linear matrix inequalities and robust control · 7th elgersburg school on mathematical systems...

Linear Matrix Inequalities and Robust Control

Carsten Scherer and Siep Weiland

7th Elgersburg School on Mathematical Systems Theory

Class 4 (version March 6, 2015)

Carsten Scherer and Siep Weiland (Elgersburg)Linear Matrix Inequalities and Robust ControlClass 4 (version March 6, 2015) 1 /

45

Outline

1 Robust Stability Against UncertaintiesQuadratic stabilityRate-bounded uncertaintiesRobust performance

2 Robust Controller SynthesisRobust State-Feedback SynthesisRobust EstimationSummary

3 Linear Parametrically-Varying Controller Synthesis


45

Outline





45

Time-Invariant Parametric Uncertainty

Consider the linear time-invariant (LTI) system

x(t) = A(δ)x(t)

where A(·) is a continuous function of a (constant) parameter vector

δ = ( δ1 . . . δp )>

which is known to be contained in an (commonly closed) uncertainty set

δ ⊂ Rp.

Robust stability analysis

Is this system asymptotically stable for all δ ∈ δ?


45

Example: Load variation in a mechanical system

Differential eq. (kernel) model:

Mx+Bx+Kx = 0

State-space model:(xv

)=

(0 I

−M−1K −M−1B

)(xv

)where x is position and v = x isvelocity.

Uncertain parameter: loadchanges in the mass matrix

M ∈M

m��BB��BB

b

k-

x

load variation in an MSDsystem


45

Example: Academic

Academic example with rational parameter-dependence

x =

−1 2δ1 2δ2 −2 1

3 −1 δ3−10δ1+1

x

where the parameters δ1, δ2, δ3 are bounded as

δ1 ∈ [−0.5, 1], δ2 ∈ [−2, 1], δ3 ∈ [−0.5, 2].

Hence δ is actually a polytope (box) with eight generators:

δ = [−0.5, 1]× [−2, 1]× [−0.5, 2] =

= co

δ1

δ2δ3

: δ1 ∈ {−0.5, 1}, δ2 ∈ {−2, 1}, δ3 ∈ {−0.5, 2}

.


45

Quadratic Stability

Definition

The uncertain system x = A(δ)x with δ ∈ δ is said to be quadraticallystable if there exists X � 0 with

A>(δ)X +XA(δ) ≺ 0 for all δ ∈ δ.

Why is the name? V (x) = x>Xx is a quadratic Lyapunov function.

Why this is relevant? Implies that A(δ) is Hurwitz for all δ ∈ δ.

How to check? Easy, if A(δ) is affine in δ and δ = co{δ1, . . . , δN} is apolytope with moderate number of generators: Verify whether

X � 0, A>(δk)X +XA(δk) ≺ 0, k = 1, . . . , N

is feasible.


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Example: Academic cont’d

If A(δ) is not affine in δ, a parameter transformation often helps!

Regarding the previous example, introduce δ4 =δ3 − 10

δ1 + 1+ 12.5. Test

quadratic stability of −1 2δ1 2δ2 −2 13 −1 δ4 − 12.5

, (δ1, δ2, δ4) ∈ δ := [−0.5, 1]×[−2, 1]×[−172 ,

172 ].

What’s the price?

LMI-Toolbox: System quadratically stable for

(δ1, δ2, δ4) ∈ rδ with largest possible factor r ≈ 0.49.

Quadratically stable for a deflated set 0.49 δ. Not for rδ with r > 0.49.This critical factor is called the quadratic stability margin.


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Example - implementation LMI toolbox

LMI-toolbox commands: quadstab, psys and pvec

Define system matrices

>> S0 = ltisys([-1 0 2 ; 0 -2 1 ; 3 -1 -12.5]);

>> S1 = ltisys([ 0 2 0 ; 0 0 0 ; 0 0 0 ], zeros(3));

>> S2 = ltisys([ 0 0 0 ; 1 0 0 ; 0 0 0 ], zeros(3));

>> S4 = ltisys([ 0 0 0 ; 0 0 0 ; 0 0 1 ], zeros(3));

Define parameter ranges

>> pv = pvec(’box’,[-0.5 1; -2 1; -8.5 8.5]);

Define affine parameter dependent system

>> pdsys = psys( pv, [S0,S1,S2,S4] );

Find quadratic stability margin

>> [tau,X] = quadstab(pdsys, [1 1 1e8]);


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Time-Varying (Dynamic) Uncertainties

Now, assume that the parameters δ(t) vary with time, and that they areknown to satisfy δ(t) ∈ δ for all t. Check stability of

x(t) = A(δ(t))x(t), δ : R→ δ.

