Robust Adaptive Model Predictive Control: Performance and Parameter Estimation

For systems with uncertain linear models, bounded additive disturbances and state and control constraints, a robust model predictive control algorithm incorporating online model adaptation is proposed. Sets of model parameters are identified online and employed in a robust tube MPC strategy with a nominal cost. The algorithm is shown to be recursively feasible and input-to-state stable. Computational tractability is ensured by using polytopic sets of fixed complexity to bound parameter sets and predicted states. Convex conditions for persistence of excitation are derived and are related to probabilistic rates of convergence and asymptotic bounds on parameter set estimates. We discuss how to balance conflicting requirements on control signals for achieving good tracking performance and parameter set estimate accuracy. Conditions for convergence of the estimated parameter set are discussed for the case of fixed complexity parameter set estimates, inexact disturbance bounds and noisy measurements.


Introduction
Model Predictive Control (MPC) repeatedly solves a finite-horizon optimal control problem subject to input and state constraints. At each sampling instant a model of the plant is used to optimize predicted behaviour and the first element of the optimal predicted control sequence is applied to the plant [17]. Any mismatch between model and plant causes degradation of controller performance [4]. As a result, the amount of model uncertainty strongly affects the bounds of the achievable performance of a robust MPC algorithm [12].
To avoid the disruption caused by intrusive plant tests [4], adaptive Model Predictive Control attempts to improve model accuracy online while satisfying operating constraints and providing stability guarantees. Although the literature on adaptive control has long acknowledged the need for persistently exciting inputs for system identification [20], few papers have explored how to incorporate Persistency of Excitation (PE) conditions with feasibility guarantees within adaptive MPC [18]. In addition, adaptive MPC algorithms must balance conflicting requirements for system identification accuracy and computational complexity [18,21].
Various methods for estimating system parameters and meeting operating constraints are described in the adaptive MPC literature. Depending on the assumptions on model parameters, parameter identification methods such as recursive least squares [10], comparison sets [3], set membership identification [14,25] and neural networks [2,23] have been proposed. Heirung et al. [10] propose an algorithm where the unknown parameters are estimated using recursive least squares (RLS) and system outputs are predicted using the resulting parameter estimates. The use of RLS introduces nonlinear equality constraints into the optimisation. On the other hand, the comparison model approach [3] addresses the trade-off between probing for information and output regulation by decoupling these two tasks; a nominal model is used to impose operating constraints whereas performance is evaluated via a model learnt online using statistical identification tools. However the use of a nominal model implies that [3] cannot guarantee robust constraint satisfaction.
Tanaskovic et al. [25] consider a linear Finite Impulse Response (FIR) model with measurement noise and constraints. This approach updates a model parameter set using online set membership identification; constraints are enforced for the entire parameter set and performance is optimized for a nominal prediction model. The paper proves recursive feasibility but does not show convergence of the identified parameter set to the true parameters. To avoid the restriction to FIR models, Lorenzen et al. [14] consider a linear state space model with additive disturbance. An online-identified set of possible model parameters is used to robustly stabilize the system. However the approach suffers from a lack of flexibility in its robust MPC formulation, which is based on homothetic tubes [22], allowing only the centers and scalings of tube cross-sections to be optimized online, and it does not provide convex and recursively feasible conditions to ensure persistently exciting control inputs.
In this paper we also consider linear systems with parameter uncertainty, additive disturbances and constraints on system states and control inputs. Compared with [14], the proposed algorithm reduces the conservativeness in approximating predicted state tubes by adopting more flexible cross-section representations. Building on [16], we take advantage of fixed complexity polytopic tube representations and use hyperplane and vertex representations interchangeably to further simplify computation. We use, similarly to [10], a nominal performance objective, whereas constraints are imposed robustly on all possible models within the identified model set. We prove that the closed loop system is input-to-state stable (ISS). In comparison with the min-max approach of [16], the resulting performance bound takes the form of an asymptotic bound on the 2-norm of the sequence of closed loop states in terms of the 2-norms of the additive disturbance and parameter estimate error sequences. In addition, we propose a convex condition to ensure persistence of excitation (PE). This is included via a term in the cost function that allows the relative importance of the two objectives, namely controller performance and convergence of model parameters, to be specified.
Bai et al. [5] consider a particular set membership identification algorithm and show that the parameter set estimate converges with probability 1 to the actual parameter vector (assumed constant) if: (a) a tight bound on disturbances is known; (b) the input sequence is persistent exciting and (c) the minimal parameter set estimate is employed. However the minimal set estimate can be arbitrarily complex, and to provide computational tractability various non-minimal parameter set approximations have been proposed, such as n-dimensional balls [1] and bounded complexity polytopes [25]. The current paper allows the use of parameter set estimates with fixed complexity and proves that, despite their approximate nature, such parameter sets converge with probability 1 to the true parameter values. We also derive lower bounds on convergence rates for the case of inexact knowledge of the disturbance bounding set and for the case that model states are estimated in the presence of measurement noise. This paper has five main parts. Section 2 defines the problem and basic assumptions. Section 3 gives details of the parameter estimation, robust constraint satisfaction, nominal cost function, convexified PE conditions and the MPC algorithm. Section 4 proves recursive feasibility and input-to-state stability of the proposed algorithm. Section 5 proves the convergence of the parameter set in various conditions and Section 6 illustrates the approach with numerical examples.
Notation: N and R denote the sets of integers and reals, and N ≥0 = {n ∈ N : n ≥ 0}, N [p,q] = {n ∈ N : p ≤ n ≤ q}. The ith row of a matrix A and ith element of a vector a are denoted [A] i and [a] i . Vectors and matrices of 1s are denoted 1, and I is the identity matrix. For a vector a, a is the Euclidean norm and a 2 P = a ⊤ P a; the largest element of a is max a and [a] ≥0 = max{0, a}. The absolute value of a scalar s is |s| and the floor value is ⌊s⌋. |S| is the number of elements in a set S. A ⊕ B is Minkowski addition for sets A and B, and A ⊕ B = {a + b : a ∈ A, b ∈ B}. The matrix inequality A 0 (or A ≻ 0) indicates that A is positive semidefinite (positive definite) matrix. The k steps ahead predicted value of a variable x is denoted x k , and the more complete notation x k|t indicates the k steps ahead prediction at time t. A continuous function σ : R ≥0 → R ≥0 is a K-function if it is strictly increasing with σ(0) = 0, and is a K ∞ -function if in addition σ(s) → ∞ as s → ∞. A continuous function φ : R ≥0 × R ≥0 → R ≥0 is a KL-function if, for all t ≥ 0, φ(·, t) is a K-function, and, for all s ≥ 0, φ(s, ·) is decreasing with φ(s, t) → 0 as t → ∞. For functions σ a and σ b we denote σ a • σ b (·) = σ a σ b (·) , and σ k+1 a (·) = σ a • σ k a (·) with σ 1 a (·) = σ a (·).

