ece7850: hybrid systems:theory and applications lecture ...zhang/hybridsystemsclass/...ece7850:...

ECE7850: Hybrid Systems:Theory and Applications

Lecture Note 7: Switching Stabilization viaControl-Lyapunov Function

Wei Zhang

Assistant ProfessorDepartment of Electrical and Computer Engineering

Ohio State University, Columbu, Ohio, USA

Spring 2017

Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 1 / 28

Outline

• Classical Control-Lyapunov Function Approach

• Switching Stabilization Problem

• Switching Stabilization via Control Lyapunov Function

• Special Case: Quadratic Switching Stabilization

• Special Case: Piecewise Quadratic Switching Stabilization

Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 2 / 28

Stabilization Problem I

• Consider general nonlinear control system:

x = f(x, u)

- f : X × U → X , where X ⊂ Rn is state space and U ⊂ Rm is control space

- assume f is locally Lipschitz in (x, u)

- for simplicity and without loss of generality, we assume zero control input willresult in equilibrium at origin, i.e., f(0, 0) = 0

Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 3 / 28

Stabilization Problem II• Roughly speaking, there are two classes of stabilization questions:

- Open-loop: whether there is a control input signal u : R+ → U such that the

time-varying system x(t) = f(x(t), u(t)) is asymptotically stable?• This is often referred to as the Asymptotic Controllability problem.

- Feedback: whether there is a state-feedback control law µ : X → U under whichthe closed-loop system x(t) = f(x(t), µ(x(t))) is asymptotically stable?

• Precise definitions of asymptotic controllability and feedback stabilizabilitycan be quite involved and depend on many additional assumptions. We willnot get into those technical details. (See [Son89; SS95; PND99])

• Under some conditions, these two types of stabilization problems areequivalent


Control-Lyapunov Function I

• Feedback stabilization is concerned with constructing control law to ensureclosed-loop stability.

• Lyapunov function is the most important tool for stability analysis.

• Under a given feedback law µ(x(t)), the closed-loop system is stable if ∃ a

PD C1 function V such that (Vx(x))Tf(x, µ(x)) < 0, ∀x 6= 0

• By the converse Lyapunov function theorems, if the closed-loop system isstable, it must have certain kind of Lypuanov functions V

• Stabilization can be thought of as finding the control law µ that minimizesthe Lie derivative of some Lypunov function of the closed-loop system

• ⇒ Control-Lyapunov Function Approach


Control-Lyapunov Function IIDefinition 1 ((Smooth) Control-Lyapunov Function).

A C1 PD function V : Rn → R+ is called a Control Lyapunov Function (CLF) if ∃a PD function W such that

infu∈U

(∇V (x))Tf(x, u) < −W (x), ∀x

• Control Lyapunov function V can be used to generate stabilizing control law:

µ∗(x) = argminu∈U

l(u) : (∇V (x))

Tf(x, u) < −W (x)

(1)

• when l(u) = ‖u‖, the resulting controller is called pointwise min-normcontroller [PND99]

• Classical results on control-Lyapunov functions are mostly based on inputaffine system: f(x, u) = f(x) + g(x)u, for which general formula (Sontag’sformula) of µ∗ exists which does not involve solving optimizationproblems [Son89]


Control-Lyapunov Function IIITheorem 1.

Assume (i) x = f(x, u) has a control Lyapunov function V ; and (ii) the correspondingµ∗ defined in (1) makes f(x, µ∗(x)) locally Lipschitz. Then µ∗ asymptotically stabilizethe system.

• This theorem follows immediately by applying the standard Lypapunovstability theorem to the closed-loop system

• The result can be easily extended to obtain stronger stability result.

- e.g.: If β1‖x‖α ≤ V (x) ≤ β2‖x‖α and W can be chosen as cV (x) with c > 0,then µ∗ exponentially stabilizes the system.


Switching Stabilization Problem I

Consider a switched nonlinear system

x(t) = fσ(t)(x(t)) (2)

• σ(t) ∈ Q, where Q is finite

• fi locally Lipschitz, for i ∈ Q

• origin is a common equilibrium, i.e. fi(0) = 0, i ∈ Q

• different classes of admissible switching signals:

- Sm: set of all measurable switching signals

- Sp: set of all piecewise constant switching signals

- Sp[τ ]: set of switching signals with dwell time no smaller than τ

Switching Stabilization Problem Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 8 / 28

Switching Stabilization Problem II• Switching Stabilization Problem: view σ(t) in (2) as the control input and

design the switching control to stabilize the continuous state x(t).

