Design method and device of finite time state feedback controller
A state feedback, limited time technology, applied in the direction of adaptive control, general control system, control/regulation system, etc., can solve problems such as degree difficulty, fuzzy set can not obtain good effect, etc., to achieve the effect of simple method
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example 1
[0165] Example 1. In this example, consider the interval type 2 T-S fuzzy time-delay model (2), and the system matrix is (42).
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[0167] The membership function of interval type II T-S fuzzy time-delay model is chosen as follows:
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[0169] To find the controller gain matrix that makes the closed-loop system stable in finite time, let τ=0.4,c 1 =0.4,γ=9,c 2 =0.8, T=50, ρ 1 =0.1626,ρ 2 =0.4661, let α=10 -5 ,σ=0.4, use the cone compensation linearization algorithm to get the controller gain matrix:
[0170] K 1 =[-8.7864-1.7715],K 2 =[-8.7918-1.7743] (43)
[0171] When the initial condition is t∈[-τ,0], the state response x(t) of the closed-loop system is as follows image 3 shown. From image 3 It can be seen that when t→∞, the value of the state variable converges and maintains at the equilibrium point. Figure 4 Represents the norm of the state vector of the closed-loop system. From Figure 4 can be seen in Therefore, it can be conclu...
example 2
[0172] Example 2. Consider the following inverted pendulum model:
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[0174] in, x 2 (t) ∈ [-5,5], g=9.8, L=0.5. m p ∈[2,3],m c ∈[8,16] is an uncertain parameter, a=1 / (m p +m c ).
[0175] The above inverted pendulum model is subject to parameter uncertainties, so the type-one fuzzy finite-time stability theory cannot be applied in this example. The above inverted pendulum model is transformed into a four-rule interval type II T-S fuzzy time-delay model in the form of (2). The system matrix is as follows:
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[0177] The upper membership degree and lower membership degree of each rule of the interval type 2 T-S fuzzy time-delay model are defined in Table 1.
[0178] Table 1 Upper membership degree and lower membership degree
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[0180] Let ρ 1 = ρ 2 = ρ 3 = ρ 4 = 0.1, τ = 0.2, c 1 =0.4, γ=9, c 2 =0.53, T=100, using the cone compensation linearization algorithm to obtain the controller gain matrix as follows:
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