Robust Control - \(\mathcal{H}_\infty\) Synthesis

1. Introduction to the Control Methodology - Model Based Control
2. Some Background: From Classical Control to Robust Control
3. The \(\mathcal{H}_\infty\) Norm
4. \(\mathcal{H}_\infty\) Synthesis
5. The Generalized Plant
6. Problem Formulation
7. Classical feedback control and closed loop transfer functions
8. From a Classical Feedback Architecture to a Generalized Plant
9. Modern Interpretation of the Control Specifications +
- 9.1. Introduction
+
10. Resources

+ +

1 Introduction to the Control Methodology - Model Based Control

+The typical methodology when applying Model Based Control to a plant is schematically shown in Figure 1. +It consists of three steps: +

Identification or modeling: \(\Longrightarrow\) mathematical model
Translate the specifications into mathematical criteria: +
- Specifications: Response Time, Noise Rejection, Maximum input amplitude, Robustness, …
- Mathematical Criteria: Cost Function, Shape of TF
Synthesis: research of \(K\) that satisfies the specifications for the model of the system

+ + +

Figure 1: Typical Methodoly for Model Based Control

+ +

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

+ +

2 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

+ +

3 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 2. +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. +

+ + +

Figure 2: Example of the \(\mathcal{H}_\infty\) norm of a SISO system

+ +

4 \(\mathcal{H}_\infty\) Synthesis

+Optimization problem: +\(\hinf\) 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 \(\hinf\) norms of defined transfer functions are minimized. +

+ +

+Engineer work: +

Write the problem as standard \(\hinf\) problem
Translate the specifications as \(\hinf\) norms
Make the synthesis and analyze the obtain controller
Reduce the order of the controller for implementation

+ +

+Many ways to use the \(\hinf\) Synthesis: +

Traditional \(\hinf\) Synthesis
Mixed Sensitivity Loop Shaping
Fixed-Structure \(\hinf\) Synthesis
Signal Based \(\hinf\) Synthesis

+ +

5 The Generalized Plant

+ +

+ + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Table 1: Notations for the general configuration
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

+ +\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} +

+ +

6 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} + +

+ + +

Figure 4: General Control Configuration

+ + +

7 Classical feedback control and closed loop transfer functions

+ +

Figure 5: Classical Feedback Architecture

+ + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Table 2: Notations for the 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

+ +

8 From a Classical Feedback Architecture to a Generalized Plant

+The procedure is: +

define signals of the generalized plant
Remove \(K\) and rearrange the inputs and outputs

+ +

+Let’s find the Generalized plant of corresponding to the tracking control architecture shown in Figure 6 +

+ + +

Figure 6: 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 7. +

+ + +

Figure 7: 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]
+

+ +

9 Modern Interpretation of the Control Specifications

9.1 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

+ +

+** +

+ +

10 Resources

+ +

Robust Control - \(\mathcal{H}_\infty\) Synthesis

Table of Contents

1 Introduction to the Control Methodology - Model Based Control

2 Some Background: From Classical Control to Robust Control

3 The \(\mathcal{H}_\infty\) Norm

4 \(\mathcal{H}_\infty\) Synthesis

5 The Generalized Plant

6 Problem Formulation

7 Classical feedback control and closed loop transfer functions

8 From a Classical Feedback Architecture to a Generalized Plant

9 Modern Interpretation of the Control Specifications

9.1 Introduction

10 Resources