Problem formulation and preliminaries
This paper considers a linear system with linear state and input constraints and unknown additive disturbance: where x t ∈ R nx is the system state, u t ∈ R nu is the control input, w t ∈ R nx is an unknown disturbance input, and t is the discrete time index. The system matrices A(θ * ) and B(θ * ) depend on an unknown but constant parameter θ * ∈ R p . The disturbance sequence {w 0 , w 1 , . . .} is stochastic and (w i , w j ) is independent for all i = j. States and control inputs are subject to linear constraints, defined for F ∈ R nc×nx , G ∈ R nc×nu by Assumption 1 (Additive disturbance). The disturbance w t lies in a convex and compact polytope W, where with Π w ∈ R nw×nx , π w ∈ R nw and π w > 0.
Assumption 2 (Parameter uncertainty). The system matrices A and B are affine functions of the parameter vector θ ∈ R p : for known matrices A j , B j , j ∈ N [1,p] , and θ * lies in a known, bounded, convex polytope Θ 0 given by Assumption 3 (State and control constraints). The set is compact and contains the origin in its interior.
To obtain finite numbers of decision variables and constraints in the MPC optimization problem, the predicted control sequences are assumed to have the dual mode form: where v 0|t , . . . , v N −1|t are optimization variables at time t and N is the prediction horizon. The gain K is designed offline and is assumed to robustly stabilize the uncertain system x t+1 = (A(θ) + B(θ)K)x t , ∀θ ∈ Θ 0 in the absence of constraints.
for all x ∈ {x : T x ≤ 1} and θ ∈ Θ 0 . The representation X = {x : T x ≤ 1} is assumed to be minimal in the sense that it contains no redundant inequalities.