Theorem

The uncertain system with time-varying uncertainties is exponentiallystable if there exists X � 0 with

A>(δ)X +XA(δ) ≺ 0 for all δ ∈ δ.

The proof will be given for a more general result in full detail.

Quadratic stability does in fact imply robust stability for arbitrary fasttime-varying uncertainty.

If bounds on the rate of variations of the parameters are known, thistest is conservative.


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Rate-Bounded Uncertainties

Let us hence assume that the trajectories δ(.) are continuouslydifferentiable and are only known to satisfy

δ(t) ∈ δ and δ(t) ∈ v for all time instances.

Here δ ⊂ Rp and v ⊂ Rp are given compact sets (e.g., polytopes).

Robust stability analysis

Verify whether the linear time-varying system

x(t) = A(δ(t))x(t)

is exponentially stable for all trajectories δ(.) that satisfy the abovedescribed bounds on value and variation.

Key idea: Search for a suitable (quadratic) Lyapunov function.


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Main Stability Result

Theorem

Suppose X(δ) is continuously differentiable w.r.t. δ and satisfies

X(δ) � 0,

p∑k=1

∂kX(δ)vk +A>(δ)X(δ) +X(δ)A(δ) ≺ 0

for all δ ∈ δ and v ∈ v. Then, there exist constants K > 0, a > 0 suchthat all state trajectories of the uncertain time-varying system satisfy

‖x(t)‖ ≤ Ke−a(t−t0)‖x(t0)‖ for all t ≥ t0.

Covers many tests in literature. Study the proof to derive variants!

In general, this condition is only sufficient!

It is necessary in case v = {0}: Time-invariant (static) uncertainty.

Note that, in the Theorem, the dependency structure is not restricted!


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Proof

Continuity/compactness ⇒ exist α, β, γ > 0 such that for all δ ∈ δ, v ∈ v:

αI 4 X(δ) 4 βI,

p∑k=1

∂kX(δ)vk +A>(δ)X(δ) +X(δ)A(δ) 4 −γI.

Suppose that δ(t) is an admissible parameter trajectory and let x(t) denotea compatible state-trajectory of the system. Here is the crucial point:

d

dtx>(t)X(δ(t))x(t)︸︷︷︸

ξ(t)

= x>(t)

[p∑

k=1

∂kX(δ(t))δk(t)

]x(t)+

+ x>(t)[A>(δ(t))X(δ(t)) +X(δ(t))A(δ(t))

]x(t).

Since δ(t) ∈ δ and δ(t) ∈ v we can hence conclude

α‖x(t)‖22 ≤ ξ(t) ≤ β‖x(t)‖22,d

dtξ(t) ≤ −γ‖x(t)‖22.


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Proof

The conclusion of the proof follows by recognizing that

‖x(t)‖22 ≤1

αξ(t), ξ(t) ≤ β‖x(t)‖22, ξ(t) ≤ −γ

βξ(t).

The latter inequality leads to

ξ(t) ≤ ξ(t0) e−γβ(t−t0) for all t ≥ t0.

With the former inequalities we infer

‖x(t)‖22 ≤β

αe− γ

β(t−t0) ‖x(t0)‖22 for all t ≥ t0.

Choose K =√β/α and a = γ/(2β) to complete proof.


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Extreme Cases

Parameters are time-invariant: v = {0}.We need to find a X(δ) satisfying

X(δ) � 0, A>(δ)X(δ) +X(δ)A(δ) ≺ 0 for all δ ∈ δ.

Parameters vary arbitrarily fast:

We need to find a parameter-independent X satisfying

X � 0, A>(δ)X +XA(δ) ≺ 0 for all δ ∈ δ.

This is identical to the quadratic stability test!

Can apply the subsequently suggested numerical techniques in both cases!


