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ECE7850 Wei Zhang

ECE7850 Lecture 8

Nonlinear Model Predictive Control: Theoretical Aspects

• Model Predictive control (MPC) is a powerful control design method for constrained dynam-

ical systems.

• The basic principles and theoretical results for MPC are almost the same for most nonlinear

systems, including discrete-time hybrid systems.

• The particular underlying model (e.g. linear or switched affine, or piecewise affine, or hybrid

systems), mainly affects the computational aspects for MPC.

• This lecture focuses on principles for general nonlinear MPC; next lecture will cover compu-

tational aspects, especially for MPC of hybrid systems.

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ECE7850 Wei Zhang

Lecture Outline

• Formulation and Related Definitions for General MPC

• Persistent Feasibility of MPC

• Stability Analysis of MPC

• Analysis Without Terminal Constraint/Cost

• Other Selected Topics

2

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Formulation and Related Definitions

• General discrete-time nonlinear systems:

⎧⎪⎪⎨⎪⎪⎩x(t + 1) = f(x(t), u(t))y(t) = h(x(t))

, t ∈ Z+ (1)

• State and Control constraints:

x(t) ∈ X u(t) ∈ U (2)

– In general, X ⊂ Rn × Q and U ⊂ R

m × Σ have both continuous and discrete components.

– To simplify presentation, we assume X ⊂ Rn (but u can have both continuous and dis-

crete components)

• Assume full state information available, unless otherwise stated. (e.g. y(t) = x(t) or h(·) is

bijection)

Formulation and Related Definitions 3

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• Basic ideas for MPC: Receding Horizon Control (RHC)

– At time t, solves a finite horizon optimal control problem based on the system model

– Apply the first step of the optimal control sequence

– At time t + 1, horizon is shifted and the optimal control problem is solved again using

newly obtained state measurements

Formulation and Related Definitions 4

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• Formalizing the idea:

– At time t: solve the following N -horizon optimal control problem:

PN(x(t)) : VN(x(t)) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

minu J(x(0), u) � Jf(xN) + ∑N−1k=0 l(xk, uk)

subj. to: xk+1 = f(xk, uk), k = 0, . . . , N − 1xk ∈ X, uk ∈ U, k = 0, . . . , N − 1xN ∈ Xf, x0 = x(t)

(3)

– Xf ⊆ X: terminal state constraint set

– Assume nonnegative cost functions: Jf : X → R+ and l : X × U → R+

– Given x(0), optimal control sequence u∗0, . . . , u∗

N−1 can be found via numerical optimiza-

tion

Formulation and Related Definitions 5

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• Problem (3) can also be solved using DP, leading to optimal control laws

μ∗j(z) = argminu∈U(z){l(z, u) + Vj−1(f(z, u))}, j = 1, . . . , N

• Finding {u∗k} is much easier than finding μ∗

j(·);

• Why we care about {μ∗j(·)}?

– For many cases, μ∗j(z) may have appealing analytical structures that can simplify the

“online” computation for {u∗k} (e.g.: explicit MPC)

– Enable various analysis for feasibility, performance, and stability of MPC

• At any time t, if the state is x(t), then MPC controller will apply ut = μ∗N

(x

(t

))

Formulation and Related Definitions 6

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• Two fundamental issues for MPC

– Persistent Feasibility: Problem P(x(t)) remains feasible for all t

– Closed-Loop Stability: x(t + 1) = f(x(t), μ∗N(x(t))) is stable

• We need to introduce some key concepts and definitions:

– One-step backward reachable set: P re(S) = {x ∈ Rn : ∃u ∈ U s.t. f(x, u) ∈ S}

– One-step forward reachable set:

Reach(S) = {x ∈ Rn : ∃u ∈ U, z ∈ S s.t. x = f(z, u) ∈ S}

Formulation and Related Definitions 7

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– N -step backward reachable set subject to system constraints:

R−j+1(S) = P re

(R−

j (S))

∩ X, R−0 (S) = S (4)

– N -step forward reachable set subject to system contratins:

R+j+1(S) = Reach

(R+

j (S))

∩ X, R+0 (S) = S (5)

– Positive Invariant Set: Given an constrained autonomous system

x(t + 1) = fa(x(t)), t ∈ Z+, x(t) ∈ X (6)

a set O ⊂ X is called a positive invariant set if

x(0) ∈ O ⇒ x(t) ∈ O, ∀t ∈ Z+

Formulation and Related Definitions 8

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– Maximal Positive Invariant Set O∗: positive invariant + contains all invariant sets con-

tained in X

– C ⊆ X is called a Control Invariant Set for the constrained system (1) if

x(t) ∈ C ⇒ ∃u(t) ∈ U such that f(x(t), u(t)) ∈ C, ∀t ∈ Z+

– Maximal control invariant set C∗: control invariant + contains all control invariant sets

contained in X

– Efficient algorithms to compute (control) invariant sets are available for particular classes

of constrained systems

Formulation and Related Definitions 9

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Persistent Feasibility of MPC