Adaptive Robust MPC
In this section a parameter estimation scheme based on [9,26] is introduced. We then discuss the construction of tubes to bound predicted model states and associated constraints.

Set-based parameter estimation
At time t we use observations of the system state x t to determine a set ∆ t of unfalsified model parameters. The set ∆ t is then combined with the parameter set estimate Θ t−1 to construct a new parameter set estimate Θ t . Unfalsified parameter set: Define D t and d t as the matrix and vector Then, given x t , x t−1 , u t−1 and the disturbance set W in (2.3), the unfalsified parameter set at time t is given by Parameter set update: Let M Θ ∈ R r×p be an a priori chosen matrix. The estimated parameter set Θ t is defined by where µ t is updated online at times t ∈ N ≥0 . The complexity of Θ t is controlled by fixing M Θ , which fixes the directions of the half-spaces defining the parameter set. We assume that M Θ is chosen so that Θ t is compact for all µ t such that Θ t = ∅. Using a block recursive polytopic update method [9], Θ t is defined as the smallest set (3.4) containing the intersection of Θ t−1 and unfalsified sets ∆ j over a window of length N u : (where ∆ j = R for all j ≤ 0). We refer to N u as the PE window. Note that N u is independent of the MPC prediction horizon N . Using linear conditions for polyhedral set inclusion [8] µ t in (3.5) can be obtained by solving a linear program for each i ∈ N [1,r] : Lemma 1. If θ * ∈ Θ 0 and Θ t is defined by (3.4), (3.5), then θ * ∈ Θ t and Θ t ⊇ Θ t+1 ⊇ (Θ t ∩ ∆ t+1 ) for all t ∈ N ≥0 .

Polytopic tubes for robust constraint satisfaction
This section considers predicted state and control trajectories. To simplify notation, we omit the subscript t indicating the time at which state and control predictions are made; thus the k steps ahead predictions x k|t , v k|t are denoted x k , v k . To ensure that the predicted state and control sequences satisfy the operating constraints (2.2) robustly for the given uncertainty bounds, we construct a tube (sequence of sets) X 0 , X 1 , . . . satisfying, for all x ∈ X k , w ∈ W, θ ∈ Θ t , Hyperplane form: For given T ∈ R nα×nx satisfying Assumption 4 and α k ∈ R nα , let X k ⊂ R nx denote the k steps ahead cross section of the predicted state tube: We assume (without loss of generality) that α k is defined so that, for each i ∈ N [1,nα] , [T ] i x = [α k ] i holds for some x ∈ X k . Then (3.6) is equivalent to, for all x ∈ X k and θ ∈ Θ t , wherew is the vector with ith element [w] i = max w∈W [T w] i for all i ∈ N [1,nα] . Substituting D(x, u) and d(x, u) from (3.1), (3.2), this implies linear conditions on θ, for all x ∈ X k , θ ∈ Θ t : and, for a given initial state x, the constraint x ∈ X 0 requires Vertex form: X k has an equivalent representation in terms of its vertices, which we denote as x Since T is constant, the index set R j associated with active inequalities at the vertex x (j) is independent of α k and can be computed offline. Therefore, for each j ∈ N [1,m] , we have where the matrix U j ∈ R nx×nα can be computed offline given knowledge of R j using the property that Using the vertex representation (3.10), the condition that Furthermore condition (3.6) can be expressed equivalently as for all θ ∈ Θ t , j ∈ N [1,m] and k ∈ N ≥0 . This is equivalent [8,Prop. 3.31] to the requirement that there exist matrices Λ k,j satisfying, for each prediction time step k ∈ N ≥0 and each vertex j ∈ N [1,m] , the conditions Given the dual mode predicted control law (2.5), we introduce the terminal conditions that A(θ) + B(θ)K x + w ∈ X N and (F + GK)x ≤ 1 for all x ∈ X N , w ∈ W and θ ∈ Θ. Then (2.2) and (3.6) are satisfied if (3.12), (3.13) hold for all j ∈ N [1,m] and k ∈ N [0,N −1] , and there exist matrices Λ N,j satisfying the conditions, for all

