Question

5. (20 pts) Consider the first order system \(\dot{x} = x + u\) with initial condition \(x(0) = 1\). An optimal state feedback controller is to be designed such that \(u = -kx\), to minimize the performance index \(PI = \int_0^\infty x^2 + u^2 dt\) (a) (5 pts) Substitute \(u = -kx\) and solve the differential equation for \(x(t)\). For which values of \(k\) is the system stable? (b) (15 pts) Assume \(k\) is chosen such that the system is stable. Substitute your solution \(x(t)\) and \(u = -kx(t)\) in the integrand and determine the value of the performance index in terms of \(k\). For which value(s) of \(k\) is this performance index minimized? Write down the reduced Riccati equation for this problem and demonstrate that it gives the same solution as yours.

          5. (20 pts) Consider the first order system
\(\dot{x} = x + u\)
with initial condition \(x(0) = 1\). An optimal state feedback controller is to be designed such that
\(u = -kx\), to minimize the performance index
\(PI = \int_0^\infty x^2 + u^2 dt\)
(a) (5 pts) Substitute \(u = -kx\) and solve the differential equation for \(x(t)\). For which values of \(k\) is
the system stable?
(b) (15 pts) Assume \(k\) is chosen such that the system is stable. Substitute your solution \(x(t)\) and
\(u = -kx(t)\) in the integrand and determine the value of the performance index in terms of \(k\).
For which value(s) of \(k\) is this performance index minimized? Write down the reduced Riccati
equation for this problem and demonstrate that it gives the same solution as yours.

Show more…

5. (20 pts) Consider the first order system
ẋ = x + u
with initial condition x(0) = 1. An optimal state feedback controller is to be designed such that
u = -kx, to minimize the performance index
PI = ∫0^∞ x^2 + u^2 dt
(a) (5 pts) Substitute u = -kx and solve the differential equation for x(t). For which values of k is
the system stable?
(b) (15 pts) Assume k is chosen such that the system is stable. Substitute your solution x(t) and
u = -kx(t) in the integrand and determine the value of the performance index in terms of k.
For which value(s) of k is this performance index minimized? Write down the reduced Riccati
equation for this problem and demonstrate that it gives the same solution as yours.

Added by Ryan Z.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Consider the first-order system with initial condition z0=1. An optimal state feedback controller is to be designed such that u=-kx to minimize the performance index z^2+12at. a. Substitute u=-kr and solve the differential equation for z. For which values of k is the system stable? b. Assume k is chosen such that the system is stable. Substitute your solution zt and u=rt in the integrand and determine the value of the performance index in terms of k. For which values of k is this performance index minimized? Write down the reduced Riccati equation for this problem and demonstrate that it gives the same solution as yours.

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00:01 Hi, in this question, given that x dot equals 6x plus u and v equals integral over 0 to 1 x square plus 2u square into dt.

00:14 In part a, we have to determine the state function of boundary gain h.

00:20 So, which can be written as h equals lambda x cap plus x square plus 2u square and where x cap equals 6x plus u.

00:33 On substituting in this, then we get h equals 6 lambda x plus u lambda plus x square plus 2u square.

00:45 On differentiating with respect to u, then we get 0 plus lambda plus 0 plus 4u.

00:53 Hence, conclude that dou h by dou u equals lambda plus 4u.

01:05 Hence, we can write it as u dash equals minus lambda by 4.

01:16 Next, move on to part b.

01:18 Here, we need to determine the optimal input u naught.

01:22 We know that u dash equals minus lambda by 4.

01:26 So, h of x comma lambda comma t equals 6 lambda x plus u into lambda plus x square plus 2u square.

01:36 On substituting in this, then we get 6 lambda u plus u into lambda s lambda u s minus lambda by 4 plus x square plus 2 into minus lambda by 4 the whole square which is equal to 6 lambda x minus lambda square divided by 8 plus x square.

02:07 Let this as equation number 2.

02:11 In part c, we have to determine the equation governing the control system in terms of v and lambda.

02:19 So, x equals dou h into x comma h comma t divided by dou x.

02:27 So, here x can be written as 6x minus 2 lambda divided by 8 plus 0.

02:34 X dash equals 6x minus lambda by 4 which implies here capital x dash equals minus dou h dash x comma h lambda comma t divided by dou x.

02:52 From equation 2, we can write it as x dash equals minus 6 lambda minus 0 plus 2u which is equal to minus 6 lambda minus 2u...

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