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Remarks

We have derived general results based on Lyapunov functions which stilldepend quadratically on the state (which is restrictive), but which allowfor non-linear (smooth) dependence on the uncertain parameters.

Pure algebraic test which does not involve state nor δ(t) trajectories.

Not easy to apply:

Have to find a function satisfying a partial differential LMI.

Have to make sure that inequality holds for all δ ∈ δ, v ∈ v.

Allows to easily derive special cases which are or can be implemented withLMI solvers. (Affine dependence is just around the bend)

We will only consider a couple of examples.


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Example: Affine Dependence - Affine Lyapunov Matrix

Suppose A(δ) depends affinely on the parameters:

A(δ) = A0 + δ1A1 + · · ·+ δpAp.

Parameter- and rate-constraints are intervals (box structure):

δ = {δ ∈ Rp : δk ∈ [δk, δk] }, v = {v ∈ Rp : vk ∈ [vk, vk] }

Identical to convex hulls of

δg = {δ ∈ Rp : δk ∈ {δk, δk} }, vg = {v ∈ Rp : vk ∈ {vk, vk} }

Search for an affine parameter dependent X(δ):

X(δ) = X0 + δ1X1 + · · ·+ δpXp and hence ∂kX(δ) = Xk.


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With δ0 = 1 observe thatp∑

k=1

∂kX(δ)vk +A>(δ)X(δ) +X(δ)A(δ) =

=

p∑k=1

Xkvk +

p∑ν=0

p∑µ=0

δνδµ(AνXµ +XµAν).

Is affine in X1, . . . , Xp and v1, . . . , vp but quadratic in δ1, . . . , δp.

Relaxation

Include an additional constraint: A>ν Xν +XνAν < 0.

Implies that it suffices to guarantee the required inequality at thegenerators. Why? Partially convex function over the box!

The extra condition renders the test conservative, but numericallyverifiable. Still sufficient for robust stability!


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Robust exponential stability guaranteed if

There exist X0, . . . , Xp with A>ν Xν +XνAν < 0, ν = 1, . . . , p, and

p∑k=0

Xkδk � 0

p∑k=1

Xkvk +

p∑ν=0

p∑µ=0

δνδµ(A>ν Xµ +XµAν) ≺ 0

for all δ ∈ δg and v ∈ vg and δ0 = 1.

This test is implemented in the LMI toolbox (within RCT).

For rate-bounded uncertainties, it is often much less conservativethan the quadratic stability test.

Useful to understand the proof and derive your own generalization(s).


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Robust Quadratic Performance

Given an uncertain system described as

x(t) = A(δ(t))x(t) +B(δ(t))w(t)

z(t) = C(δ(t))x(t) +D(δ(t))w(t)

with continuously differentiable parameter trajectories δ(.) that satisfy

δ(t) ∈ δ and δ(t) ∈ v (δ, v ⊂ Rp compact).

Robust quadratic performance property

Robust stability and existence of an ε > 0 such that, for all solutiontrajectories (w(t), z(t), δ(t)) with x(0) = 0, it holds that∫ ∞

0

(w(t)z(t)

)>(Qp SpS>p Rp

)︸︷︷︸

Pp

(w(t)z(t)

)dt ≤ −ε‖w‖22.

L2-gain, passivity, . . .Carsten Scherer and Siep Weiland (Elgersburg)Linear Matrix Inequalities and Robust Control

Class 4 (version March 6, 2015) 20 /45

Sufficient Condition for Robust Quadratic Performance

Theorem

Assume that Rp < 0. Suppose that there exists a continuouslydifferentiable symmetric-valued X(δ) such that X(δ) � 0 and

p∑k=1

∂kX(δ)vk+A>(δ)X(δ)+X(δ)A(δ) X(δ)B(δ)

B>(δ)X(δ) 0

+

+

(0 I

C(δ) D(δ)

)>(Qp SpS>p Rp

)(0 I

C(δ) D(δ)

)≺ 0

for all δ ∈ δ, v ∈ v. Then, the uncertain system satisfies the robustquadratic performance property.

Numerical search for X(δ): like for stability!

Can be easily extended to other LMI performance specifications!


