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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
Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 4 / 28
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
Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 5 / 28
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]
Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 6 / 28
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.
Classical Control-Lyapunov Function Approach Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 7 / 28
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 Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 9 / 28
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 Stabilization Problem Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 10 / 28
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 Stabilization Problem Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 11 / 28
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 Problem Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 12 / 28
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)
Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 14 / 28
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)
Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 15 / 28
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.
Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 16 / 28
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)
Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 17 / 28
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].
Switching Stabilization via CLF Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 18 / 28
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 Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 20 / 28
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 Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 21 / 28
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: Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 22 / 28
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]
Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 24 / 28
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]
Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 25 / 28
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].
Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 26 / 28
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
Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 27 / 28
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).
Piecewise Quadratic Switching Stabilization Lecture 7 (ECE7850 Sp17) Wei Zhang(OSU) 28 / 28