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Stewart Platform - Vibration Isolation

Table of Contents

1 HAC-LAC (Cascade) Control - Integral Control

1.1 Introduction

In this section, we wish to study the use of the High Authority Control - Low Authority Control (HAC-LAC) architecture on the Stewart platform.

The control architectures are shown in Figures 1 and 2.

First, the LAC loop is closed (the LAC control is described here), and then the HAC controller is designed and the outer loop is closed.

control_arch_hac_iff.png

Figure 1: HAC-LAC architecture with IFF

control_arch_hac_dvf.png

Figure 2: HAC-LAC architecture with DVF

1.2 Initialization

We first initialize the Stewart platform. 

stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'none');

The rotation point of the ground is located at the origin of frame \(\{A\}\). 

ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');

1.3 Identification

We identify the transfer function from the actuator forces \(\bm{\tau}\) to the absolute displacement of the mobile platform \(\bm{\mathcal{X}}\) in three different cases:

  • Open Loop plant
  • Already damped plant using Integral Force Feedback
  • Already damped plant using Direct velocity feedback

1.3.1 HAC - Without LAC

controller = initializeController('type', 'open-loop');

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'input');      io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]

%% Run the linearization
G_ol = linearize(mdl, io);
G_ol.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_ol.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

1.3.2 HAC - IFF

controller = initializeController('type', 'iff');
K_iff = -(1e4/s)*eye(6);

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'input');      io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]

%% Run the linearization
G_iff = linearize(mdl, io);
G_iff.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_iff.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

1.3.3 HAC - DVF

controller = initializeController('type', 'dvf');
K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6);

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'input');      io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]

%% Run the linearization
G_dvf = linearize(mdl, io);
G_dvf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

1.4 Control Architecture

We use the Jacobian to express the actuator forces in the cartesian frame, and thus we obtain the transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\). 

Gc_ol = minreal(G_ol)/stewart.kinematics.J';
Gc_ol.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

Gc_iff = minreal(G_iff)/stewart.kinematics.J';
Gc_iff.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

Gc_dvf = minreal(G_dvf)/stewart.kinematics.J';
Gc_dvf.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

We then design a controller based on the transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\), finally, we will pre-multiply the controller by \(\bm{J}^{-T}\).

1.5 6x6 Plant Comparison

hac_lac_coupling_jacobian.png

Figure 3: Norm of the transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (png, pdf)

1.6 HAC - DVF

1.6.1 Plant

hac_lac_plant_dvf.png

Figure 4: Diagonal elements of the plant for HAC control when DVF is previously applied (png, pdf)

1.6.2 Controller Design

We design a diagonal controller with equal bandwidth for the 6 terms. The controller is a pure integrator with a small lead near the crossover. 

wc = 2*pi*300; % Wanted Bandwidth [rad/s]

h = 1.2;
H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));

Kd_dvf = diag(1./abs(diag(freqresp(1/s*Gc_dvf, wc)))) .* H_lead .* 1/s;

hac_lac_loop_gain_dvf.png

Figure 5: Diagonal elements of the Loop Gain for the HAC control (png, pdf)

Finally, we pre-multiply the diagonal controller by \(\bm{J}^{-T}\) prior implementation. 

K_hac_dvf = inv(stewart.kinematics.J')*Kd_dvf;

1.6.3 Obtained Performance

We identify the transmissibility and compliance of the system. 

controller = initializeController('type', 'open-loop');
[T_ol, T_norm_ol, freqs] = computeTransmissibility();
[C_ol, C_norm_ol, ~] = computeCompliance();

controller = initializeController('type', 'dvf');
[T_dvf, T_norm_dvf, ~] = computeTransmissibility();
[C_dvf, C_norm_dvf, ~] = computeCompliance();

controller = initializeController('type', 'hac-dvf');
[T_hac_dvf, T_norm_hac_dvf, ~] = computeTransmissibility();
[C_hac_dvf, C_norm_hac_dvf, ~] = computeCompliance();

hac_lac_C_T_dvf.png

Figure 6: Obtained Compliance and Transmissibility (png, pdf)

1.7 HAC - IFF

1.7.1 Plant

hac_lac_plant_iff.png

Figure 7: Diagonal elements of the plant for HAC control when IFF is previously applied (png, pdf)

