02-LQR and LQG Controllers

8/3/2019 02-LQR and LQG Controllers

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1 LQR, LQG and H∞ Controllers.

1.1 Linear Quadratic Regulator

Consider the system

u B Ax x +=&

And the performance criteria

,0,0,][)]([0

∫ ∞

>≥+=⋅ RQdt RuuQx xu J T T

K

Problem: Calculate function such that is minimized. p

u ℜ∞ a],0[: ][u J

Remarks:

1. LQR can be considered for final times2. LQR can be considered for time varying matrices

3. LQR can be extended in several ways to nonlinear systems (e.g. State Dependent

Riccati Equations)4. LQR assumes full knowledge of the state

The LQR controller has the following form

)()(1

t Px B Rt uT −−=

Where is given by the positive (symmetric) semi definite solution of nn

P×ℜ∈

P BPBRQP APA T T 10 −+−+=

This equation is called Ricatti equation. It is solvable iff the pair is controllable

and is detectable

),( B A

),( AQ

LQR controller design

1. is given by “design” and can not be modified at this stage),( B A

2. are the controller design parameters. LargeQ

penalizes transients of ),( RQ x , large

penalizes usage of control action R u

1.2 Linear Quadratic Gaussian Regulator

In LQR we assumed that the whole state is available for control at all times (see formulafor control action above). This is unrealistic as the very least there is always measurement

noise. One possible generalization is to look at



vCx y

wu B Ax x

+=

++=&

Where are stochastic processes called measurement and process noise, respectively.wv,

For simplicity one assumes these processes to be white noise (ie zero mean, uncorrelated,Gaussian distribution)

Now only is available for control. It turns out that for linear systems a separation

principle holds

)(t y

1. First, calculate estimate the full state using the available information)(ˆ t x )(t x

2. Secondly, apply the LQR controller, using the estimation in place of the true

(now unknown) state .

)(ˆ t x

)(t x

Observer design (Kalman Filter)

The estimation is calculated by integrating in real time the following ODE)(ˆ t x

)ˆ(ˆˆ xC y L Bu x A x −++=&

With the following matrices calculated offline (ie before hand)

)(),(

,0,0 1

1

T T

Y T T

T

vv E Rww E Q

PQCP RPC PA AP

RPC L

==

≥+−+=

=−

−

The Riccati equation above has its origin in the minimization of the cost functional

∫ ∞−−−=⋅

0

])ˆ)(ˆ[()](ˆ[ dt x x x x x J

T

1.3 H∞ Control

H∞ norm of a transfer function is defined byq p

C sG×∈)(

))((max)(max)( 1

2ω σ ω

ω ω jG jGsG ℜ∈ℜ∈∞

==

Where 1σ denotes the largest singular value of the complex matrix )( jG .

Consider the system depicted below.

w xC y

w D xC z

u Bw B Ax x

+=

+=

++=

2

121

21&

)(t w is a disturbance acting on the system, while is the control action. Further,

is to be understood as performance index.

)(t u )(t z



Problem H∞ Control is about finding the stabilizing control (not necessarilymemoryless) law

))(()( t yF t u =

That stabilizes the system above and minimizes the effect of the disturbance on the

performance index . This goal can be achieved by minimizing the H∞ norm of the

transfer function that is generated once a controller design has been chosen.

)(t w

z

zwT a:

H∞ Control Design

In practice one does not seek the optimal controller (ie the ones that produces the absolute

minimal value for ∞T but the design is made using an iterative procedure that seeks for

reducing the norm while still looking at other performance measures. Indeed, for any

sufficiently large given 0>γ one can calculate a controller that makes γ <∞

T using

the following formulae

i) Find that satisfies the ARE0≥ X

X B B B B X C C X A XAT T T T )2211/1(110

2 −+−+− γ

And

X B B B B AT T )2211/1( 2 −− γ is stable

ii) ii) Find that satisfies the ARE0≥Y

Y C C C C Y B BYA AY T T T T )2211/1(110 2 −+−+− γ

And

Y C C C C AT T )2211/1( 2 −− γ is stable

iii) (lowest singular value)2

)( γ ρ < XY

Then the dynamic H∞ controller has the form

)(ˆ)(

)(ˆ1)(

))()(ˆ2()(2)(1ˆˆ

2

t xF t u

t x X Bt w

t yt xC ZLt u Bt w B x A x

opt

T worst

opt worst

=

=

−+++=

γ

&

Where

X BF

YC L

XY I Z

T

2

2

)/1( 12

−=

−=

−= −γ

Remarks.

1. For very small values of γ , these equations will have no solution while for very

large γ , these equations reduce to the LQG case.



2. The best design is made by an iterative procedure reducing γ that keeps an eye on the

disturbance rejection properties at all interesting frequencies (and not jus the

maximum value).

1.4 General Robustness Considerations

1. LQR produces very robust controllers

2. LQG robustness not all guaranteed. It is thus it is important to look at the stability

radius and spectral values sets of the closed loop design. This is related to theinteraction of the transients of the true state, the controller action and the observer.

3. H∞ design provides robustness at the cost of a pessimistic control law: it assumes that

the worst possible perturbation is acting on the system at all times.

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02-LQR and LQG Controllers