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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
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ECE7850 Wei Zhang
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
ECE7850 Wei Zhang
• 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
ECE7850 Wei Zhang
– 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
ECE7850 Wei Zhang
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
ECE7850 Wei Zhang
• 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
ECE7850 Wei Zhang
• 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
ECE7850 Wei Zhang
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