• Conceptually, this is similar to the classical nonlinear stabilization problem.But there are some key differences: the control set Q is discrete, and fσ(x) isnot (locally) Lipschitz in (x, u). Most existing results in classical nonlinearstabilization cannot be directly used.

• Open-Loop Switching Stabilization: Find an admissible switching signal σ sothat the solution to x(t) = fσ(t)(x(t)) is asymptotically stable

- depends on the assumption on admissible switching signals, e.g. Sm or Sp orSp[τ ] for some τ > 0

- this is similar to the asymptotic controllability problem for classical nonlinearsystem


Switching Stabilization Problem III• Feedback Switching Stabilization: Design a (state-dependent) switching lawν : Rn → Q so that the cl-system x = fν(x)(x) is asymp. (or exp.) stable.

- Note: Even when all the subsystem vector fields fi is smooth, the closed-loopvector field is typically discontinuous

- The stabilization problem relies crucially on the solution notion adopted for theclosed-loop discontinuous system

- Feedback stabilization in Fillipov Sense: Filippov solution notion is adopted forthe discontinuous CL-system

- Feedback stabilization in sample-and-hold sense: sample-and-hold solution notionis adopted for the discontinuous CL-system


Switching Stabilizability I

When is a given switched system stabilizable?

• Switching Stabilizable with Sm (or Sp): if ∃σ ∈ Sm (or ∈ Sp) so that

x(t) = fσ(t)(x(t)) asymp. stable.

• Feedback Stabilizable in Filippov Sense: if ∃ a switching law ν : Rn → Q sothat all Filippov solutions of the cl-system x = fν(x)(x) is asymp. stable.

• Feedback Stabilizable in Sample-and-Hold Sense: if ∃ a switching lawν : Rn → Q so that all sample-and-hold solutions of the cl-systemx = fν(x)(x) is asymp. stable.

• These stabilizability concepts are different. Their relations are unknown ingeneral except for switching linear systems [LZ16]


Switching Stabilizability IITheorem 2 (Switching Stabilizability Theorem for SLS [LZ16]).

For switched linear system: x(t) = fσ(t)(x(t)) with finite subsystems. The followingstatements are equivalent

• open-loop switching stabilizable with measurable switching input (i.e. σ ∈ Sm)

• open-loop switching stabilizable with piecewise constant switching input (i.e.σ ∈ Sp)

• feedback switching stabilizable in Filippov sense

• feedback switching stabilizable in sample-and-hold sense


Switching Stabilization via Control Lyapunov Function I

• We will focus on feedback stabilization in Filippov sense via acontrol-Lyapunov function approach

• Question: can we use the classical control Lyapunov function approach todesign switching control?

- Yes, but there are additional challenges

- Challenge 1: smooth control Lyapunov function is too restrictive forswitched/hybrid systems

• need to extend the framework to enable the use of nonsmooth control Lyapunovfunctions

- Challenge 2: the generated switching law ν∗ (even with a smooth controlLyapunov function) will make the cl-system vector field fν∗(x)(x) discontinuous;

• need to analyze possible sliding motions

Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 13 / 28

Piecewise Smooth Control-Lyapunov Function I

• Focus on a large class of nonsmooth control Lyapunov functions: piecewisesmooth functions.

• A function g : Rn → R is called piecewise smooth if it is continuous andthere exists a finite collection of disjoint and open sets Ω1, . . . ,Ωm ⊆ Rn,such that (i) ∪jΩj = Rn; (ii) g is C1 on Ωj ; (iii) ∂Ωj is a differentiablemanifold

• Important properties of piecewise smooth function

- directional derivative exists everywhere- nonsmooth surface ∂Ωj is of measure 0

• One-sided directional derivative of V along η:

V (x; η) = lim∆t↓0

V (x+ ∆tη)− V (x)

∆t(3)


Piecewise Smooth Control-Lyapunov Function II• Derivative along vector field direction in mode i ∈ Q: V (x; fi(x))

• If V is differentiable, then

V (x; η) = (∇V (x))Tη and V (x; fi(x)) = ∇V (x)T fi(x)

- NOTE: in this case, the V (x; η) is linear w.r.t η, which may not hold fornonsmooth V

• More general definition of Control Lyapunov Function: a PD function V (notnecessarily C1) is called a control Lypapunov function for switched system (2)if ∃ a PD function W s.t.

