MAE 280B
Linear Control Design – Winter 2019
Final
Instructions:
• Due on 03/18/2019 by 9:00 AM via TritonEd; • Use Matlab or Mathematica;
• You get marks for clarity;
• You loose marks for obscurantism;
• Good luck!
In class you learned how to design a robust state-feedback controller by solving LMIs. In this exam we will modify those LMIs so as to obtain a gain-scheduled controller. The setup is as follows. Consider the time-varying linear system
x ̇(t) = A(ξ(t))x(t) + Buu(t) + Bww(t) z(t) = Czx(t) + Dzuu(t),
where A(ξ) = Ni=1 ξiAi and ξ(t) is a vector of measurable parameters in N
Ξ:= ξ: ξ∈RN ξi =1, ξi ≥0foralli=1,…,N i=1
(1) (2)
.
The dimensions are as follows: the state x(t) ∈ Rn, the control input u ∈ Rm, the input
noise w(t) ∈ Rp, and the controlled output z ∈ Rr. 1. Gain Scheduling Stabilizability
(a) (4 points) In this part we are interested only in stability. That is, we have w(t) ≡ 0 and x(0) = x0 ̸= 0. Show that if the LMIs
X≻0, AiX+XATi +BuLi+LTiBuT ≺0, i=1,…,N, (3) have a feasible solution in terms of the variables X, Li, i = 1,…,N, then the
gain-scheduled state-feedback control
N
K(ξ) = ξiKi, Ki := LiX−1 (4)
i=1
is such that all matrices [A(ξ)+BuK(ξ)] are Hurwitz (i.e. eigenvalues have negative real part) for all ξ ∈ Ξ.
(b) (2 points (bonus)) Show that the above control guarantees that the zero equilib- rium point of the time-varying system
x ̇(t) = [A(ξ(t)) + BuK(ξ(t))]x(t) is asymptotically stable for all ξ(t) ∈ Ξ.
Hint: Use Lyapunov stability.
(c) (2 points (bonus)) Show that the LMIs (3) are equivalent to
X≻0, (BuT)⊥T AiX+XATi (BuT)⊥ ≺0,
i=1,…,N, (5)
on the single variable X. Use this equivalence to justify the following statement: There exists a gain-scheduled state-feedback control of the form (4) if and only if there exists X and M such that
X≻0, AiX+XATi +BuM+MTBuT ≺0, i=1,…,N, (6) that is, if and only if (Ai,Bu) are quadratically stabilizable.
Hint: Use Finsler’s Lemma.
Total for Question 1: 4 Total for Question 1: 4 (bonus)
2. Gain Scheduling Performance
(a) (4 points) In this part assume x(0) = 0 and w(t) is a Gaussian white-noise with
zero mean and covariance matrix W ≻ 0. Show that if the LMIs
trace(Zi) ≤ μ (7)
Zi CzX + DzuLi
XCT +LTDT X ≽0, (8)
z i zu
AiX+XATi +BuLi+LTiBuT+BwWBwT≼0 (9)
have a feasible solution in terms of the variables X ≻ 0, Li, Zi, i = 1,…,N, for all i = 1, . . . , N , then the gain-scheduled state-feedback control (4) is such that all matrices [A(ξ) + BuK(ξ)] are Hurwitz and
lim E{z(t)T z(t)} ≤ μ (10) t→∞
for all ξ ∈ Ξ.
(b) (2 points (bonus)) Does the above bound hold when ξ(t) is allowed to be time- varying? Explain.
Total for Question 2: 4 Total for Question 2: 2 (bonus)
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3. Gain Scheduling State-Feedback Controller Design
The data for this problem is based on the paper “Self-scheduled Control of Linear Parameter-varying Systems: a Design Example,” by Apkarian, Gahinet and Beckers, Automatica, pp 1251–1261, 1995.
The linearized pitch-axis dynamics of a missile is described by the state-space model α ̇ − β 1 α 0
q ̇ = −γ 0 q + 1 (u+w) (11)
where α is the angle of attack, q the pitch rate, u the fin deflection, and w actuator noise. The parameters β and γ are functions of the flight conditions (altitude, speed, etc) which are measured during flight. This means that β and γ are available for feedback during flight. When the missile speed is between Mach 0.5 and 4, the parameters β and γ assume values in the range β ∈ [0.35, 4.35] and γ ∈ [−365, 380]. Variations of β and γ are assumed to be independent. The actuator noise is Gaussian white-noise with zero-mean and variance 0.1.
(a) (2 points) Construct a model of the form (1)-(2) that can represent all possible variations of β and γ, and where z(t) = (x(t), u(t)).
(b) (4 points) Solve one LQR optimal control problem (that is find a controller that minimizes limt→∞ E{z(t)T z(t)}) for each vertex of the polytopic model you con- structed in part (a). You will obtain N different control gains Ki, i = 1,…N. Now use each control gain Ki, i = 1, . . . N, to close the loop at each vertex of the poly- topic model. Compute limt→∞ E{z(t)T z(t)} at each of the N2 possible combinations of vertex and control. Make a table that shows the value of limt→∞ E{z(t)T z(t)} you obtained for each combination.
(c) (2 points) Does any of the controllers you obtained in part (b) stabilize all vertices? Can you prove whether any of the gains you obtained is a robust control using the concept of quadratic stability?