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Sketch of the Proof

Exponential stability: Left-upper block isp∑

k=1

∂kX(δ)vk +A>(δ)X(δ) +X(δ)A(δ) + C>(δ)RpC(δ)︸︷︷︸<0

≺ 0.

Hence, we can apply our general result on robust exponential stability.

Performance: Adding εI (small ε > 0) to the right-lower block(compactness). Left- & right-multiply the inequality with col(x(t), w(t)):

d

dtx>(t)X(δ(t))x(t) +

(w(t)z(t)

)>Pp

(w(t)z(t)

)+ εw>(t)w(t) ≤ 0.

Integrate over [0, T ] and use x(0) = 0 to obtain

x>(T )X(δ(T ))x(T ) +

∫ T

0

(w(t)z(t)

)>Pp

(w(t)z(t)

)dt ≤ −ε

∫ T

0w>(t)w(t)dt.

Since X(δ(T )) � 0, x>(T )X(δ(T ))x(T ) can be dropped and the limitT →∞ is taken to obtain the required quadratic performance inequality.


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Outline





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Configuration for Robust Controller Synthesis

Design a controller guaranteeing:

robust stability

robustly a desired performancespecification on w → z

System(δ(t))

Controller

z w

uy

Consider the following approach:

System matrices depend affinely on parameter δ

Parameter varies in polytope: δ ∈ co{δ1, . . . , δN}Employ parameter-independent storage function

Goal: Robust stability and quadratic performance for controlled system.


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System Descriptions

Uncontrolled uncertain system:

x = A(δ(t))x+B1(δ(t))w +B(δ(t))uz = C1(δ(t))x+D1(δ(t))w + E(δ(t))uy = C(δ(t))x+ F (δ(t))w

Controller:

xc = Acxc +Bcyu = Ccxc +Dcy

Controlled uncertain system:

ξ = A(δ(t))ξ + B(δ(t))wz = C(δ(t))ξ +D(δ(t))w

We consider static state-feedback synthesis and estimator synthesis.


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Static State-Feedback Synthesis

Find X � 0 and Dc such that for δ ∈ co{δ1, . . . , δN}:((A(δ) +B(δ)Dc)

>X + X (A(δ) +B(δ)Dc) XB1(δ)B1(δ)

>X 0

)+

+

(∗∗

)>Pp

(0 I

C1(δ) + E(δ)Dc D1(δ)

)≺ 0

Variable change Y = X−1 and M := DcX−1 as earlier:

Find Y � 0 and M such that for δ ∈ co{δ1, . . . , δN}:((A(δ)Y +B(δ)M)> + (A(δ)Y +B(δ)M) B1(δ)

B1(δ)> 0

)+

+

(∗∗

)>Pp

(0 I

C1(δ)Y + E(δ)M D1(δ)

)≺ 0

This is an LMI problem! Does not work for output-feedback synthesis!Carsten Scherer and Siep Weiland (Elgersburg)Linear Matrix Inequalities and Robust Control


Robust Estimation

Uncertain system: xzy

=

A(δ(t)) B1(δ(t))C1(δ(t)) D1(δ(t))C(δ(t)) F (δ(t))

( xw

)Suppose A(δ) is quadratically stable on parameter polytope.

Robust estimator: (xcz

)=

(Ac Bc

Cc Dc

)(xcy

)

Design problem: For given γ > 0 check existence of an estimator suchthat Ac is Hurwitz and

sup0<‖w‖2<∞

‖z − z‖2‖w‖2

< γ


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Robust Estimation: Is Output-Feedback Synthesis

Uncertain system with estimator output as control input: xz − zy

=

A(δ(t)) B1(δ(t)) 0C1(δ(t)) D1(δ(t)) −IC(δ(t)) F (δ(t)) 0

xwu

Robust estimator viewed as controller:(

xcu

)=

(Ac Bc

Cc Dc

)(xcy

)

Design problem: For given γ > 0 check existence of an estimator suchthat Ac is Hurwitz and

sup0<‖w‖2<∞

‖z − z‖2‖w‖2

< γ


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Modified Transformation

Interconnection is xξz

=

A(δ(t)) 0 B1(δ(t))BcC(δ(t)) Ac BcF (δ(t))

C1(δ(t))−DcC(δ(t)) −Cc D1(δ(t))−DcF (δ(t))

xξ

w

.

