## 非线性MPC的稳定性

x ˙ = f ( x ( t ) , u ( t ) ) , x ( 0 ) = x 0 , u ( t ) ∈ U \dot<\mathbf>=\bm\big(\mathbf(t),\mathbf(t)\big),\quad \mathbf(0)=\rm_0,\\ \quad\mathbf(t)\in U x ˙ = f ( x ( t ) , u ( t ) ) , x ( 0 ) = x 0 , u ( t ) ∈ U

## 2 Assumptions

- f \bm f is twice continuously differentiable and f ( 0 , 0 ) = 0 \bm(
*MT4自带的30个指标说明及调用*\bm,\bm)=\bm f ( 0 , 0 ) = 0 , i.e. x = 0 \mathbf=\bm x = 0 is an MT4自带的30个指标说明及调用 equilibrium under zero input. - U U U is compact, convex and 0 \bm 0 is contained in MT4自带的30个指标说明及调用 MT4自带的30个指标说明及调用 U U U .
- The jacobian linearization of system ( 1 ) (1) ( 1 ) is stabilizable.

## 3. Standard MPC formulation

min u ( ⋅ ) J ( x ( t ) , u ( ⋅ ) ) \min _ J(\mathbf(t), \mathbf(\cdot)) u ( ⋅ ) min J ( x ( t ) , u ( ⋅ ) )

with

J ( x ( t ) , u ( ⋅ ) ) = ∫ t t + T p ( ∥ MT4自带的30个指标说明及调用 MT4自带的30个指标说明及调用 MT4自带的30个指标说明及调用 x ( τ ) ( x ( t ) , t ) ∥ Q 2 + ∥ u ( τ ) ∥ R 2 ) d τ + ∥ x ( t + T p ; x ( t ) , t ) ∥ P 2 (2) J(\mathbf(t), \mathbf(\cdot))=\int_^>\left(MT4自带的30个指标说明及调用 \|\mathbf(\tau)(\mathbf(t), t)\|_^+\|\mathbf(\tau)\|_^\right) \mathrm \tau \\ \quad\quad\quad\quad+\left\|\mathbf\left(t+T_<\mathrm

> ; \mathbf(t), t\right)\right\|_^\tag J ( x ( t ) , u ( ⋅ ) ) = ∫ t t + T p ( ∥ x ( τ ) ( x ( t ) , t ) MT4自带的30个指标说明及调用 ∥ Q 2 + ∥ u ( τ ) ∥ R 2 ) d τ + ∥ x ( t MT4自带的30个指标说明及调用 + T p ; x ( t ) , t ) ∥ P 2 ( 2 )

subject to

x ˙ = f ( x , MT4自带的30个指标说明及调用 u ) , x ( t ; x ( t ) , t ) = x ( t ) u ( τ ) ∈ U , τ ∈ [ t , t + T p ] x ( t + T p ; x ( t ) , t ) ∈ Ω (3) \dot<\mathbf>=\mathbf(\mathbf, \mathbf), MT4自带的30个指标说明及调用 MT4自带的30个指标说明及调用 \quad \mathrm(t ; \mathbf(t), t)=\mathbf(t)\\ \mathbf(\tau) \in U, \quad \tau \in\left[t, t+T_<\mathrm

>\right]\\ \mathbf\left(t+T_<\mathrm

> ; \mathbf(t), t\right) \in \Omega \tag x ˙ = f ( x , u ) , x ( t ; x ( t ) , t ) = x ( t ) u ( τ ) ∈ U , τ ∈ [ t , t + T p ] x ( t + T p ; x ( t ) , t ) ∈ Ω ( 3 )

## 4. Main ideas

- Since the finite horizon optimization property of MPC can not guarantee the stability, while the infinite may do more, such as the LQR.
- By introducing terminal state cost and terminal inequality constraint, the terminal states can be forced to be in the terminal region, which is selected as the attraction region around origin and is invariant under local linear feedback controllers.
- When trajectory under optimization goes into the terminal region at t + T p t+T_p t + T p , the states will stay in the region when imposing designed linear state feedback.
- It can be proved that the infinite integral cost ∫ s ∞ ( ∥ x ( s ) ( x ( t ) , t ) ∥
*MT4自带的30个指标说明及调用*Q 2 + ∥ u ( s ) ∥ R 2 ) d s \int_^<\infty>\left(\|\mathbf(s)(\mathbf(t), t)\|_^+\|\mathbf(s)MT4自带的30个指标说明及调用**MT4自带的30个指标说明及调用**\|_^\right) \mathrm s ∫ s ∞ ( ∥ x ( s ) ( x ( t ) , t ) ∥ Q 2 + ∥ u ( s ) ∥ R 2 ) d s starting from invariant set Ω \Omega Ω is always less equal than the terminal cost in ( 2 ) (2) ( 2 ) . - The optimal value function can thus be proved
**MT4自带的30个指标说明及调用**to be non-increasing. This further leads to the stability. - Although the local state feedback is used to demonstrate the stability, MPC is implemented in online manner at every time step and the linear feedback control law MT4自带的30个指标说明及调用 is never used. The finite horizon control can thus be proved in a quasi-infinite way.

## 5. Method

Solve the linear state feedback control problem to find a locally stabilizing feedback gain K after jacobian linearization around origin.

x ˙ = A x + B u \dot<\mathbf>=A\mathbf+B\mathbf x ˙ = A x + B u A = MT4自带的30个指标说明及调用 ∂ f ∂ x ( 0 , 0 ) , B = ∂ f ∂ u ( 0 , 0 ) A=\frac<\partial<\bm

Solve the Lyapunov equation with a suitable κ ∈ [ 0 , + ∞ ] \kappa\in[0,+\infty] κ ∈ [ 0 , + ∞ ] . κ \kappa κ should be selected (usually depends on the eigenvalues of A k A_k A k ) to make sure that the equation has unique positive definite and symmetric MT4自带的30个指标说明及调用 MT4自带的30个指标说明及调用 solutions P P P .

Find the largest possible set Ω β = < x ∣ x ⊤ P x ≤ β >\Omega_\beta=\<\mathbf|\mathbf^P\mathbf\leq\beta\> Ω β = < x ∣ x ⊤ P x ≤ β >, such that K x ∈ U K\mathbf\in U K x ∈ U for all x ∈ Ω β \mathbf\in\Omega_\beta x ∈ Ω β . This ensures that the state feedback control respects the control constraints.

Find the largest possible set Ω α = < x ∣ x ⊤ P x ≤ α , α ≤ β >\Omega_\alpha=\<\mathbf|\mathbf^P\mathbf\leq\alpha, \alpha\leq\beta\> Ω α = < x ∣ x ⊤ P x ≤ α , α ≤ β >, such that the condition of non-increasing-value-function MT4自带的30个指标说明及调用 is satisfied in Ω α \Omega_\alpha Ω α .

## 6. Feasibility and stability

The feasibility at any t > 0 MT4自带的30个指标说明及调用 t>0 t > 0 can be guaranteed if the feasibility is satisfied at t = 0 t=0 t = 0 . The stability is guaranteed by the feasibility and non-increasing property mentioned above.

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