Objective function
Consider the nominal cost defined for Q, R ≻ 0 by wherex k andū k are elements of predicted state and control sequences generated by a nominal parameter vector θ t :x Define P (θ) as the solution of the Lyapunov matrix equation We assume knowledge of an initial nominal parameter vectorθ 0 ∈ Θ 0 , which could be estimated using physical modeling or offline system identification, alternativelyθ 0 could be defined as the Chebyshev centre of Θ 0 . For t > 0, we assume thatθ t is updated by projectingθ t−1 onto the parameter set estimate Θ t , i.e.
Remark 2. For the stability analysis in Section 4 it is essential thatθ t ∈ Θ t . However, subject to this constraint, alternative update laws forθ t are possible; for example the weighted RLS estimate projected onto Θ t [14].

Augmented objective function and persistent excitation
The regressor D t in (3.1) is persistently exciting (PE) if for some PE window N u ∈ N >0 , some β 2 ≥ β 1 > 0, and all times t 0 [20]. Although the upper bound in (3.20) implies convex constraints on x t and u t , the lower bound is nonconvex. In [16] the PE condition (3.20) is defined on an interval {t − N u + 1, . . . , t}, where t is current time, which means the PE constraint is dependent on the first element of the control sequence u t . Here the PE condition is defined instead over predicted trajectories (from k = 0 to k = N u − 1 steps ahead), and we therefore require, for some β 1 > 0, Here k ) is a positive semidefinite matrix so by omitting this term we obtain sufficient conditions for (3.21) as a set of linear matrix inequalities in α k and v k : for all j ∈ N [1,m] . The bounds on convergence of the parameter set Θ t derived in Section 5 suggest faster convergence as β 1 in the PE condition (3.21) increases. In order to maximize the value of β, we therefore modify the MPC cost function: where γ ≥ 0 is a weight that controls the relative priority given to satisfaction of the PE condition (3.21) and tracking performance. The constraint (3.22) is not enforced if γ = 0. The relationship between γ and β is explored in Section 6.

Proposed algorithm
Offline: 1. Choose suitable T defining the predicted state tube and compute the corresponding U j in (3.11).
3. Minimise the contracitivity factor λ satisfying (2.6) and obtain a feedback gain K.
1. Obtain the current state x t and set x = x t .

Implement the current control input
Remark 3. At t = 0, the reference sequencesx = {x 0 , . . . ,x N } andû = {û 0 , . . . ,û N −1 } may be computed by solving Remark 4. The online computation of the proposed algorithm may be reduced by updating Θ t only once every In Section 4 we use the property that Θ t ⊆ Θ t−1 to show that the solution, v * t−1 , of P at time t − 1 forms part of a feasible solution of P at time t. As a result, the reference sequencesx,û in Step 3 are feasible for problem P at all times t > 0.

