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

The typical methodology when applying Model Based Control to a plant is schematically shown in Figure 1. 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

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

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

Let’s consider the test-system shown in Figure 2. The notations used are listed in Table 1.

Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure 2 into a classical feedback form as shown in Figure 4.

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

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

• 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.

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

G =

Specifications:

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

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 6.

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

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

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

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

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})

• 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

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