V (x; fi(x)) exists , ∀x, imini∈Q V (x; fi(x)) < −W (x), ∀x

(4)


Piecewise Smooth Control-Lyapunov Function III• Switching law generation:

ν∗(x) = argmini∈Q V (x; fi(x)) (5)

• If V is a control-Lyapunov function, then under the switching law ν∗, allclassical solutions (excluding sliding motion) to the cl-system fν∗(x)(x) isasymp. stable

• When W in (4) can be chosen as cV (x) for some c > 0, then exponentialstability (excluding sliding motions) can be concluded.


Piecewise Smooth Control-Lyapunov Function IV• What happens if there are sliding motions?

- suppose x(t) involves sliding motion during [t1, t2].

- For any t ∈ [t1, t2], x(t) =∑i∈Ism αifi(x(t)) with

∑i∈Ism αi = 1

- We should require

V(x(t);

∑i∈Ism

αifi(x(t))≤ −W (x(t)) (6)


Piecewise Smooth Control-Lyapunov Function V- If V smooth, then (4) ⇒ (6)

- If V nonsmooth, (6) is not easy to check/guarantee in general

• CLF-based switching stabilization approach: In many cases, control-Lyapunovfunction conditions (4) can be translated into LMIs or BMIs, so one cansearch for CLF by solving LMIs or BMIs. Once a CLF is found, the switchinglaw (5) ensures Filippov solution of CL-system asymp. stable. If needed,sliding motion stability can be guaranteed if the CLF also satisfies (6)

• General results about piecewise smooth CLF can be found in [LZ17].


Special Case: Quadratic Switching Stabilization I

• We now focus on a simple special case for which the CLF is chosen to be ofquadratic form.

Definition 2.

The system is called quadratically stabilizable if there exists a quadratic controlLyapunov function

• The most extensively studied class of switching stabilization problems

• A quadratic control Lyapunov function: V (x) = xTPx needs to satisfy

P 0 and mini∈Q∇V (x)T fi(x) < 0 (7)

• The corresponding switching law: ν∗(x) = argmini∈Q∇V (x)T fi(x)

Special Case: Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 19 / 28

Special Case: Quadratic Switching Stabilization IITheorem 3.

For switched nonlinear system (2), if ∃P satisfying (7), then cl-trajectory (includingsliding motion) under ν∗ is stable.

• Stable sliding motion is automatically guaranteed

• If exponential stability is desired: then should require

mini∈Q∇V (x)T fi(x) < −αV (x)


Special Case: Quadratic Switching Stabilization III• For switched linear systems x = Aσx, quadratically switching stabilizable

requiresmini∈Q

xT(ATi P + PAi

)x < 0, ∀x

- Checking the condition is NP-hard

- A sufficient condition is the existence of convex combination∑i αiAi that is

stable• proof: stable convex combination means(

l∑i=1

αiAi

)TP + P

(l∑i=1

αiAi

)≺ 0 (8)

• (8) is a bilinear matrix inequality; still NP-hard to solve

• but good numerical algorithms exist: e.g. path-following method (see HHB99)


Special Case: Quadratic Switching Stabilization IV• Extension to co-design of switching and continuous controls

- Model: x = Aix+Biu, i ∈ Q- Assume u = Kix for mode i- BMI (8) becomes:(∑l

i=1 αi · (Ai +BKi))T

P + P(∑l

i=1 αi · (Ai +BiKi))≺ 0

- change of variable: X = P−1 and Yi = KiP−1⇒∑

i

αi[XATi + Y Ti B

Ti +AiX +BiYi

]≺ 0 (9)

- numerical algorithm for (8) can be used to solve (9) as well.