1.7.2 Controller Design

We design a diagonal controller with equal bandwidth for the 6 terms. The controller is a pure integrator with a small lead near the crossover. 

wc = 2*pi*300; % Wanted Bandwidth [rad/s]

h = 1.2;
H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));

Kd_iff = diag(1./abs(diag(freqresp(1/s*Gc_iff, wc)))) .* H_lead .* 1/s;

hac_lac_loop_gain_iff.png

Figure 8: Diagonal elements of the Loop Gain for the HAC control (png, pdf)

Finally, we pre-multiply the diagonal controller by \(\bm{J}^{-T}\) prior implementation. 

K_hac_iff = inv(stewart.kinematics.J')*Kd_iff;

1.7.3 Obtained Performance

We identify the transmissibility and compliance of the system. 

controller = initializeController('type', 'open-loop');
[T_ol, T_norm_ol, freqs] = computeTransmissibility();
[C_ol, C_norm_ol, ~] = computeCompliance();

controller = initializeController('type', 'iff');
[T_iff, T_norm_iff, ~] = computeTransmissibility();
[C_iff, C_norm_iff, ~] = computeCompliance();

controller = initializeController('type', 'hac-iff');
[T_hac_iff, T_norm_hac_iff, ~] = computeTransmissibility();
[C_hac_iff, C_norm_hac_iff, ~] = computeCompliance();

hac_lac_C_T_iff.png

Figure 9: Obtained Compliance and Transmissibility (png, pdf)

1.8 Comparison

hac_lac_C_full_comparison.png

Figure 10: Comparison of the norm of the Compliance matrices for the HAC-LAC architecture (png, pdf)

hac_lac_T_full_comparison.png

Figure 11: Comparison of the norm of the Transmissibility matrices for the HAC-LAC architecture (png, pdf)

hac_lac_C_T_comparison.png

Figure 12: Comparison of the Frobenius norm of the Compliance and Transmissibility for the HAC-LAC architecture with both IFF and DVF (png, pdf)

2 MIMO Analysis

Let’s define the system as shown in figure 13.

general_control_names.png

Figure 13: General Control Architecture

Table 1: Signals definition for the generalized plant
  Symbol Meaning
Exogenous Inputs \(\bm{\mathcal{X}}_w\) Ground motion
  \(\bm{\mathcal{F}}_d\) External Forces applied to the Payload
  \(\bm{r}\) Reference signal for tracking
Exogenous Outputs \(\bm{\mathcal{X}}\) Absolute Motion of the Payload
  \(\bm{\tau}\) Actuator Rate
Sensed Outputs \(\bm{\tau}_m\) Force Sensors in each leg
  \(\delta \bm{\mathcal{L}}_m\) Measured displacement of each leg
  \(\bm{\mathcal{X}}\) Absolute Motion of the Payload
Control Signals \(\bm{\tau}\) Actuator Inputs

2.1 Initialization

We first initialize the Stewart platform. 

stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'none');

The rotation point of the ground is located at the origin of frame \(\{A\}\). 

ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');

2.2 Identification

2.2.1 HAC - Without LAC

controller = initializeController('type', 'open-loop');

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'input');      io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]

%% Run the linearization
G_ol = linearize(mdl, io);
G_ol.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_ol.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

2.2.2 HAC - DVF

controller = initializeController('type', 'dvf');
K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6);

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'input');      io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]

%% Run the linearization
G_dvf = linearize(mdl, io);
G_dvf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

2.2.3 Cartesian Frame

Gc_ol = minreal(G_ol)/stewart.kinematics.J';
Gc_ol.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

Gc_dvf = minreal(G_dvf)/stewart.kinematics.J';
Gc_dvf.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

2.3 Singular Value Decomposition

freqs = logspace(1, 4, 1000);

U_ol = zeros(6,6,length(freqs));
S_ol = zeros(6,length(freqs));
V_ol = zeros(6,6,length(freqs));

