LQR-LMI control applied to convex-bounded domains

Figures & data

Figure 1. Uncertainty regions for polytopic and norm-bounded models, respectively.

Figure 2. Schematic drawing of 3-DOF helicopter (Quanser, 2002).

Table 1. Helicopter parameters

Power constant of the propeller (found experimentally)	$k_f$	0.1188
Mass of the helicopter body (kg)	$m_h$	1.158
Mass of counterweight (kg)	$m_w$	1.87
Mass of the whole front of the propeller (kg)	$m_f$	$m_h/2$
Mass of the whole back of the propeller (kg)	$m_b$	$m_h/2$
Distance between each axis of pitch and motor (m)	$l_h$	0.1778
Distance between the lift axis and the body of the helicopter (m)	$l_a$	0.6604
Distance between the axis of elevation and the counterweight (m)	$l_w$	0.4699
Gravitational constant (m/${\text{s}}^2$)	g	9.81

Figure 3. Feasibility regions to polytopic uncertain ($\circ$: Theorem 3.1; $\times$: Theorem 4.1; grey color: Theorem 5.1).

Figure 4. Eigenvalues behavior of the uncertain closed-loop system (black color: Theorem 3.1; red color: Theorem 4.1 and blue color: Theorem 5.1).

Figure 5. Practical implementation of the designed K, (a) Theorem 3.1 and (b) Theorem 4.1.

Table 2. Numerical complexity to polytopic uncertains

Method	U	V	L
Theorem 3.1	1	53	53
	2	53	116
Theorem 4.1	1	153	86
	2	225	172
Theorem 5.1	1	153	102
	2	225	204

Figure 6. Mass-spring-damper system.

Figure 7. Number of feasibility regions for uncertainty models.

Table 3. Numerical complexity for uncertainty models

Method	U	V	L
Theorem 3.1	2	6	25
Theorem 3.2	2	8	14
Theorem 4.2	2	15	28

Figure 8. Responses to control and states signals in function of the decay rate (dark gray surface: Theorem 3.2; white surface: Theorem 4.2).

Quanser . (2002). 3-DOF helicopter. Reference manual . Retrieved from http://www.quanser.com/english/html/products/fs_product_echallenge.asp?

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