lecture-h-infinity/index.org

Robust Control - $\mathcal{H}_\infty$ Synthesis

• Introduction
• Introduction to the Control Methodology - Model Based Control
  • Control Methodology
  • Some Background: From Classical Control to Robust Control
  • Example System
• Classical Open Loop Shaping
  • Introduction ot Open Loop Shaping
    • Explain why the Loop gain si an important "value"
    • END
  • Example of Open Loop Shaping
  • $\mathcal{H}_\infty$ Loop Shaping Synthesis
  • Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
• The $\mathcal{H}_\infty$ Norm
• $\mathcal{H}_\infty$ Synthesis
• The Generalized Plant
• Problem Formulation
• Classical feedback control and closed loop transfer functions
• From a Classical Feedback Architecture to a Generalized Plant
• Modern Interpretation of the Control Specifications
  • Introduction
• Resources

Introduction   ignore

Introduction to the Control Methodology - Model Based Control

Control Methodology

The typical methodology when applying Model Based Control to a plant is schematically shown in Figure fig:control-procedure. It consists of three steps:

1. Identification or modeling: $\Longrightarrow$ mathematical model

2. Translate the specifications into mathematical criteria:

   • Specifications: Response Time, Noise Rejection, Maximum input amplitude, Robustness, …
   • Mathematical Criteria: Cost Function, Shape of TF
3. Synthesis: research of $K$ 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}

Typical Methodoly for Model Based Control

In this document, we will mainly focus on steps 2 and 3.

Some Background: From Classical Control to Robust Control

Classical Control (1930)

• Tools:

  • TF (input-output)
  • Nyquist, Bode, Black, \ldots
  • P-PI-PID, Phase lead-lag, \ldots

• Advantages:

  • Stability
  • Performances
  • Robustness

• Disadvantages:

  • Manual Method
  • Only SISO

Modern Control (1960)

• Tools:

  • State Space
  • Optimal Command
  • LQR, LQG

• Advantages:

  • Automatic Synthesis
  • MIMO
  • Optimisation problem

• Disadvantages:

  • Robustness
  • Rejection of Perturbations

Robust Control (1980)

• Tools:

  • Disk Margin
  • Systems and Signals norms ($\mathcal{H}_\infty$ and $\mathcal{H}_2$ norms)
  • Closed Loop Transfer Functions
  • Loop Shaping

• Advantages:

  • Stability
  • Performances
  • Robustness
  • Automatic Synthesis
  • MIMO
  • Optimization Problem

• Disadvantages:

  • Requires the knowledge of specific tools
  • Need a reasonably good model of the system

Example System

Let's consider the test-system shown in Figure fig:mech_sys_1dof_inertial_contr. The notations used are listed 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}

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.

Notation	Description	Value	Unit
$m$	Payload's mass to position / isolate		[kg]
$k$	Stiffness of the suspension system		[N/m]
$c$	Damping coefficient of the suspension system		[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]

Example system variables

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:

1. List all applied forces ot the mass: Actuator force, Stiffness force (Hooke's law), …
2. Apply the Newton's Second Law on the payload \[ m \ddot{y} = \Sigma F \]
3. Transform the differential equations into the Laplace domain: \[ \frac{d\ \cdot}{dt} \Leftrightarrow \cdot \times s \]
4. Write $y(s)$ as a function of $u(s)$ and $w(s)$

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 form 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}

Block diagram corresponding to the example system

  k = 1e6; % Stiffness [N/m]
  c = 4e2; % Damping [N/(m/s)]
  m = 16; % Mass [kg]

  G = 1/(m*s^2 + c*s + k);
  Gd = (c*s + k)/(m*s^2 + c*s + k);

Classical Open Loop Shaping

Introduction ot Open Loop Shaping

Usually, the controller $K(s)$ is designed such that the loop gain $L(s)$ has desirable shape. This technique is called Open Loop Shaping.

TODO Explain why the Loop gain si an important "value"

For instance example all the specifications can usually be explained in terms of the open loop gain.

END

  \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}

Classical Feedback Architecture

This is usually done manually has the loop gain $L(s)$ depends linearly of $K(s)$:

\begin{equation} L(s) = G(s) K(s) \end{equation}

• where $L(s)$ is called the Loop Gain Transfer Function

$K(s)$ then consists of a combination of leads, lags, notches, etc. such that its product with $G(s)$ has wanted shape.

Example of Open Loop Shaping

  k = 1e-6;
  m = 10;
  c = 10;

  G =

Bode plot of the plant $G(s)$

Specifications:

• Performance: Bandwidth of approximately 50Hz
• Noise Attenuation: Roll-off of -40dB/decade past 250Hz
• Robustness: Gain margin > 5dB and Phase margin > 40 deg

Using SISOTOOL, design a controller that fulfill the specifications.

  sisotool(G)

In order to have the wanted Roll-off, two integrators are used, a lead is also added to have sufficient phase margin.

The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure fig:loop_gain_manual_afm.

  K = 6e4 * ... % Gain
      1/(s^2) * ... % Double Integrator
      (1 + s/111)/(1 + s/888); % Lead

Bode Plot of the obtained Loop Gain $L(s) = G(s) K(s)$

And we can verify that we have the wanted stability margins:

  [Gm, Pm, ~, Wc] = margin(G*K)

	Value
Gain Margin [dB]	7.2
Phase Margin [deg]	48.1
Crossover [Hz]	50.7

$\mathcal{H}_\infty$ Loop Shaping Synthesis

The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$ Synthesis.