Proceed as for output-feedback synthesis with factorization

YTX = Z where YT =

(I Y −1VI 0

), Z =

(Y −1 0X U

).

With the suggested factorization consider(YT (XA)Y YT (XB)

CcY D

)=

=

(Y −1 0X U

)(A 0BcC Ac

)(I I

V TY −1 0

) (Y −1 0X U

)(B1

BcF

)(C1 −DcC −Cc

)( I IV TY −1 0

)D1 −DcF

=


45

Modified Transformation

=

( Y −1A 0XA+ UBcC UAc

)(I I

V TY −1 0

) (Y −1B1

XB1 + UBcF

)(C1 −DcC − CcV

TY −1 C1 −DcC)

D1 −DcF

=

=

( Y −1A Y −1AXA+ UBcC + UAcV

TY −1 XA+ UBcC

) (Y −1B1

XB1 + UBcF

)(C1 −DcC − CcV

TY −1 C1 −DcC)

D1 −DcF

=

=

(ZA ZA

XA+ LC +K XA+ LC

) (ZB1

XB1 + LF

)(C1 −NC −M C1 −NC

)D1 −NF

for

Z := Y −1 and

(K LM N

)=

(UAcV

TY −1 UBc

CcVTY −1 Dc

).

The latter involves no system parameters!


45

Robust Estimator Synthesis

The synthesis inequalities with quadratic performance spec read as(Z ZZ X

)� 0

and

?

0 0 0 I 0 00 0 0 0 I 00 0Qp 0 0 SpI 0 0 0 0 00 I 0 0 0 00 0 STp 0 0Rp

I 0 00 I 00 0 I

ZA(δ) ZA(δ) ZB1(δ)XA(δ)+LC(δ)+K XA(δ)+LC(δ) XB1(δ)+LF (δ)C1(δ)−NC(δ)−M C1(δ)−NC(δ)D1(δ)−NF (δ)

≺0

Find X, Z and K, L, M , N such these hold for δ ∈ co{δ1, . . . , δN}.Choose invertible U , V with U>V = I −XZ−1.

Invert transformation on previous slide to find Ac, Bc, Cc, Dc.


45

Summary

Robustness Analysis and Synthesis

Techniques to search for Lyapunov functions if the state-space modelof the system depends affinely on the uncertain parameters.

Did not reveal role of linear fractional representations and multipliersif the system depends rationally on uncertain parameters.

Did not sketch extension to nonlinearities / non-linear uncertainties.

Robust Controller Synthesis

Robust state-feedback controller design is convex!

Robust estimator design is convex!

Output feedback controller design non-convex! Multipliers allow forheuristic algorithms. Many open problems!

If parameters are measurable on-line: LPV design ...


45

Outline





45

Configuration for LPV Synthesis

Open-loop system: x

zy

=

A(δ(t)) B1(δ(t)) B(δ(t))

C1(δ(t)) D1(δ(t)) E(δ(t))C(δ(t)) F (δ(t)) 0

x

wu

Controller: (

xcu

)=

(Ac(δ(t)) Bc(δ(t))

Cc(δ(t)) Dc(δ(t))

)(xcy

)Controlled System:(

ξ

z

)=

(A(δ(t)) B(δ(t))C(δ(t)) D(δ(t))

)(ξ

w

)

Parameter trajectories satisfy δ(t) ∈ δ = co{δ1, . . . , δN}.Carsten Scherer and Siep Weiland (Elgersburg)Linear Matrix Inequalities and Robust Control


Analysis

If there exists a X � 0 such that

[∗]>

0 X 0 0X 0 0 0

0 0 Qp Sp0 0 S>p Rp

I 0A(δ) B(δ)0 IC(δ) D(δ)

≺ 0 for all δ ∈ δ

one has achieved robust quadratic performance for the controlled system.

Reduces to LMIs in generators δ1, . . . , δN if(A(δ) B(δ)C(δ) D(δ)

)is affine in δ.

How can we guarantee that? What is the resulting design procedure?


45

Hypotheses on the System Description

Hypothesis on the open-loop system: A(δ) B1(δ) B

C1(δ) D1(δ) EC F 0

is affine in δ.