• Feasible set Xk: the set of feasible state xk at prediction step k for which (3) is feasible

Xk = {x ∈ X : ∃u ∈ U, s.t. f(x, u) ∈ Xk+1}, with XN = Xf (7)

• can be equivalently defined as: Xk = P re(Xk+1) ∩ X, with XN = Xf

• Persistent Feasibility:

– start from any x(0) ∈ X0

– evolve under MPC control law, i.e., x(t + 1) = f(x(t), μ∗N(x(t)))

– feasibility is guaranteed at all time, i.e. x(t) ∈ X0, for all t ∈ Z+.

Persistent Feasibility of MPC 10

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• Lemma 1 If X1 is control invariant, then receding horizon control problem PN(x(t)) is persis-

tently feasible.

Proof:

– X1 ⊆ P re(X1) ∩ X = X0

– ∀x ∈ X0, apply the first-step of MPC control u∗0. We have x+ = f(x, u∗

0) ∈ X1 ⊆ X0

• However, the above condition is hard to verify as X1 is defined recursively from Xf . A

condition directly imposed on Xf is desirable.

Persistent Feasibility of MPC 11

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Theorem 1 If Xf is control invariant, then the receding horizon optimization PN(x(t)) is per-

sistently feasible.

Persistent Feasibility of MPC 12

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Stability of MPC

• Stability analysis of MPC:

– closed-loop system: x(t + 1) = f(x(t), μ∗N(x(t)) persistently feasible and stable

– characterize domain of attraction

• Assumption 1 (i) X and U contains origin in their interior.

(ii) x = 0, u = 0 is an equilibrium f(0, 0) = 0;

(iii) β−l ‖z‖ ≤ l(z, u) ≤ β+

l ‖z‖, for all z ∈ X, u ∈ U .

Stability of MPC 13

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• Theorem 2 Assume conditions in Assumption 1 hold and

1. Xf ⊆ X is control invariant

2. Terminal cost function Jf is a local control Lyapunov function satisfying

Jf(z) ≤ βf‖z‖, and ∃μ such that: Jf(z) − Jf(f(z, μ(z))) ≥ l(z, μ(z)), ∀z ∈ Xf (8)

Then the origin of the closed-loop system under MPC control is exponentially stable with

region of attraction X0.

Sketch of proof

– Pick an arbitrary z ∈ X0, let u∗0, . . . , u∗

N−1 and x∗0, x∗

1, . . . , x∗N be the corresponding optimal

control and trajectory.

Stability of MPC 14

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– Need to show VN(z) is a control Lyapunov function on X0

Let μ be the law satisfying (8). Construct a new control sequence

u = {u∗1, u∗

2, . . . , u∗N−1, μ(x∗

N)}.

Starting from x∗1, the control sequence u results in state trajectory {x∗

1, x∗2, . . . , x∗

N, f(x∗N, μ(x∗

N))}.By optimality of VN , we have

VN(x∗1) ≤ JN(x∗

1, u) =N−1∑k=1

l(x∗k, u∗

k) + l(x∗N, μ(x∗

N)) + Jf(f(x∗N, μ(x∗

N)))

= VN(x∗0) − l(x∗

0, u∗0) − Jf(x∗

N) + l(x∗N, μ(x∗

N)) + Jf(f(x∗N, μ(x∗

N)))

Notice that x∗0 = z, x∗

1 = f(z, μ∗N(z)), and x∗

N ∈ Xf . By (8), we have

VN(z) − VN(f(z, μ∗N(z))) ≥ l(z, u∗

0) ≥ β−l ‖z‖

The conditions on l and Jf , along with the boundedness of state trajectory within the N

prediction steps clearly indicates that VN(z) ≤ β‖z‖, for z ∈ X0 and some β < ∞. The

stability the follows from the ECLF theorem.

Stability of MPC 15

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Analysis Without Terminal Constraint/Cost

• Computation of control invariant set is very challenging (except for some simple cases such

as linear systems with polytopic constraints)

• Even the control invariant set is available, using it as a terminal constraint set can lead to

poor numerical performance for both online and offline optimization algorithms

• Using control invariant set as terminal constraint set also shrinks feasible set X 0.

