end{scope}
\end{tikzpicture}
The order (resp. number of state) of the controller given by the $\mathcal{H}_\infty$ synthesis is equal to the order (resp. number of state) of the weighted generalized plant. It is thus equal to the sum of the number of state of the non-weighted generalized plant and the number of state of all the weighting functions.
Two approaches can be used to obtain controllers with reasonable order:
- use simple weights (usually first order)
- perform a model reduction on the obtained high order controller
Mixed-Sensitivity $\mathcal{H}_\infty$ Control - Example
<<sec:h_infinity_mixed_sensitivity>>
Control Problem
- Control Diagram
-
Inputs Signals
- Reference steps of 1mm
- Measurement noise is considered to be a white noise with a power spectral density of …
-
Specifications
- Follow reference steps with a response time of 0.1s and static error less than $1 \mu m$
- Maximum control signal of 10N
- Robustness
- Reduce the effect of measurement noise on the position
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
G = 1/(m*s^2 + c*s + k); % Plant
Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance % Generate Input Signals
t = 0:1e-3:1;
r = zeros(size(t));
r(t>0.1) = 1e-3;
Fs = 1e3; % Sampling Frequency [Hz]
Ts = 1/Fs; % Sampling Time [s]
n = sqrt(Fs/2)*randn(1, length(t)); % Signal with an ASD equal to one
n = n*1e-6;
d = zeros(size(t));
d(t>0.5) = 5e-4;Control Design Procedure
| Response Time | ||
| Robustness | ||
| Limitation of the Command Amplitude | ||
| Filtering of the measurement noise |
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, right=0.8 of G] (addw) {};
\node[block, above right=0.4 and 0.8 of addw] (W3) {$W_3$};
\node[block, above=0.2 of W3] (W2) {$W_2$};
\node[block, above=0.2 of W2] (W1) {$W_1$};
\coordinate (Gin) at ($(G.west)+(-0.5, 0)$);
\begin{scope}[on background layer]
\node[fit={(Gin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=0.8 of W1] (z);
\coordinate[above left=1.4 and 1.3 of G] (w);
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west);
\draw[->] (Gin)node[branch]{} |- (W2.west);
\draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W3.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (W3.east) -- (W3-|z) node[above left](z3){$z_3$};
\draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west);
\draw[->] (w) node[above right]{$w$} -| (addw.north);
\end{tikzpicture}
- $W_1$ is used to shape $S$
- $W_2$ is used to shape $KS$
- $W_3$ is used to shape $T$
P = [1 -G
0 1
0 G
1 -G];Step 1 - Shaping of $S$
<<sec:step_shaping_S>>
We start with the shaping of $S$ alone. To not constrain $KS$ and $T$, we set small values for $W_2$ and $W_3$
W2 = tf(1e-8);
W3 = tf(0.1); W1 = generateWeight('G0', 1e3, ...
'G1', 1/2, ...
'Gc', sqrt(2), 'wc', 2*pi*10, ...
'n', 1); Pw = blkdiag(W1, W2, W3, 1)*P; K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
Test bounds: 0.5 <= gamma <= 0.552
gamma X>=0 Y>=0 rho(XY)<1 p/f
5.25e-01 0.0e+00 0.0e+00 6.061e-16 p
5.13e-01 0.0e+00 0.0e+00 0.000e+00 p
5.06e-01 -2.5e+00 # -6.4e-14 1.440e-15 f
5.09e-01 -5.5e+00 # -4.1e-14 9.510e-16 f
Limiting gains...
5.14e-01 0.0e+00 0.0e+00 1.039e-25 p
5.14e-01 0.0e+00 0.0e+00 1.040e-25 p
Best performance (actual): 0.514
'org_babel_eoe'
ans =
'org_babel_eoe'
Z1 = lft(P, K1); % Create model with r and n as inputs and y and u as outputs
Psim = [0 0 Gd G
0 0 0 1
1 -1 -Gd -G];
Z1sim = lft(Psim, K1); z = lsim(Z1sim, [r;n;d], t);
y1 = z(:,1);
u1 = z(:,2); figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
hold on;
plot(t, y1);
hold off;
xlabel('Time [s]'); ylabel('Output $y$ [m]');
nexttile;
hold on;
plot(t, u1);
hold off;
xlabel('Time [s]'); ylabel('Control Input $u$ [N]');Step 2 - Shaping of $KS$
<<sec:step_shaping_KS>>
W2 = generateWeight('G0', 5e-7, ...
'G1', 1e-3, ...
'Gc', 1e-6, 'wc', 2*pi*100, ...
'n', 1); Pw = blkdiag(W1, W2, W3, 1)*P; K2 = hinfsyn(Pw, 1, 1, 'Display', 'on');K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
Test bounds: 0.51 <= gamma <= 1.2
gamma X>=0 Y>=0 rho(XY)<1 p/f
7.82e-01 0.0e+00 0.0e+00 0.000e+00 p
6.32e-01 -2.2e+00 # -3.0e-14 -1.136e-32 f
7.03e-01 -5.1e+00 # -2.9e-14 5.128e-17 f
7.41e-01 -1.1e+01 # -1.0e-14 1.431e-22 f
7.61e-01 -2.6e+01 # -9.1e-15 1.215e-21 f
7.72e-01 -6.9e+01 # -1.9e-14 2.828e-16 f
7.77e-01 -3.6e+02 # -1.2e-14 1.213e-15 f
Best performance (actual): 0.782
'org_babel_eoe'
ans =
'org_babel_eoe'
Z2 = lft(P, K2); Z2sim = lft(Psim, K2); z = lsim(Z2sim, [r;n;d], t);
y2 = z(:,1);
u2 = z(:,2); figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
hold on;
plot(t, y1);
plot(t, y2);
hold off;
xlabel('Time [s]'); ylabel('Output $y$ [m]');
nexttile;
hold on;
plot(t, u1);
plot(t, u2);
hold off;
xlabel('Time [s]'); ylabel('Control Input $u$ [N]');Step 3 - Shaping of $T$
<<sec:step_shaping_T>>
W3 = generateWeight('G0', 1e-1, ...
'G1', 1e4, ...
'Gc', 1, 'wc', 2*pi*30, ...
'n', 3); Pw = blkdiag(W1, W2, W3, 1)*P; K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
Test bounds: 0.578 <= gamma <= 1.66
gamma X>=0 Y>=0 rho(XY)<1 p/f
9.78e-01 -1.3e+01 # -6.2e-15 1.141e-18 f
1.27e+00 0.0e+00 0.0e+00 1.524e-15 p
1.12e+00 0.0e+00 0.0e+00 4.481e-15 p
1.04e+00 -4.0e+01 # -1.0e-13 1.102e-42 f
1.08e+00 -6.6e+02 # -4.4e-15 3.641e-18 f
1.10e+00 0.0e+00 0.0e+00 6.052e-21 p
1.09e+00 0.0e+00 0.0e+00 5.005e-15 p
Best performance (actual): 1.09
'org_babel_eoe'
ans =
'org_babel_eoe'
Z3 = lft(P, K3); Z3sim = lft(Psim, K3); z = lsim(Z3sim, [r;n;d], t);
y3 = z(:,1);
u3 = z(:,2); figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
hold on;
plot(t, y1);
plot(t, y2);
plot(t, y3);
hold off;
xlabel('Time [s]'); ylabel('Output $y$ [m]');
nexttile;
hold on;
plot(t, u1);
plot(t, u2);
plot(t, u3);
hold off;
xlabel('Time [s]'); ylabel('Control Input $u$ [N]');Conclusion
<<sec:conclusion>>
Resources
yt:?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin
yt:?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o