Observe independence of B, E and C, F from δ (can be guaranteed viasimple filters).

Hypothesis on the controller:(Ac(δ) Bc(δ)Cc(δ) Dc(δ)

)is affine in δ.

Then

(A(δ) B(δ)C(δ) D(δ)

)is affine in δ.


45

From Analysis to Synthesis

We achieve robust quadratic performance for the controlled system if thereexists X � 0 such that for all k = 1, . . . , N :

[∗]>

0 X 0 0X 0 0 0

0 0 Qp Sp0 0 S>p Rp

I 0A(δk) B(δk)

0 IC(δk) D(δk)

≺ 0.

How to proceed? Apply our general convexifying transformation

(X , Ac(δk), Bc(δ

k), Cc(δk), Dc(δ

k))→ (X,Y ,Kk, Lk,Mk, Nk).


45

LPV Synthesis Inequalities

Synthesis inequalities for k = 1, . . . , N

(Y II X

)� 0

?

0 0 0 I 0 00 0 0 0 I 00 0Qp 0 0 SpI 0 0 0 0 00 I 0 0 0 00 0 STp 0 0Rp

I 0 00 I 00 0 I

A(δk)Y +BMk A(δk)+BNkC B1(δk)+BNkF

Kk XA(δk)+LkC XB1(δk)+LkF

C1(δk)Y +EMk C1(δ

k)+ENkC D1(δk)+ENkF

≺0


45

LPV Controller Construction

Solve inequalities for (X,Y ,Kk, Lk,Mk, Nk), k = 1, . . . , N .

Construct X and the extreme-point controllers(Ac,k Bc,k

Cc,k Dc,k

), k = 1, . . . , N

as in the standard synthesis procedure.

Let δ ∈ δ, represented as δ =N∑k=1

λkδk with λk ≥ 0,

N∑k=1

λk = 1.

Then, the analysis inequalities are satisfied with(Ac Bc

Cc Dc

)=

N∑k=1

λk

(Ac,k Bc,k

Cc,k Dc,k

).


45

Comments

For controller simulation and implementation one has to proceed asfollows: At time t, find convex combination coefficients in

δ(t) =N∑k=1

λk(t)δk and use

N∑k=1

λk(t)

(Ac,k Bc,k

Cc,k Dc,k

)to define the dynamics of the LPV controller.

Determination of λk(t) requires the solution of an LP. In order toensure uniqueness (e.g. to assure continuity in time) one couldenforce, in addition, that, e.g.,

∑Nk=1 λ

2k(t) is minimized.

This (simplest) procedure for designing LPV controllers isimplemented in the Robust Control Toolbox in Matlab.


45

High-Performance Aircraft System

nz

q

u

® u: Control inputα: Measurable parameternz: Tracked output

Nonlinear system description with aerodynamic effects:

α = KM[(anα

2+bnα+cn(2−M/3))α+dnu

]+q

q = M2[(amα

2+bmα−cm(7−8M/3))α+dmu

]nz = M2

[(anα

2+bnα+cn(2−M/3))α+dnu

]


45

Main Idea

Rewrite as linear parameter-varying system

α = Kδ1[(anδ2

2+bnδ2+cn(2−δ1/3))α+dnu

]+q

q = δ12[(amδ2

2+bmδ2−cm(7−8δ1/3))α+dmu

]nz = δ1

2[(anδ2

2+bnδ2+cn(2−δ1/3))α+dnu

]with bounds 2 ≤ δ1(t) ≤ 4 and −20 ≤ δ2(t) ≤ 20.

Design good controller scheduled with δ1(t), δ2(t)

→ Is good controller for nonlinear system


45

Interconnection Structure

Model-Matching

Let controlled system approximately act like ideal model wideal


45

Synthesis with Convex Hull Relaxation

M(t) decreases in 5 seconds from 4 to 2.

Normalacceleration

ReferenceResponse

Application to Aircraft Model

42/50

Carsten Scherer

M(t) decreases in 5 seconds from 4 to 2.

Normal

acceleration

Reference

Response

0 1 2 3 4 5−20

−10

0

10

20

30

40

TimeN

orm

al a

ccel

erat

ion


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That’s all folks !!


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linear matrix inequalities and robust control · 7th elgersburg school on mathematical systems...

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