U_dvf = zeros(6,6,length(freqs));
S_dvf = zeros(6,length(freqs));
V_dvf = zeros(6,6,length(freqs));

for i = 1:length(freqs)
    [U,S,V] = svd(freqresp(Gc_ol, freqs(i), 'Hz'));
    U_ol(:,:,i) = U;
    S_ol(:,i) = diag(S);
    V_ol(:,:,i) = V;

    [U,S,V] = svd(freqresp(Gc_dvf, freqs(i), 'Hz'));
    U_dvf(:,:,i) = U;
    S_dvf(:,i) = diag(S);
    V_dvf(:,:,i) = V;
end

3 Diagonal Control based on the damped plant

From (Skogestad and Postlethwaite 2007), a simple approach to multivariable control is the following two-step procedure:

  1. Design a pre-compensator \(W_1\), which counteracts the interactions in the plant and results in a new shaped plant \(G_S(s) = G(s) W_1(s)\) which is more diagonal and easier to control than the original plant \(G(s)\).
  2. Design a diagonal controller \(K_S(s)\) for the shaped plant using methods similar to those for SISO systems.

The overall controller is then: \[ K(s) = W_1(s)K_s(s) \]

There are mainly three different cases:

  1. Dynamic decoupling: \(G_S(s)\) is diagonal at all frequencies. For that we can choose \(W_1(s) = G^{-1}(s)\) and this is an inverse-based controller.
  2. Steady-state decoupling: \(G_S(0)\) is diagonal. This can be obtained by selecting \(W_1(s) = G^{-1}(0)\).
  3. Approximate decoupling at frequency \(\w_0\): \(G_S(j\w_0)\) is as diagonal as possible. Decoupling the system at \(\w_0\) is a good choice because the effect on performance of reducing interaction is normally greatest at this frequency.

3.1 Initialization

We first initialize the Stewart platform. 

stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'none');

The rotation point of the ground is located at the origin of frame \(\{A\}\). 

ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');

3.2 Identification

controller = initializeController('type', 'dvf');
K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6);

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'input');      io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]

%% Run the linearization
G_dvf = linearize(mdl, io);
G_dvf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

3.3 Steady State Decoupling

3.3.1 Pre-Compensator Design

We choose \(W_1 = G^{-1}(0)\). 

W1 = inv(freqresp(G_dvf, 0));

The (static) decoupled plant is \(G_s(s) = G(s) W_1\). 

Gs = G_dvf*W1;

In the case of the Stewart platform, the pre-compensator for static decoupling is equal to \(\mathcal{K} \bm{J}\):

\begin{align*} W_1 &= \left( \frac{\bm{\mathcal{X}}}{\bm{\tau}}(s=0) \right)^{-1}\\ &= \left( \frac{\bm{\mathcal{X}}}{\bm{\tau}}(s=0) \bm{J}^T \right)^{-1}\\ &= \left( \bm{C} \bm{J}^T \right)^{-1}\\ &= \left( \bm{J}^{-1} \mathcal{K}^{-1} \right)^{-1}\\ &= \mathcal{K} \bm{J} \end{align*}

The static decoupled plant is schematic shown in Figure 14 and the bode plots of its diagonal elements are shown in Figure 15.

control_arch_static_decoupling.png

Figure 14: Static Decoupling of the Stewart platform

static_decoupling_diagonal_plant.png

Figure 15: Bode plot of the diagonal elements of \(G_s(s)\) (png, pdf)

3.3.2 Diagonal Control Design

We design a diagonal controller \(K_s(s)\) that consist of a pure integrator and a lead around the crossover. 

wc = 2*pi*300; % Wanted Bandwidth [rad/s]

h = 1.5;
H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));

Ks_dvf = diag(1./abs(diag(freqresp(1/s*Gs, wc)))) .* H_lead .* 1/s;

The overall controller is then \(K(s) = W_1 K_s(s)\) as shown in Figure 16. 

