King Fahd University of Petroleum and Minerals

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EE 653 - Robust Control - Semester 182 Homework 2

Problem 1. Let (S ) be a single-input-single output linear system where u is the control input and y is the system output. Let G(s) be the transfer function that relates the input u to the output y. Show that kG(s)kH∞ ≥

ky(t)kL2 · ku(t)kL2

(1)

Problem 2. Consider the uncertain linear system having the state space dynamics:   0 1 0 0 1+α  x x˙ =  0 −3 −3 −2 + β

(2)

where x ∈ IR3 is the state vector, α and β are positive uncertain constants. • Write the necessary and the sufficient conditions that must satisfy the constants α and β in order to ensure the stability of system (2). • Formulate a convex-optimization problem that permits to calculate the maximum and the minimum values of α and β that guarantee the system stability. Discuss the conservatism of the obtained conditions. Provide justifications and Matlab codes.

Problem 3. Consider the uncertain system governed by the state-space equations:      1 0 1 0 0   2    1 0 0  u, x˙ =  0 −2 x + ξ + 1  0 0 1 0 θ −3   y= 1 0 0 x 

−1

where θ is an uncertain parameter that is upper and lower bounded by control variable and ξ ∈ IR2 is a time-varying uncertainty input.

(3)

1 3 ≤ θ ≤ , u ∈ IR is the 2 2

• Let Gξ,y (s) be the transfer function that relates the input ξ to the output y. Find a tight upper 1 3 bound for the H∞ norm and H2 norm of Gξ,y when θ is a constant verifying ≤ θ ≤ . 2 2 Justify your response. • Using convex-optimization technique, construct a Laypunov function and show that the system is stable for ξ = 0 and u = 0. Justify your steps.

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• Determine a feedback control law u = Kx that minimizes the H∞ norm of the closed-loop transfer function Gξ,y (s) that relates the input ξ to the output y. • Determine a feedback control law u = Kx that minimizes the H2 norm of the closed-loop transfer function Gξ,y (s) that relates the input ξ to the output y.

Problem 4. Let (Σ) be an LTI uncertain system described by the state-space dynamics: (Σ) :

(4)

x˙ = A(θ) x + B u,

where x ∈ IRn is the state vector and u ∈ IRm is the control input. The state matrix A(θ) is dependent on the uncertain vector θ. Additionally, we assume that A(θ) :=

µ X i=1

αi Ai ,

µ X

αi = 1,

αi ≥ 0, ∀i.

(5)

i=1

• Show that the system is stable if there exist a positive definite matrix P ∈ IRn×n and an arbitrary matrix H ∈ IRn×n such that   0 Ai H + H 0 Ai P − H 0 + A0i H < 0, 1 ≤ i ≤ µ. (6) ? −H − H 0 • Based on the previous conditions (6), develop a set of conditions that guarantee the system stabilizability by the static feedback u = Kx where K is matrix gain to be determined.

King Fahd University of Petroleum and Minerals.

EE 652