Input-to-state stability (ISS)
Throughout this section we set γ = 0 in problem P. Therefore the objective of P is J(x, v,θ t ) where J is the nominal cost (3.18). To simplify notation we define a stage cost L(x, v) and terminal cost φ(x, θ) as Denoting the actual state at next time step as x + , we define the function f (x, v, w, θ * ) as Lemma 6 (ISS-Lyapunov function [13]). The system with control law v = v(x,θ t , t) is ISS with region of attraction X ∈ R n if the following conditions are satisfied. (i). X contains the origin in its interior and is a robust positively invariant set for (4.1), i.e. f x, v(x,θ t , t), w, θ ∈ X for all x ∈ X , w ∈ W, θ ∈ Θ and t ∈ N ≥0 . (ii). There exist K ∞ -functions ς 1 , ς 2 , ς 3 , K-functions σ 1 , σ 2 and a function V : X × N ≥0 → R ≥0 such that for all t ∈ N ≥0 , V(·, t) is continuous, and for all (x, t) ∈ X × N ≥0 , In the following we define X P as the set of states x such that problem P is feasible and assume that X P is non-empty. In addition, for a given state x, nominal parameter vectorθ and parameter set Θ, we denote v * (x,θ, Θ) as the optimal solution of problem P, and let V * (x,θ, Θ) be the corresponding optimal value of the cost in (3.18), so that V * (x,θ, Θ) = J(x, v * (x,θ, Θ),θ).
Theorem 7. Assume that γ = 0 and the nominal parameter vectorθ t is not updated, i.e.θ t =θ 0 for all t ∈ N ≥0 . Then for all initial conditions x 0 ∈ X P , the system (2.1) with control law is the first element of v * (x t ,θ t , Θ t ), robustly satisfies the constraint (2.2) and is ISS with region of attraction X P .
Proof. We first show that condition (i) of Lemma 6 is satisfied with X = X P . If P is feasible at t = 0, then (3.14) implies that α N exists such that is feasible for P for all x 0 ∈ X N , and hence X N ⊆ X P . Furthermore, the robust invariance of X N implied by (4.4) ensures that W ⊆ X N , and since 0 ∈ int(W) due to Assumption 1, X P must contain the origin in its interior. Proposition 5 shows that if P is initially feasible, then it is feasible for all t ≥ 0. It follows that condition (i) of Lemma 6 is satisfied if X = X P . We next consider the bounds (4.2) in condition (ii) of Lemma 6. For a given state x, nominal parameter vectorθ and parameter set Θ, problem P with γ = 0 and Q, R ≻ 0 is a convex quadratic program. Therefore V * (x,θ, Θ) is a continuous positive definite, piecewise quadratic function of x [7] for eachθ ∈ Θ 0 and Θ ⊆ Θ 0 . Furthermore Θ 0 and X P are compact due to Assumptions 2 and 3, and it follows that there exist K ∞ -functions ς 1 , ς 2 such that (4.2) holds with V(x, t) = V * (x,θ t , Θ t ).
To show that the bound (4.3) in condition (ii) of Lemma 6 holds, let F P denote the set {v : (x, Kx + v) ∈ Z, x ∈ X P }. Then, given the linear dependence of the system (2.1), the model parameterisation (2.4) and the predicted control law (2.5) on the state x, disturbance w and parameter vector θ, and since W, Θ 0 , X P and F P are compact sets by Assumptions 1, 2 and 3, there exist K ∞ functions σ x , σ w , σ θ , σ L , σ φ such that, ∀x, x 1 , x 2 ∈ X P , ∀v ∈ F P , ∀w, w 1 , w 2 ∈ W, ∀θ, From the triangle inequality we have Collecting these bounds and using the weak triangle inequality for K-functions [24] we obtain •2σ θ (( s )), and both σ 1 and σ 2 are K-functions. Sincev is a feasible but suboptimal solution of P at x + , and sinceθ t+1 =θ t by assumption, the optimal cost function satisfies V(x + , t + 1) = V * (x + ,θ t+1 , Θ t+1 ) ≤ J(x + ,v,θ t ) and hence Thus all conditions of lemma 6 are satisfied.
Corollary 8. Assume that γ = 0 and the nominal parameter vectorθ t is updated at each time t ∈ N ≥0 using (3.19). Then for all initial conditions x 0 ∈ X P , the system (2.1) with control law u t = Kx t + v * 0|t , where v * 0|t is the first element of v * (x t ,θ t , Θ t ), robustly satisfies the constraint (2.2) and is ISS with region of attraction X P .
Proof. It can be shown that condition (i) of Lemma 6 and the bounds (4.2) in condition (ii) of Lemma 6 are satisfied with X = X P and V(x, t) = V * (x,θ t , Θ t ) for some K ∞ -functions ς 1 , ς 2 using the same argument as the proof of Theorem 7. To show that (4.3) is also satisfied and hence complete the proof we use an argument similar to the proof of Theorem 7. In particular, as before we definex * = {x * 0 , . . . ,x * N } using the optimal solution of P, v * (x,θ t , Θ t ) = {v * 0 , . . . , v * N −1 }, and However, here we define z = {z 0 , . . . , z N } as the sequence (4.5) The update law (3.19) ensures that θ t+1 −θ t ≤ θ t − θ * since θ * ∈ Θ t+1 . Hence, for all k ∈ N [1,N −1] , we have and it follows that, for all k ∈ N [1,N −1] , Furthermore (3.17) is linear in P (θ) and the solution P (θ) is unique for all θ ∈ Θ t (see e.g. [11]) since A(θ) + B(θ)K is by assumption stable. Therefore, by the implicit function theorem, P (θ) is Lipschitz continuous and