Special Case: Piecewise Quadratic Switching Stabilization I

• Even for switched linear systems, quadratic stabilizability is much weakerthan switching stabilizability

• Piecewise quadratic control Lyapunov function is a natural extension

• The most important class of such nonsmooth control Lyapunov functions isobtained by taking pointwise minimum of a finite number of quadraticfunctions:

Vmin(x) = minj∈J

xTPjx (10)

- # of quadratic functions (i.e. |J |) is different from # of subsystems (i.e. |Q|) ingeneral

Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 23 / 28

Special Case: Piecewise Quadratic Switching Stabilization II

• Vmin is nonconvex and nonsmooth (piecewise smooth). Level set of Vmin:

• Switching strategy is simply: ν∗ = argmini∈Q Vmin(x; fi(x))

• It can be verified that: for switched linear systems Ai and exponentialstability (i.e. W (x) = cVmin(x) for some c > 0). The CLF conditions (4) canguaratneed by the following BMIs

∃P, c > 0, βjk ≥ 0, αij ∈ [0, 1],

∑i αij = 1, for each j,

s.t.(∑

i∈Q αijAi

)T Pj + Pj

(∑i∈Q αijAi

)∑

k∈J βjk(Pj − Pk)− cPj , for each j ∈ J

• The conditions are only sufficient and can be conservative. Derivations andinsights can be found in [HML08; LZ17]


Special Case: Piecewise Quadratic Switching Stabilization III

• Why considering pointwise minimum functions?

Theorem 4 (pointwise minimum CLF automatically guarantee stable sliding motions).

For switched nonlinear system (2), let Vmin(x) = minj∈J Vj(x), where Vj is a smoothfunction (not necessarily quadratic). If Vmin(x) is a CLF (i.e. it satisfies condition (4)),then the CL-sys under switching law (5) is asymp. stable including sliding motions.

• Therefore, if system has a pointwise-min CLF, then stable sliding motion isautomatically guaranteed (no need to further check condition (6))

• Proof can be found in [LZ17]


Special Case: Piecewise Quadratic Switching Stabilization IV

• Why is pointwise minimum piecewise quadratic CLF important?

Theorem 5 (Converse CLF for switched linear system).

A switched linear system is switching stabilziable (in any open-loop or closed-loopstabilizability sense) if and only if there exists a piecewise quadratic CLF of form (10).

• To study switching stabilization of switched linear systems (regardless ofopen-loop or feedback based control), it suffices to consider piecewisequadratic CLFs that can be written as a pointwise minimum of a finitenumber of quadratic functions.

• Proof can be found in [LZ16].


Conclusion

• Control-Lyapunov function (CLF) is an effective approach to designstabilizing control law.

• Once a CLF is found, stabilizing law can be constructed by minimizing theLie derivative of CLF over admissible control inputs

• For switched systems, one needs to consider nonsmooth CLF and needs toanalyze sliding motion

• Two important cases: if a switched nonlinear system has a quadratic orpointwise-min CLF, then stable sliding motion is automatically guaranteed.

• For switched linear systems, open-loop and feedback switching stabilizability(in Filippov or sample-and-hold sense) are all equivalent to the existence of apointwise-min piecewise quadratic CLF.

• Further reading: [LZ16; LZ17; HML08]

• Next Lecture: Discret-time Optimal Control


References

[Son89] Eduardo D Sontag. “A fffdfffdfffduniversalfffdfffdfffdconstruction of Artstein’stheorem on nonlinear stabilization”. In: Systems & control letters 13.2 (1989).

[SS95] Eduardo Sontag and Hector J Sussmann. “Nonsmooth control-Lyapunov functions”.In: Decision and Control, 1995., Proceedings of the 34th IEEE Conference on. Vol. 3.IEEE. 1995.

[PND99] James A Primbs, Vesna Nevistic, and John C Doyle. “Nonlinear optimal control: Acontrol Lyapunov function and receding horizon perspective”. In: Asian Journal ofControl 1.1 (1999).

[HML08] Tingshu Hu, Liqiang Ma, and Zongli Lin. “Stabilization of switched systems viacomposite quadratic functions”. In: IEEE Transactions on Automatic Control 53.11(2008).

[LZ16] Yueyun Lu and Wei Zhang. “On switching stabilizability for continuous-timeswitched linear systems”. In: IEEE Transactions on Automatic Control 61.11 (2016).

[LZ17] Yueyun Lu and Wei Zhang. “A piecewise smooth control-Lyapunov functionframework for switching stabilization”. In: Automatica 76 (2017).


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