Analysis Without Terminal Constraint/Cost 16

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• Lemma 2 (i) If Xf = Rn, then C∗ ⊆ X0 ⊆ X1 · · · ⊆ XN = Xf ;

(ii) If Xf is control invariant, then X ⊇ C∗ ⊇ X0 ⊇ X1 · · · ⊇ XN = Xf

• Persistent feasibility and closed-loop stability can also be guaranteed without terminal con-

straint set

Analysis Without Terminal Constraint/Cost 17

ECE7850 Wei Zhang

• We consider MPC with no terminal constraint:

PntN (z) : VN(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

minu JN(z, u) � Jf(xN) + ∑N−1k=0 l(xk, uk)

subj. to: xk+1 = f(xk, uk), k = 0, . . . , N − 1xk ∈ X, uk ∈ U, k = 0, . . . , N − 1x0 = z

(9)

• With slight abuse of notation, let μN be the MPC law, i.e., μN(z) = argminu∈U(z){l(z, u) +

VN−1(f(z, u))}, where VN is defined above.

• CL-system under MPC: x(t + 1) = f(x(t), μN(x(t)))

• Define the infinite-horizon version of the above problem:

V ∗(z) = infu0,u1,...

⎧⎨⎩

∞∑k=0

l(xk, uk) : subj. to constraint in (9)⎫⎬⎭

Analysis Without Terminal Constraint/Cost 18

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• We want

⎧⎪⎪⎨⎪⎪⎩Pnt

N (x(t)) persistently feasible along cl-traj

The cl-system is exponentially stable(10)

• We shall establish conditions to ensure (10) for two different cases:

– Case I: Jf(z) ≡ 0, namely, MPC with neither terminal constraint nor terminal cost.

– Case II: Jf(z) is an ECLF (MPC without terminal constraint but with nontrivial terminal

cost)

• Assumption 2 (i) Jf(z) ≤ β+f ‖z‖; (i) β−

l ‖z‖ ≤ l(z, u) ≤ β+l ‖z‖; (iii) V ∗(z) ≤ β∗‖z‖, ∀z ∈ X∗

– As discussed in Lecture Note 7, condition (iii) is “almost equivalent” to exponential stabi-

lizability of the constrained system (1).

– Of course, Assumption 1 is always assumed as well.

Analysis Without Terminal Constraint/Cost 19

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• The key property for both cases is the convergence of the optimal trajectory and value

iteration as horizon length N increases

• Lemma 3 (Convergence of Optimal Trajectory): Under Assumption 2,

– ‖x(t; z, π∗N)‖ ≤ cxγt

x‖z‖, for all z ∈ X∗ and t = 0, . . . , N − 1

– if additionally Jf(z) ≥ β−l ‖z‖, then the above inequality also holds for t = N

• Without condition Jf(z) ≥ β−l ‖z‖, the final state x(N ; z, π∗

N) may be arbitrarily large.

Example 1 x(t + 1) = x(t) + u(t), for t = {0, 1, . . . , N − 1}, with X = U = R. Let L(x, u) = x2,ψ ≡ 0. Fix an initial state x(0) = z. It can be easily verified that the N -horizon controlsequence is of the form {−z, 0, . . . , 0, c} is optimal for all c ∈ R. Therefore, the terminal stateof the corresponding optimal trajectory is equal to c and can be made arbitrarily large.

Analysis Without Terminal Constraint/Cost 20

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• Lemma 4 (Value Iteration Convergence): Under Assumption 2,

– Value iteration converges exponentially, |VN(z) − V ∗(z)| ≤ cV γNV ‖z‖;

– ∃N0 such that VN(z) is an ECLF for all N ≥ N0;

sketch of proof:

– Part I: show special case with Jf ≡ 0. General case can be found in zhang09

Analysis Without Terminal Constraint/Cost 21

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– Part II: show VN eventually becomes an ECLF for sufficiently large N

Analysis Without Terminal Constraint/Cost 22

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• Theorem 3 (Main Result for Case I): Under Assumption 2 with Jf(z) ≡ 0, there exists

N0 < ∞ such that (10) is guaranteed for all N ≥ N0 with region of attraction Lα = {z ∈ Rn :

VN(z) ≤ α} for any α that ensures Lα ⊆ X.

sketch of proof:

Analysis Without Terminal Constraint/Cost 23

ECE7850 Wei Zhang

• Remarks for Main Result I:

– With neither terminal constraint nor terminal cost, persistent feasibility and cl-stability can

still be guaranteed as long as the prediction horizon N is sufficiently large

– N can be determined by checking whether VN is an ECLF, which can be done through

LMIs in some special cases

– Large N may cause issues for both online optimization and offline explicit MPC solutions.

– As a compromise, we can add the terminal cost back while still omitting the terminal

constraint, which leads us to Case II.

