Stability of LQ and LQG
Formulation of Nyquist’s stability theorem for
multivariable systems
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Stability for Linear Quadratic control with state feedback
Stability for Linear Quadratic control with feedback from
observed states.
Opt lecture 5 – p. 1/
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Formulation MIMO transferfunctions
For SISO system the closed loop poles are the zeros of the characteristic polynomial, which
may be formed the return difference
R(s) = 1 + T (s)
We will use the MIMO return difference to see the connection to MIMO Nyquist criterion
System
:
ẋ(t)
= Ax(t) + Bu(t)
y(t)
= Cx(t)
Regulator :
u(t) = −Lx(t)
Using Laplace transform we have
x(s)
=
(sIn − A)−1 Bu(s) = F(s)Bu(s)
u(s)
=
−Lx(s)
where we have introduced the shorthand notation F(s) = (sIn − A)−1
Opt lecture 5 – p. 2/
Return difference Matrix
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Opening loop at the input (I), the loop transfer matrix is T(s) = LF(s)B. The return
difference matrix will here be of dimension r which is the number of inputs
RI (s) = Ir + LF(s)B
Opening the loop at the output (II ) give us the loop gain T(s) = F(s)BL, and the return
difference matrix is of dimension n equal to the number of state variables
RII (s) = In + F(s)BL
Opt lecture 5 – p. 3/
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Poles in terms of return difference
We will now find a relation between the return difference matrix and the characteristic
polynomial for open and closed loop.
We start with the closed loop characteristic polynomial
det(sI − A + BL)
=
det[(sI − A)(I + (sI − A)−1 BL)]
=
det(sI − A) det(I + F(s)BL)
=
det(sI − A) det(RII (s))
=
det(sI − A) det(RI (s))
This give us the relation
det(R(s))
=
=
Pc (s)
det(sI − A + BL)
=
det(sI − A)
Po (s)
Characteristic polynomial f or closedloop
Characteristic polynomial f or openloop
We write the relation as
Pc (s) = Po (s) det(R(s))
Opt lecture 5 – p. 4/
MIMO Nyquist theorem
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The relation Pc (s) = Po (s) det(R(s)) makes it possible to formulate the Nyquist criterion
including MIMO systems: The condition for closed loop stability that the mapping of
det(RI (jω) = det(Ir + LF(jω)B)
encircles the origin one time counter clock wise for each open loop pole in the right half
plane when s goes round the Nyquist path which encircles all the unstable poles.
In figure 5 this is illustrated for an open loop stable system for which the map must not
encircle the origin
Opt lecture 5 – p. 5/
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Nyquist in terms of eigenvalues of R(jω)
The determinant of a matrix is the product of the eigenvalues of the matrix, which in this case
are functions l(jω)
r
Y
li (jω)
det(R(jω)) =
i=1
The closed loop stability for an open loop stable system can also be formulated such that
none of the eigenvaluefunctions li (jω) for the return difference matrix R(jω) must encircle
origo of the complex plane when s traverses the Nyquist path.
This may also be illustrated for a stable system
Opt lecture 5 – p. 6/
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Nyquist in terms of eigenvalues for T(jω)
Since R(s) = Ir + LF(s)B the eigenvalues of R(s) can also be expressed as
li (s) = 1 + ki (s)
where ki (s) ar ethe eigenvalues of the loop transfer matrix T(s) = LF(s)B
Opt lecture 5 – p. 7/
Stabilty for continous time LQ
It is possible use the MIMO stability tests from the previous section to derive stability
conditions for LQ-controlled systems.
ẋ(t)
= Ax(t) + Bu(t)
y(t)
= Cx(t)
:
I
=
:
u(t)
= −Lx(t)
with
L
T
= Q−1
2 B S, hvor
and
0
T
= CT Q1y C + AT S + SA − SBQ−1
2 B S
System
Performance
:
with
Controller
C
R∞
T
T
0 (y (t)Q1y y(t) + u (t)Q2 u(t))dt
R
= 0∞ (xT (t)Q1 x(t) + uT (t)Q2 u(t))dt
Q1 = CT Q1y C
We use the stationary LQ-controller. With the abreviation F(s) = (sIn − A)−1 we have
Open loop input output matrix
: W(s)
= CF(s)B
loop gain matrix
: T(s)
= LF(s)B
Return difference matrix
: R(s)
= Ir + T(s) = Ir + LF(s)B
Opt lecture 5 – p. 8/
Stability for continous LQ
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Making manipulations of the Riccati equation it is possible to show that all the eigenvalue
functions vli (s) of vR(s) are numerically larger than 1, which imply that
| det(R(s))| = |
r
Y
li (s)| > 1
i=1
If we use this result in Nyquist-considerations it simply implies that the plot of det(R(jω)) in
the complex plane for ω running from 0 til ∞ be strictly outside a cirle with centre in (0, 0)
and with radius 1.
Opt lecture 5 – p. 9/
Stability for continous LQ
C
With the definitions of phase margin and gain margin known from SISO systems it may be
seen from the Nyquist-curve that for a continous time LQ-controlled system holds:
Phasemargin
> 60◦
Gainmargin
=∞
Opt lecture 5 – p. 10/