K_hac_dvf = W1 * Ks_dvf;

control_arch_static_decoupling_K.png

Figure 16: Controller including the static decoupling matrix

3.3.3 Results

We identify the transmissibility and compliance of the Stewart platform under open-loop and closed-loop control. 

controller = initializeController('type', 'open-loop');
[T_ol, T_norm_ol, freqs] = computeTransmissibility();
[C_ol, C_norm_ol, ~] = computeCompliance();

controller = initializeController('type', 'hac-dvf');
[T_hac_dvf, T_norm_hac_dvf, ~] = computeTransmissibility();
[C_hac_dvf, C_norm_hac_dvf, ~] = computeCompliance();

The results are shown in figure

static_decoupling_C_T_frobenius_norm.png

Figure 17: Frobenius norm of the Compliance and transmissibility matrices (png, pdf)

3.4 Decoupling at Crossover

  • [ ] Find a method for real approximation of a complex matrix

4 Time Domain Simulation

4.1 Initialization

We first initialize the Stewart platform. 

stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'none');

The rotation point of the ground is located at the origin of frame \(\{A\}\). 

ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');

load('./mat/motion_error_ol.mat', 'Eg')

4.2 HAC IFF

controller = initializeController('type', 'iff');
K_iff = -(1e4/s)*eye(6);

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'input');      io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]

%% Run the linearization
G_iff = linearize(mdl, io);
G_iff.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_iff.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

Gc_iff = minreal(G_iff)/stewart.kinematics.J';
Gc_iff.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

wc = 2*pi*100; % Wanted Bandwidth [rad/s]

h = 1.2;
H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));

Kd_iff = diag(1./abs(diag(freqresp(1/s*Gc_iff, wc)))) .* H_lead .* 1/s;
K_hac_iff = inv(stewart.kinematics.J')*Kd_iff;

controller = initializeController('type', 'hac-iff');

4.3 HAC-DVF

controller = initializeController('type', 'dvf');
K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6);

%% Name of the Simulink File
mdl = 'stewart_platform_model';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'],              1, 'input');      io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]

%% Run the linearization
G_dvf = linearize(mdl, io);
G_dvf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

Gc_dvf = minreal(G_dvf)/stewart.kinematics.J';
Gc_dvf.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

wc = 2*pi*100; % Wanted Bandwidth [rad/s]

h = 1.2;
H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));

Kd_dvf = diag(1./abs(diag(freqresp(1/s*Gc_dvf, wc)))) .* H_lead .* 1/s;

K_hac_dvf = inv(stewart.kinematics.J')*Kd_dvf;

controller = initializeController('type', 'hac-dvf');

4.4 Results

figure;
subplot(1, 2, 1);
hold on;
plot(Eg.Time, Eg.Data(:, 1), 'DisplayName', 'X');
plot(Eg.Time, Eg.Data(:, 2), 'DisplayName', 'Y');
plot(Eg.Time, Eg.Data(:, 3), 'DisplayName', 'Z');
hold off;
xlabel('Time [s]');
ylabel('Position error [m]');
legend();

subplot(1, 2, 2);
hold on;
plot(simout.Xa.Time, simout.Xa.Data(:, 1));
plot(simout.Xa.Time, simout.Xa.Data(:, 2));
plot(simout.Xa.Time, simout.Xa.Data(:, 3));
hold off;
xlabel('Time [s]');
ylabel('Orientation error [rad]');

5 Functions

5.1 initializeController: Initialize the Controller

Function description

function [controller] = initializeController(args)
% initializeController - Initialize the Controller
%
% Syntax: [] = initializeController(args)
%
% Inputs:
%    - args - Can have the following fields:

Optional Parameters

arguments
    args.type   char   {mustBeMember(args.type, {'open-loop', 'iff', 'dvf', 'hac-iff', 'hac-dvf', 'ref-track-L', 'ref-track-X', 'ref-track-hac-dvf'})} = 'open-loop'
end

Structure initialization

controller = struct();

Add Type

switch args.type
  case 'open-loop'
    controller.type = 0;
  case 'iff'
    controller.type = 1;
  case 'dvf'
    controller.type = 2;
  case 'hac-iff'
    controller.type = 3;
  case 'hac-dvf'
    controller.type = 4;
  case 'ref-track-L'
    controller.type = 5;
  case 'ref-track-X'
    controller.type = 6;
  case 'ref-track-hac-dvf'
    controller.type = 7;
end

Bibliography

Skogestad, Sigurd, and Ian Postlethwaite. 2007. Multivariable Feedback Control: Analysis and Design. John Wiley.

Author: Dehaeze Thomas

Created: 2021-01-08 ven. 15:52