Recapitulation What do we want? v = disturbance y u System e = - - PowerPoint PPT Presentation

Recapitulation

What do we want?

System

• u

y − r + e = control error v = disturbance

◮ The control problem:

◮ Minimize e, ◮ including the effect of v, and ◮ keep u small (umin ≤ u ≤ umax).

◮ Ultimate course goal: Find the optimal solution! ◮ The controller is implemented in a computer ⇒ must be

described as a discrete-time filter.

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Recapitulation, cont’d

What have we achieved? — summary of what we have looked at so far

◮ Discrete-time systems:

◮ Difference equations, ◮ shift operator q, ◮ stability region = inside of the unit circle.

◮ Sampling of systems = zero-order hold sampling (ZOH):

◮ Exact for t = kh, ◮ sampling period = h, ◮ Nyquist frequency ωn = ωs/2 = π/h, ◮ frequency response of G(z) for z = eiωh.

◮ MIMO systems: Straightforward to use state space forms. ◮ Disturbance models:

◮ Spectral density: Φ(ω) = F[r(τ)], ◮ white noise v ⇔ Φv(ω) = Rv = const., ◮ linear filtering: y = Gu ⇒ Φy = |G|2Φu, ◮ spectral factorization: Φ = |G|2R, ◮ Lyapunov equation ⇒ Πx = ExxT .

◮ Kalman filters: Optimal observer, Riccati equations (CARE,

DARE).

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Control design

Starting point in the continuous-time case

◮ Use the “standard” state space representation

     ˙ x = Ax + Bu + Nv1, z = Mx, y = Cx + v2, η = v1 v2

• , Φη(ω) =

R1 R12 RT

12

R2

• ◮ Minimize the criterion

V = ||z||2

Q1 + ||u||2 Q2 = E

• zT Q1z + uT Q2u
• ◮ The weighting matrices,

Q1 = QT

1 ≥ 0

and Q2 = QT

2 > 0,

are design parameters.

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Control strategy

State feedback with observer

The optimal controller is conveniently represented as state feedback from estimated states.

◮ Control law: u(t) = −Lˆ

x(t) + ˜ r(t)

◮ Observer: ˙

ˆ x(t) = Aˆ x(t) + Bu(t) + K(y(t) − Cˆ x(t))

◮ The control law can also be written as

     U(s) = Fr(s) ˜ R(s) − Fy(s)Y (s), Fy(s) = L(sI − A + BL + KC)−1K, Fr(s) = I − L(sI − A + BL + KC)−1B

◮ The poles of the closed loop system are the roots of

0 = det(sI − A + BL) · det(sI − A + KC), i.e. poles from state feedback + the observer poles.

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LQG: The optimal controller

Theorem 9.1

◮ The optimal control law is u(t) = −Lˆ

x(t),

◮ ˆ

x(t) is obtained from the corresponding Kalman filter.

◮ The optimal state feedback gain is

L = Q−1

2 BT S, ◮ the matrix S = ST ≥ 0 is the solution to the continuous-time

algebraic Riccati equation (CARE) 0 = AT S + SA + MT Q1M − SBQ−1

2 BT S ◮ N.B. There are two different CAREs involved, the one above

and the one for the Kalman filter!

◮ Some technical conditions: (A, B) stabilizable, (A, C) and

(A, MT Q1M) detectable...

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LQ/LQG: Properties

◮ A − BL is always stable ◮ The control law u = −Lx(t) (pure state feedback) is optimal,

also for the deterministic case (v1 = 0 and v2 = 0) ⇔ LQ = linear quadratic control.

◮ If v1 and v2 have Gaussian distributions the controller is the

• ptimal controller (Cor. 9.1) ⇔ LQG = linear quadratic

Gaussian control.

◮ Theorem 9.1 = the separation theorem:

The optimal observer = the Kalman filter, combined with the

• ptimal state feedback (LQ) give the optimal controller! (This

is far from obvious...)

◮ The LQ/LQG controller looks exactly the same for SISO and

MIMO systems.

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The servo problem

How can the reference signal r(t) be included?

◮ General solution: Characterize r(t) by its spectrum and model

it in the same way as a disturbance, i.e. incorporate it in the model.

◮ Special case, Theorem 9.2: If r(t) is piecewise constant,

then the criterion V = ||z − r||2

Q1 + ||u − u∗(r)||2 Q2

is minimized with the control law u(t) = −Lˆ x(t) + Lrr(t), where L and ˆ x(t) are given as in Theorem 9.1, and Lr is chosen so that I = M(sI − A + BL)−1BLr

• s=0 = Gc(0)Lr.

(Then u∗(r) = Lrr.)

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Example: LQ control of a DC-motor

The effect of Q1 and Q2

◮ The DC-motor: Y (s) = 1 s(s+1)U(s)

Design parameters: Q1 = 1, Q2 = 1 / 0.1 / 0.01 LQ control ⇒ pure state feedback: u(t) = −Lx(t) + mr(t)

◮ Simulations: Step responses for the closed loop systems.

◮ The outputs, y:

Q2 = 1 Q2 = 0.1 Q2 = 0.01

◮ The inputs, u:

Q2 = 1 Q2 = 0.1 Q2 = 0.01

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