L-5 Introduction to robust control.pdf

EMT 3104
Dr.-Ing. Jackson G. Njiri L-5: Introduction to robust control
Sem. 2
2019/2020
1
Review of Nyquist stability
 Nyquist stability criterion is a graphical way to determine stability of a feedback
system
 Originally developed as a method to determine stability without having to solve for
closed-loop poles
 Nyquist still useful for several reasons:
‒ Gain margin and phase margin—measures of stability and robustness—readily
determined from plot.
‒ Can be applied to systems with time delays (Routh cannot)
‒ Modifications to controller frequency response which improve gain and phase-
margin may be readily observed from Nyquist map
‒ Nyquist is primarily used to observe GM and PM-Indicators of robustness
The Nyquist test
 First, compute the loop transfer function L(s)=G(s)K(s)
 Plot the loop frequency response L(jɷ)=L(s)|s=jɷ as a polar plot. ie., plot the set of
points (R{L(jɷ)}, I{L(jɷ)}) for -∞<ɷ < ∞
 Resulting plot is a “Nyquist plot,” or “Nyquist map”
 Np is the number of open-loop unstable poles in L(s)
 Nc is the number of clockwise encirclements of -1 in the Nyquist map
 Nz=Np+Nc is the number of closed-loop unstable poles. System stable iff Nz=0
 Special cases when L(s) has poles on the jɷ-axis
Introduction to robust control

EMT 3104
Sem. 2
2019/2020
2
Example
 Given
or
 No encirclements of -1, Nc=0
 No open-loop unstable poles Np=0
 Nz=Nc+Np=0. Closed-loop system is stable

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Sem. 2
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Gain margin
 Uncertainty in plant may be modeled as
 Where G0(s) is the nominal plant, g is an
unknown gain
 Nyquist stability may be applied to the
system
− By plotting map for every admissible
gain g
− By realizing that g only magnifies/
shrinks the map and by revising the
stability rule.
´
 Rule: Np is the number of open-loop
unstable poles in L(s). Nc is the number
of clockwise encirclements of -1/g and Nz
is number of closed-loop unstable
poles. System stable iff
Definitions
 The gain margin GM+ is the minimum
gain >1that results in an unstable
closed-loop system
 The downside gain margin GM- is the
maximum positive gain <1that results
in an unstable closed-loop system
Example
 Consider a plant (with two open-loop
unstable poles) and controller
 The gain g is uncertain but bounded

EMT 3104
Sem. 2
2019/2020
4
Example (cont’)
 The Nyquist map for this system is
plotted (for g=1)
 The map crosses the real axis at -0.25
and at -1.8.
 The system is therefore “robustly
stable”
Phase margin
 Uncertainty may also be modeled as an
uncertain phase shift
 This is a simplified model of an
uncertain delay or
, which is frequency
dependent and harder to analyze
 Multiplying a frequency response by e-j
results in a phase shift/rotation of  for
each point in the Nyquist map
 System becomes unstable when
rotation causes map to go through the
point -1

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Sem. 2
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5
 Definition: The phase margin PM is
the minimum amount of phase shift
added to L(s) that results in an unstable
closed-loop system.
 Phase margin is the amount of rotation
required to cause instability
➠Determined by intersection point
between Nyquist map and unit circle.
 GM=1/(distance between origin and
place where Nyquist map crosses real
axis
Robustness of LQR
 Analyzing controllers in frequency-
domain robustness can be determined
 Loop transfer function is expressed as
 The cost function which describes how
the control inputs influence the cost
(states) is given by
 The two transfer functions are related
in a special way called Kalman
frequency-domain equality (KFDE)

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Sem. 2
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 Considering the case with only one
input, L(s) is a scalar and s=jɷ
 R is a scalar, R>0, so
 Since , then the following
inequality holds
 1+L(jɷ) is the vector from the point (-1,0)
to L(jɷ) in the s-plane
 The KFDE says the L(jɷ) is excluded
from the unit circle centered at (-1,0)
 Recall that the point (-1,0) is the critical
point for stability analysis using Nyquist
 The LQR guarantees that L(jɷ) is some
distance from the critical point, so it
will take quite large changes for the
system to change number of
encirclements
 The optimal LQR controller has very
large gain/phase margins:
‒ If open-loop system is stable, then
any yields a stable closed-
loop system
‒ If open-loop system is unstable, then
any yields a stable closed-
loop system
‒ LQR phase margin of at least ±60o