Remark 9.
The input-to-state stability property implies that there exists a KL-function η(·, ·) and K-functions ψ(·) and ζ(·) such that for all feasible initial conditions x 0 ∈ X P , the closed loop system trajectories satisfy, for all t ∈ N ≥0 ,

Convergence of the estimated parameter set
In terms of D and d defined in (3.1) and (3.2), the system model x t+1 = A(θ * )x t + B(θ * )u t + w t can be rewritten as where x t+1 , D(x t , u t ) and d(x t , u t ) are known at time t + 1. Thus, the system is linear with regressor D t , uncertain parameter vector θ * and additive disturbance w t ∈ W. Bai et al. [5] show that, for such a system, the diameter of the parameter set constructed using a setmembership identification method converges to zero with probability 1 if the uncertainty bound W is tight and the regressor D t is persistently exciting. We extend this result and prove convergence of the estimated parameter set in more general cases. Specifically, in this paper we avoid the problem of computational intractability arising from a minimum volume update law of the form Θ t+1 = Θ t ∩ ∆ t . Instead, we derive stochastic convergence results for parameter sets with fixed complexity and update laws of the form Θ t+1 ⊇ Θ t ∩ ∆ t .
In this section we first discuss relevant results for an update law that gives a minimal parameter set estimate for a given sequence of states (but whose representation has potentially unbounded complexity), before considering convergence of the fixed-complexity parameter set update law of Section 3.1. We then compute bounds on the parameter set diameter if the bounding set for the additive disturbances is overestimated. Lastly, we demonstrate that similar results can be achieved when errors are present in the observed state, as would be encountered for example if the system state were estimated from noisy measurements.
In common with Bai et al. [5,6], we do not assume a specific distribution for the disturbance input. However, the set W bounding the model disturbance is assumed to be tight in the sense that there is non-zero probability of a realisation w t lying arbitrarily close to any given point on the boundary, ∂W, of W.
Assumption 6 (Persistent Excitation). There exist positive scalars τ , β and an integer N u ≥ ⌈p/n x ⌉ such that, for each t ∈ N ≥0 we have D t ≤ τ and We further assume throughout this section that the rows of M Θ are normalised so that [M Θ ] i = 1 for all i.