Even though we will not go into details, we will provide one example.

Using Matlab, the $\mathcal{H}_\infty$ synthesis of a controller based on the wanted open loop shape can be performed using the loopsyn command:

  K = loopsyn(G, Gd);

where:

• G is the (LTI) plant
• Gd is the wanted loop shape
• K is the synthesize controller

Matlab documentation of loopsyn (link).

Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis

Let's re-use the previous plant.

Translate the specification into the wanted shape of the open loop gain.

  G = tf(16,[1 0.16 16]);

  Gd = 3.7e4*1/s*(1 + s/2/pi/20)/(1 + s/2/pi/220)*1/(s + s/2/pi/500);

  [K,CL,GAM,INFO] = loopsyn(G, Gd);

  bodeFig({K})

The $\mathcal{H}_\infty$ Norm

The $\mathcal{H}_\infty$ norm is defined as the peak of the maximum singular value of the frequency response

\begin{equation} \|G(s)\|_\infty = \max_\omega \bar{\sigma}\big( G(j\omega) \big) \end{equation}

For a SISO 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 define a plant dynamics:

   w0 = 2*pi; k = 1e6; xi = 0.04;

   G = 1/k/(s^2/w0^2 + 2*xi*s/w0 + 1);

And compute its $\mathcal{H}_\infty$ norm using the hinfnorm function:

  hinfnorm(G)

1.0013e-05

The magnitude $|G(j\omega)|$ of the plant $G(s)$ as a function of frequency is shown in Figure fig:hinfinity_norm_siso_bode. 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.

Example of the $\mathcal{H}_\infty$ norm of a SISO system

$\mathcal{H}_\infty$ Synthesis

Optimization problem: $\mathcal{H}_\infty$ synthesis is a method that uses an algorithm (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.

Engineer work:

1. Write the problem as standard $\mathcal{H}_\infty$ problem
2. Translate the specifications as $\mathcal{H}_\infty$ norms
3. Make the synthesis and analyze the obtain controller
4. Reduce the order of the controller for implementation

Many ways to use the $\mathcal{H}_\infty$ Synthesis:

• Traditional $\mathcal{H}_\infty$ Synthesis
• Mixed Sensitivity Loop Shaping
• Fixed-Structure $\mathcal{H}_\infty$ Synthesis
• Signal Based $\mathcal{H}_\infty$ Synthesis

The Generalized Plant

  \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

Notations for the general configuration

\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}

Problem Formulation

The $\mathcal{H}_\infty$ Synthesis objective is to find all stabilizing controllers $K$ which minimize

\begin{equation} \| F_l(P, K) \|_\infty = \max_{\omega} \overline{\sigma} \big( F_l(P, K)(j\omega) \big) \end{equation}

  \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}

General Control Configuration

Classical feedback control and closed loop transfer functions

  \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] (addu){};
    \node[block, right=0.8 of addu] (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) -- (addu.west) node[above left]{$u$};
    \draw[->] (addu.east) -- (G.west);
    \draw[<-] (addu.north) -- ++(0, 0.8) node[below right]{$d$};
    \draw[->] (G.east) -- ++(1.2, 0);
    \draw[->] ($(G.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -0.8) -| (addfb.south);
  \end{tikzpicture}

Classical Feedback Architecture

Notation	Meaning
$G$	Plant model
$K$	Controller
$r$	Reference inputs
$y$	Plant outputs
$u$	Control signals
$d$	Input Disturbance
$\epsilon$	Tracking Error

Notations for the Classical Feedback Architecture

From a Classical Feedback Architecture to a Generalized Plant

The procedure is:

1. define signals of the generalized plant
2. Remove $K$ and rearrange the inputs and outputs

  \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 right] at (P.north west) {Generalized Plant $P(s)$};
    \end{scope}
  \end{tikzpicture}

Let's find the Generalized plant of corresponding to the tracking control architecture shown in Figure fig:classical_feedback_tracking

Classical Feedback Control Architecture (Tracking)

First, define the signals of the generalized plant:

• Exogenous inputs: $w = r$
• Signals to be minimized: $z_1 = \epsilon$, $z_2 = u$
• Control signals: $v = y$
• Control inputs: $u$

Then, Remove $K$ and rearrange the inputs and outputs. We obtain the generalized plant shown in Figure fig:mixed_sensitivity_ref_tracking.

Generalized plant of the Classical Feedback Control Architecture (Tracking)

Using Matlab, the generalized plant can be defined as follows:

  P = [1 -G;
       0  1;
       1 -G]

Modern Interpretation of the Control Specifications

Introduction

• Reference tracking Overshoot, Static error, Setling time

  • $S(s) = T_{r \rightarrow \epsilon}$

• Disturbances rejection

  • $G(s) S(s) = T_{d \rightarrow \epsilon}$

• Measurement noise filtering

  • $T(s) = T_{n \rightarrow \epsilon}$

• Small command amplitude

  • $K(s) S(s) = T_{r \rightarrow u}$

• Stability

  • $S(s)$, $T(s)$, $K(s)S(s)$, $G(s)S(s)$
• Robustness to plant uncertainty (stability margins)
• Controller implementation

**

Resources

yt:?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin

yt:?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o