Analysis Without Terminal Constraint/Cost 24

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• Theorem 4 (Main Result for Case II): Under Assumption 2 with Jf(z) being an ECLF satis-

fying condition (8) over some neighborhood of origin XJf, there exists N0 < ∞ such that (10)

is guaranteed for all N ≥ N0 with region of attraction Lα = {z ∈ Rn : VN(z) ≤ α} for any α

that ensures Lα ⊆ X

sketch of proof:

– Select a sublevel set: Xf = {x ∈ Rn : Jf(x) ≤ α} ⊂ XJf

– Xf is control invariant.

– Need to guarantee the optimal prediction trajectory always hits Xf at the end, i.e., x∗N ∈

Xf . This can be achieved by choosing a sufficiently large N .

Analysis Without Terminal Constraint/Cost 25

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Relationship With Unconstrained Problem

• State and control constraint sets X and U often cause significant challenge in finding control

invariant sets and control Lyapunov functions.

• Dropping these constraints leads to unconstrained MPC that is easier to solve.

• Solution to the unconstrained MPC can be used to generate control invariant set and control

Lyapunov function for the constrained MPC.

• Consider unconstrained system:

x(t + 1) = f(x(t), u(t)), x ∈ Rn, u ∈ R

m (11)

Relationship With Unconstrained Problem 26

ECE7850 Wei Zhang

• Unconstrained N -horizon optimal control:

V ncN (z) =

⎧⎪⎪⎨⎪⎪⎩minu JN(z, u) = ∑N−1

k=0 l(xk, uk)subj. to xk+1 = f(xk, uk), k = 0, . . . , N − 1

• By Lemma 4, we know V ncN becomes an ECLF of (11) for large N

• Suppose V ncN0 is an ECLF of (11), define

μncN0(z) = argminu∈U(z){l(z, u) + V nc

N0 (z)}

namely, V ncN0 (z) − V nc

N0

(f

(z, μnc

N0

))≥ l(z, μnc

N0(z))

Relationship With Unconstrained Problem 27

ECE7850 Wei Zhang

• Assume: μncN0(z) ≤ βμ‖z‖, z ∈ X

• Due to exponential stability, unconstrained cl-system trajectory and control satisfy:

xnc(t; z, μncN0) ≤ cxrk‖z‖, unc(t; z, μnc

N0) ≤ curk‖z‖

• Lemma 5 There exists a neighborhood around the origin X nc ⊂ X, such that the uncon-

strained cl-trajectory and control are feasible with respect to state and control constraints X

and U , i.e.,

xnc(t; z, μncN0) ∈ X, and unc(t; z, μnc

N0) ∈ U, ∀z ∈ Xnc, ∀k ∈ Z+

Relationship With Unconstrained Problem 28

ECE7850 Wei Zhang

• Definition 1 Given a control law μ, a positive invariant set Ω of the cl-system x(t + 1) =

f(x(t), μ(x(t)) is called constraint admissible if Ω ⊂ X, and {μ(z) : z ∈ Ω} ⊂ U

• Theorem 5 Consider the constrained MPC problem defined in (3). Persistent feasibility and

cl-stability can be guaranteed under either of the following two conditions:

1. Jf(z) = V ncN0 (z) and Xf is a constraint admissible positive invariant set of the uncon-

strained system under control law μncN

2. Jf(z) = V ncN0 (z), Xf = R

n, and N is sufficiently large

Relationship With Unconstrained Problem 29

ECE7850 Wei Zhang

• Remarks about Theorem 5:

– N0 is the chosen to make unconstrained value function an ECLF, while N is the horizon

size for the constrained MPC problem

– Under the first condition, Xf is control invariant and V ncN is an ECLF on Xf w.r.t. the

constrained system. The desired result follows directly from Theorem 2.

– Under the second condition, there is no terminal constraint; the desired result follows

from Theorem 4.

Relationship With Unconstrained Problem 30

ECE7850 Wei Zhang

• Summary:

– MPC: solve N -horizon constrained optimal control problem PN(z) and apply the first op-timal control action

– cl-system under MPC: x(t + 1) = f(x(t), μN(x(t)))∗ Persistent feasible: x(t) ∈ X0, where X0 denotes the set of initial state for which PN is

feasible.

∗ Stability of MPC: cl-system asymptotically (or exponentially) stable

– Persistent feasibility and cl-stability are guaranteed if either of the following holds:

∗ Xf is control invariant and Jf is a local ECLF satisfying (8) on Xf ;

∗ Jf ≡ 0, Xf = Rn, and N is sufficiently large

∗ Jf is an ECLF satisfying (8) locally and N is sufficiently large

∗ Jf = V ncN0 and Xf is a constraint admissible positive invariant set of the unconstrained

cl-system under μncN0

∗ Jf = V ncN0 , Xf = R

n, and N is sufficiently large.

Relationship With Unconstrained Problem 31

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