(d) (4 points) Solve the problem of minimizing μ subject to the LMIs (7)-(9) to obtain a gain-scheduled controller of the form (4). Compute limt→∞ E{z(t)T z(t)} at each of the N vertices. Make a table that shows the value of limt→∞ E{z(t)T z(t)} you obtained at each vertex.
(e) (4 points) Modify the LMIs (7)-(9) to obtain a robust control gain. Compute limt→∞ E{z(t)T z(t)} at each of the N vertices. Make a table that shows the value of limt→∞ E{z(t)T z(t)} you obtained at each vertex.
(f) (2 points) Compare the results of the above problems.
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Total for Question 3: 18 Total for Question 3: 0 (bonus)
4. Gain Scheduling Output-Feedback Control
As discussed in class (see notes for system data and dimensions), the optimal dynamic
output-feedback controller
that minimizes the cost function
J := lim E z(t)T z(t) ≤ μ
t→∞
is obtained by minimizing μ subject to the LMIs
AX +XAT +BuL+LTBuT A+BuRCy +QT
YBw +FDyw ≺0,
in the
matrices are given by
x ̇ c = A c x c + B c y u = Cc xc + Dc y,
trace(Z) < μ Bw +BuRDyw
⋆ ATY +YA+FCy +CyTFT
⋆ ⋆ −W−1
Z CzX+DzuL Cz+DzuRCy ⋆X I≻0,
⋆⋆Y
Dzw + DzuRDyw = 0
variables
Ac Bc V−1 −V−1YBuQ−YAX F U−1 0
L, F ,
Q, R and
symmetric variables X , Y , and Z . The controller
CD=0I LR−CXU−1I, ccy
whereU,V arematricessuchthatYX+VU=I.
(a) (2points) Under the assumptions that D = 0, D DT ≻ 0 and DT D ≻ 0
provethatDc =R=0.
(b) (4 points) Back to gain scheduling, we now seek to determine a gain scheduling full
order output-feedback controller for the system
x ̇(t) = A(ξ(t))x(t) + Bwu(t) + Bww(t) (12) y(t) = Cy(ξ(t))x(t) + Dyww(t), (13) z(t) = Czx(t) + Dzuu(t), (14)
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zw yw yw zu zu
where A(ξ) = Ni=1 ξiAi, Cy(ξ) = Ni=1 ξiCyi and ξ(t) ∈ Ξ. Show that if the LMIs trace(Zi) < μ
AiX+XATi +BuLi+LTiBuT
(15) ATY+YA+FC +CTFT YB +FD ≺0,
Ai+QTi Bw i i yi yi w yw
⋆
⋆ ⋆ −W−1
have a feasible solution in the variables Li , F , Qi and symmetric variables X , Y , and Zi for all i = 1,...,N, then the gain-scheduled full-order output-feedback controller
Ac (ξ ) Bc N Aci Cc(ξ) 0 = ξi Cci
i=1
−V−1YBuQi −YAiX Cci 0 := 0 I Li
lim E{z(t)T z(t)} ≤ μ t→∞
for all ξ(t) ∈ Ξ.
5. Gain Scheduling Output-Feedback Controller Design
Add to the missile model considered before the following noisy measurement
y1(t) ρβα(t) v1(t)
y2(t) = q(t) + v2(t) (20)
where y1 is the vertical acceleration and v(t) = (v1(t),v2(t)) is the sensor noise, and ρ = 100 is air speed. The sensor noise v(t) is a Gaussian white-noise vector with uncorrelated zero-mean channels and variance 0.1 at each channel.
(a) (2 points) Construct a model of the form (12)-(14) that can represent all possible variations of β and γ, and where z(t) = (x(t), u(t)).
where
Bc
0 , (18)
F U−1 0
0 −CyiXU−1 I , (19)
Aci Bc V−1
and Y X + V U = I, is such that the closed loop is stable and
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(16)
Zi CzX+DzuLi Cz
⋆ X I ≻ 0,
⋆⋆Y
(17)
Total for Question 4: 6 Total for Question 4: 0 (bonus)
(b) (6 points) Solve one LQG optimal control problem for each vertex of the polytopic model you constructed in part (a). You will obtain N different dynamic controllers. Now use each controller to close the loop at each vertex of the polytopic model. Compute limt→∞ E{z(t)T z(t)} at each of the N2 possible combinations of vertices and controllers. Make a table that shows the value of limt→∞ E{z(t)T z(t)} you obtained for each combination.
(c) (2 points) Does any of the controllers you obtained in part (b) stabilize all vertices? Can you prove whether any of the controllers you obtained is a robust control using the concept of quadratic stability?
(d) (6 points) Solve the problem of minimizing μ subject to the LMIs (15)-(17) to ob- tain a gain-scheduled controller of the form (18)-(19). Compute limt→∞ E{z(t)T z(t)} at each of the N vertices. Make a table that shows the value of limt→∞ E{z(t)T z(t)} you obtained at each vertex.
(e) (2 points) Compare the results of the above problems. Use the provided matlab files to simulate scheduling the LQG and the gain-scheduled controllers you designed.
Total for Question 5: 18 Total for Question 5: 0 (bonus)
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