SUBOPTIMAL FEEDBACK CONTROL DESIGN OF

CONSTRAINED PARABOLIC SYSTEMS IN UNCERTAINTY CONDITIONS

BORIS MORDUKHOVICH

Wayne State University

Talk given at the

King Fahd University of Petroleum and Minerals

Dhahran, Saudi Arabia

March 2008

Supported by NSF grants DMS-0304989 and DMS-0603846

PARABOLIC DYNAMICS

\frac{∂y}{∂t} + Ay = w(t) a.e. in Q := [0, T] × Ω,

y(0, x) = 0, x ∈ Ω,

y(t, x) = u(t), (t, x) ∈ Σ := [0, T] × ∂Ω

with controls u(·) acting in the Dirichlet boundary conditions and distributed perturbations w(·). Here A is a self-adjoint and uniformly strongly elliptic operator on L^{2}(Ω) defined by

A := − n ; ∑ ; i,j=1 \frac{∂}{∂x_{i}}(a_{ij}(x)\frac{∂}{∂x_{j}} ) − c,

where Ω ⊂ IR^{n} is an bounded domain with smooth boundary ∂Ω and where T > 0 is a fixed time bound.

HARD/POINTWISE CONSTRAINTS

ADMISSIBLE BOUNDARY CONTROLS u(·) ∈ U := {u ∈ L^{∞}[0, T]| ≤ −α ≤ u(t) ≤ α a.e. t ∈ [0, T]} ADMISSIBLE UNCERTAIN PERTURBATIONS

w(·) ∈ W := {w ∈ L^{∞}[0, T]| − β ≤ w(t) ≤ β a.e. t ∈ [0, T]} with some fixed bounds α, β > 0. POINTWISE STATE CONSTRAINTS

−η ≤ y(t, x_{0}) ≤ η a.e. t ∈ [0, T],

where x_{0} ∈ Ω is the given point of observation with the given accuracy η > 0 to stabilize the equilibrium state y ≡ 0.

Figure 1. Distributed system

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Figure 2. Transients

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Figure 3. Boundary controls

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MINIMAX FEEDBACK CONTROL

Since perturbations are uncertain, the required system performance may be achieved only by a feedback regulator f(ξ) depending on the current state position ξ = y(t, x_{0}) at the point of observation. FEEDBACK CONTROL LAW is formed by

u(t) := f(y(t, x_{0})), t ∈ [0, T].

We estimate the quality of feedback regulators by the

J(f) := max_{w∈W} { ∫_{0}^{T} |f(y(t, x_{0}))| dt},

which reflects the required control energy needed to neutralize the adverse effect of the worst perturbations to keep the state performance within the prescribed area. MINIMAX FEEDBACK CONTROL PROBLEM (P):

minimize J(f) over feasible feedbacks f .

FUNDAMENTAL PROPERTIES OF THE PARABOLIC DYNAMICS

We always have

0 ≤ λ_{1} < λ_{2} ≤ λ_{3} ≤ … ≤ ….

THE BASIC ASSUMPTION: λ_{1} > 0.

SPECTRAL REPRESENTATION OF TRANSIENTS. For any (u, w) ∈ L^{∞}[0, T] × L^{∞}[0, T] we have

y(t, x) = ∞ ; ∑ ; i=1 μ_{i}( ∫_{0}^{t} w(θ)e^{λ_{i}θ}dθ + (c + λ_{i}) ∫_{0}^{t} u(θ)e^{λ_{i}θ}dθ)e^{−λ_{i}t}ϕ_{i}(x)

with μ_{i} := ∫_{Ω} ϕ_{i}(x) dx.

MONOTONICITY PROPERTY. If u_{1}(t) ≥ u_{2}(t), w_{1}(t) ≥ w_{2}(t) on [0, T], then

y_{1}(t, x) ≥ y_{2}(t, x) on Q.

WORST PERTURBATIONS

THEOREM. Let a feasible feedback f ∈ F be a nonincreasing and odd function on IR. Then the worst perturbations w(t) providing the maximum value to the cost functional over all w ∈ W are the extreme ones from the admissible area:

w(t) ≡ β, w ≡ −β.

PROOF. Based on the Maximum Principle for the parabolic dynamics, the monotonicity property, and the convolution representation of the transients.

OPEN-LOOP OPTIMAL CONTROL PROBLEM FOR w(t) ≡ β

minimize J(u) := − ∫_{0}^{T} u(t) dt

subject to the parabolic dynamics

\{ \begin{matrix}\frac{∂y}{∂t} + Ay = β a.e. (t, x) ∈ Q, \\ y(0, x) = 0, x ∈ Ω, \\ y(t, x) = u(t) a.e. (t, x) ∈ Σ \end{matrix}

with the pointwise control and state constraints :

u(·) ∈ L^{∞}[0, T] with − α ≤ u(t) ≤ 0 a.e. t ∈ [0, T],
y(·, x_{0}) ∈ L^{2}[0, T] with y(t, x_{0}) ≤ η a.e. t ∈ [0, T].

ODE APPROXIMATION

ẏ_{i} = −λ_{i}y_{i} + μ_{i}ϕ_{i}(x_{0})(β + (c + λ_{i})u(t)) a.e. t ∈ [0, T], y_{i}(0) = 0.

Due to the Basic Assumption we can take i = 1 for the time interval large enough.