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Robustness of LQG; LTR
 LQG offers a great way to design
controllers for MIMO systems to
achieve some desired performance
specification
 However, optimal controllers can be
very sensitive to model errors
Sensitivity and robustness
 Consider the situation where we design
the controller using plant dynamics A0
but it turns out that the true dynamics
are at A0+ΔA
 Model uses , while
the compensator uses
 Then, closed-loop dynamics are (by
separation theorem)
 The actual system has dynamics given
by
 Then, the closed-loop dynamics are
given by
 The dynamics are not decoupled and
the eigenvalues are not known
Loop transfer recovery (LTR)
 The method modifies LQG controller to
recover robustness of LQR/LQE
 LTR generates suboptimal controllers
wrt LQG since any modification to
original design is suboptimal
 LTR provides a family of controllers
with a range of robustness properties
➠ Select controller with best
compromise between robustness and
performance.

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Sem. 2
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 LTR can be best understood by
considering the following system
 Make LQG “look like” LQR by cutting
feedback path to Kalman filter, making
Kalman filter “sufficiently fast” that its
dynamics may be ignored.
 This can be accomplished by adding
fictitious noise wf(t) to control input u(t)
during the design
 This noise reduces filter’s reliance on
u(t) and also makes filter faster
 LTR increases spectral density of
fictitious noise until acceptable
robustness is achieved
Alternative LTR
 An alternative form of LTR uses the
cost function
 The added term increases
robustness but results in sub-optimal
control with respect to original cost
function
 This version of LTR is used when
measurements, control inputs, and
plant is minimum-phase

EMT 3104
Sem. 2
2019/2020
9
Modelling unstructured uncertainty
 Additive (unknown dynamics in parallel
with the plant)
 Input multiplicative (unknown
dynamics in series with the plant)
 Output multiplicative (unknown
dynamics in series with the plant)
 Input feedback (uncertainty in
gain/phase/ pole locations of plant)
 Output feedback (uncertainty in gain/
phase/ pole locations of plant)

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Sem. 2
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 Analysis may be performed when model perturbations are bounded
 Where is the maximum singular value and is any of perturbation types
considered
 The bound is generally frequency-dependent, so uncertainty may vary over
frequency
 All the unstructured uncertainty models may be analyzed in a similar manner by
placing them in a common framework.
 The perturbation is normalized, so
by defining
 The plant P(s) is generally a
combination of nominal plant Go(s) and
uncertainties added together

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Sem. 2
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Modelling structured uncertainty
 Unstructured uncertainty is modeled by
connecting unknown but bounded
perturbations to the plant
 Sometimes, more information is
available ➠ New constraints on
uncertainty add “structure” to the set
of admissible perturbations
−Plant subject to multiple perturbations
−Plant has multiple uncertain
parameters
−Multiple unstructured but independent
uncertainties
 Structured uncertainty model similar to
before is block diagonal
 An individual may represent an
uncertain parameter (scalar) or an
unstructured uncertainty
 All blocks have
Example
 Consider the transfer function
 The two poles are uncertain, as given
by
 The poles can be modeled as nominal
values plus perturbation
 Where and
 The perturbation in feedback loop
around the nominal plant can be placed
as

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Sem. 2
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 The perturbations can be normalized and the system put into standard form such
that

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Formulation of the H∞ problem
 Consider the plant model with feedback as shown below
 Here, Δ(s) represents plant parameter uncertainty, P(s) represents the plant
nominal transfer function and K(s) represents feedback control
 One of the objectives is to make the system insensitive to external disturbances.
This is equivalent to make z as independent of w as possible
 If the plant perturbation Δ(s) is ignored, then the model simplifies to

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Sem. 2
2019/2020
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 With the feedback law u=K(s)y, u and y can be eliminated resulting to
 To minimize the error z due to the external inputs w, the function Fl(P, K) must be
minimized
Mixed performance and robustness objective
 The following set of characteristics are possible:
‒ Achievement of good disturbance rejection from external signals in the low-
frequency region. ➠ This can be achieved by making the sensitivity S = (I + PK)-1
small as ω → 0
‒ Make the closed loop transfer function small at high frequencies limit excitation by
noise. ➠This can be achieved by making T = I - S =I - (I + PK)-1 small as ω → ∞.
‒ Guard against instability from parameter variations. This is achieved by
minimizing K(I + PK)-1
 Then the H ∞ problem is formulated as the minimization of the function
 Where W1 and W3 are frequency-dependent matrices

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Sem. 2
2019/2020
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Glover-Doyle Algorithm in H∞ formulation
 This algorithm solves a family of stabilizing controllers such that
 The aim is to find the lowest value of for which the above equation has a solution
 One possibility is to start with the LQG solution and then to reduce it using a binary
search
 The plant equations in state-space form is
 In a packed matrix form the above equations are represented as
 The following assumptions must be satisfied to ensure a solution
‒ The pair (A, B2) must be stabilizable and the pair (C2, A) detectable
‒ With the dimensions of dim x = n, dim w = m1, dim u = m2, dim z =p1 and dim y = p2,
then the Rank D12= m2 and Rank D21=p2 to ensure that they controllers are proper
and the transfer function from w to y is non-zero at high frequencies (i.e. all-pass)

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Sem. 2
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‒ Rank for all frequencies
‒ Rank for all frequencies
‒ D11=0 and D22=0 simplifies the equations and implies that the transfer functions
from u to y and from w to z rolls oﬀ at high frequency.
 So, the problem simplifies to
Solution to H∞ problem
 This requires the solving of two Ricatti equation, one for the controller and the
other for the observer
 The control law is given by
 The state estimator equation
 Where,

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 The controller gain is Kc as for the LQG case, and the estimator gain is Z∞ke instead
of Ke as for the LQG case, with
 And
 The terms X∞ and Y∞ are solutions to the controller and estimator Ricatti equations
 Note: These calculations are not carried out by hand-MATLAB robust control
toolbox is used instead.

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Sem. 2
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Properties of H∞ controller
 The stabilizing feedback law u(s)=Ky2 minimizes the norm of the closed-loop transfer
function
‒ Optimal H2 control: min||Tyu||2
‒ Optimal H∞ control: min||Tyu||∞
‒ Standard H∞ control:
 All H∞ cost function Tyu is all-pass i.e., for all values of ɷ
 The H∞ optimal controller for an n-state augmented plant have at most n-1 states
 The H∞ sub-optimal controller for an n-state augmented plant have exactly n states
 In the weighted mixed sensitivity problem formulation, the H∞ controller always
cancels the stable poles of the plant with its transmission zeroes
 In the weighted mixed sensitivity problem formulation, the unstable poles of the
plant inside the specified bandwidth will be shifted to its mirror image once a H∞ or
H2 feedback loop is closed
Weighted mixed sensitivity problem
 This technique allows very precise frequency-domain loop shaping via suitable
weighting strategies
 If you augment the plant with frequency dependent weights W1 to W, then the
MATLAB script hinf or newer hinfsyn or mixsyn will find a controller that ”shapes”
the signals to the inverse of these weights, if it exists.

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 The MATLAB function augw.m forms the augmented plan
Example
 Consider the case of the double integrator
 This plant violates the rules for a solution (poles on imaginary axis)
 The equation set, in state space form, with the addition of a ”disturbance” term
representing uncertainty d, is

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Example (cont’)
 The equation set, in state space form,
with the addition of a ”disturbance”
term representing uncertainty d, is
 The regulated output (note the
inclusion of the control signal to bound
it) given by
 The measurement equation
 The ”noise” term n may include
measurement errors or unmodelled
high
 Note: The rank condition of D21 must be
met
 Corresponding equations
 In standard format, the block diagrams
of the double integrator is shown below

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Sem. 2
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Example (cont’)
 In H∞ format, the block diagrams of the
double integrator is shown below
 Collecting the equations in packed
matrix form
 The solution to this problem by
computer (pre-shifting the poles at the
origin and post-shifting the controller
poles back) gives
 With the closed loop poles at

L-5 Introduction to robust control.pdf

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