EXACT OPTIMAL SOLUTION TO THE STATE-CONSTRAINED APPROXIMATION PROBLEM. Optimal control is given by the two-positional piecewise constant function

\overline{u}(t) = \{ \begin{matrix}0 & if t ∈ [0, τ), \\ v := \frac{λ_{1}η − μ_{1}ϕ_{1}(x_{0})β}{μ_{1}ϕ_{1}(x_{0})(c + λ_{1})} & if t ∈ [τ, T]. \end{matrix}

PROOF. Penalty approximation of state constraints, Pontryagin Maximum Principle, singular optimal mode Figure 4. Optimal control with no state constraints

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Figure 5. Optimal control with state constraints

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OPEN-LOOP OPTIMAL CONTROL FOR PARABOLIC SYSTEM

Consider the problem:

minimize J(v, τ) := − ∫_{0}^{T} u(t) dt

over the parabolic system

\{ \begin{matrix}\frac{∂y}{∂t} + Ay = w(t) a.e. in [0, T] × Ω, \\ y(0, x) = 0, x ∈ Ω, \\ y(t, x) = u(t), (t, x) ∈ [0, T] × ∂Ω \end{matrix}

u(t) = \{ \begin{matrix}0 & if t ∈ [0, τ], \\ −v & if t ∈ (τ, T] \end{matrix}

subject to the pointwise control and state constraints: 0 ≤ v ≤ α, y(t, x_{0}) ≤ η, 0 ≤ τ ≤ T.

OPTIMAL CONTROL FOR PARABOLIC SYSTEM (cont.)

THEOREM. Optimal solution for big T > 0 is given by

v = \frac{γβ − η}{1 + cγ}

with the switching time τ = τ(T) satisfying

∞ ; ∑ ; i=1 \frac{μ_{i}ϕ_{i}(x_{0})}{λ_{i}} e^{−λ_{i}T} [(c + λ_{i})(γβ − η)e^{λ_{i}τ} − β(1 + cγ)] = 0,

Furthermore, there is the limit τ(T) ↓ \overline{τ} as T → ∞, where the asymptotically optimal switching time is computed by

τ = \frac{1}{λ_{1}} ln \frac{β(1 + cγ)}{(c + λ_{1})(γβ − η)}

and turns out to be maximal among all the switching times τ ≥ 0 ensuring the fulfillment of the state constraints.

FEEDBACK SUBOPTIMAL CONTROL FOR PARABOLIC SYSTEM

SUBOPTIMAL FEEDBACK 3-POSITIONAL STRUCTURE

f(ξ) = \{ \begin{matrix}−v & if ξ ≥ σ, \\ 0 & if − σ < ξ < σ, \\ v & if ξ ≤ −σ. \end{matrix}

OPTIMAL (IN THE MINIMAX SENSE) PARAMETERS OF THE FEEDBACK REGULATOR:

v = \frac{γβ − η}{1 + cγ}, σ(T) = β(γ − ∞ ; ∑ ; i=1 \frac{μ_{i}ϕ_{i}(x_{0})}{λ_{i}} e^{−λ_{i}τ(T)} ),

with τ(T) computed above. When T → ∞ (infinite horizon), then we have

\overline{σ} = β(γ − ∞ ; ∑ ; i=1 \frac{μ_{i}ϕ_{i}(x_{0})}{λ_{i}} [\frac{(c + λ_{1})(γβ − η)}{β(1 + cγ)} ]^{\frac{λi}{λ1}} ).

ROBUST STABILITY OF CLOSED-LOOP SYSTEM

\{ \begin{matrix}\frac{∂y}{∂t} + Ay = w(t), x ∈ Ω, t ≥ 0, \\ y(0, x) = 0, x ∈ Ω, \\ y(t, x) = f(y(t, x_{0})), x ∈ ∂Ω, t ≥ 0, \end{matrix}

where f = f(ξ) is a three-positional regulator. STABILITY IN THE LARGE of the equilibrium y ≡ 0 when

σ ≥ \frac{v}{2λ_{1}} [μ_{1}ϕ_{1}(x_{0})(c + λ_{1}) − λ_{1}(1 + cγ)].

Figure 6. Three-positional regulator

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Figure 7. Robust stability

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RELIABILITY OF CONTROL DESIGN

Compute

\begin{aligned}σ = & \frac{γβ − η + η(1 + cγ)}{2(1 + cγ)} ∞ ; ∑ ; i=1 \frac{μ_{i}ϕ_{i}(x_{0})(c + λ_{i})}{λ_{i}} (\frac{γβ − η}{γβ − η + η(1 + cγ)} )^{\frac{λi}{λ1}} \\ & +\frac{η − γβ}{2} > 0. \end{aligned}

the feedback control with

σ ≤ σ ≤ \overline{σ}

is is reliable on the infinite horizon in the sense that it is feasible by all the constraints for any perturbations enjoying simultaneously robust stability. Furthermore, the upper bound σ = \overline{σ} optimizes the suboptimal feedback structure in the minimax sense.

PROOF. VARIATIONAL APPROACH.

REFERENCES

1. B. S. MORDUKHOVICH, VARIATIONAL ANALYSIS AND GENERALIZED DIFFERENTIATION, II: APPLICA-TIONS, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 331, Springer, 2006.

2. B. S. MORDUKHOVICH, SUBOPTIMAL FEEDBACK CONTROL DESIGN OF CONSTRAINED PARABOLIC SYS-TEMS IN UNCERTAINTY CONDITIONS, preprint.