Minimal parameter set
The unfalsified parameter set at time t defined in (3.3) can be expressed as where w t is the disturbance realisation at time t and D t = D(x t , u t ). Let w 0 be an arbitrary point on the boundary ∂W, then the normal cone N W (w 0 ) to W at w 0 is defined Proposition 10. For all t ∈ N ≥0 , all ǫ > 0, and for any θ ∈ R p such that θ * − θ ≥ ǫ, under Assumptions 1, 5 and 6 we have Proof. Assumption 1 implies that there exists w 0 ∈ ∂W so that D j (θ * − θ) ∈ N W (w 0 ) for any given j ∈ ∈ W, and hence θ / ∈ ∆ j+1 from (5.2). But Hence, if θ * − θ ≥ ǫ, then there must exist some j ∈ N [t,t+Nu−1] such that If w j − w 0 < ǫ β/N u , then it follows that w j − w 0 < ǫ β/N u ≤ D j (θ * − θ) and thus θ / ∈ ∆ j+1 . Assumption 5 implies the probability of this event is at least p w ǫ β/N u .

Fixed complexity parameter set
In order to reduce computational load and ensure numerical tractability, we assume that the parameter set Θ t is defined by a fixed complexity polytope, as in (3.4) and (3.5). This section shows that, although a degree of conservativeness is introduced by fixing the complexity of Θ t , asymptotic convergence of this set to the true parameter vector θ * still holds with probability 1.
Theorem 13. If Θ t is updated according to (3.4), (3.5) and Remark 4, and Assumptions 1, 5 and 6 hold, then for all θ ∈ Θ 0 such that [M Θ ] i (θ − θ * ) ≥ ǫ for some i ∈ N [1,r] , we have, for all t ∈ N ≥0 and any ǫ > 0, Consider therefore the probability that any given θ ∈ Θ t−Nu satisfying Assumption 1 implies that, for any given g j ∈ R nx , there exists a w 0 j ∈ ∂W such that g j ∈ N W (w 0 j ). Accordingly, choose w 0 j ∈ ∂W so that g j in (5.4) satisfies g j ∈ N W (w 0 j ) for each j ∈ N [t−Nu,t−1] . Then is a necessary condition for θ ∈ ∆ j+1 due to (5.2) and (5.3). But (5.4) and Assumption 6 imply where [M Θ ] i (θ−θ * ) ≥ ǫ by assumption, and it follows from . From Assumption 5 and the independence of the sequence {w 0 , w 1 , . . .} we therefore conclude that Proof. By applying the Borel-Cantelli Lemma to Theorem 11 it can be shown (analogously to the proof of [1,r] and ǫ > 0. Since M Θ is assumed to be chosen so that Θ t is compact for all µ t such that Θ t is non-empty, it follows that Θ t → {θ * } as t → ∞ with probability 1.

Inexact disturbance bounds
We next consider the case in which the set W bounding w t in Assumption 1 does not satisfy Assumption 5. Instead, we assume that a compact set Ω providing a tight bound on w t exists but is either unknown or nonpolytopic or nonconvex. We define the unit ball B = {x : x ≤ 1} and use a scalar ρ to characterize the accuracy to which W approximates Ω.
Remark 15. Assumption 7 implies that W ⊖ Ω ⊆ ρB. As a result, every point in W can be a distance no greater than ρ from a point in Ω, i.e. maxŵ ∈W min w∈Ω ŵ − w ≤ ρ.

System with measurement noise
Consider the system model with an unknown parameter vector θ * and measurement noise s t : where y t ∈ R nx is a measurement (or state estimate) and the noise sequence {s 0 , s 1 , . . .} has independent elements satisfying s t ∈ S for all t ∈ N ≥0 .
Due to the measurement noise, the unfalsified parameter set must be constructed at each time t ∈ N ≥0 using the available measurements y t , y t−1 , the known control input u t−1 , and sets W and S bounding the disturbance and the measurement noise. To be consistent with (5.6), θ * must clearly lie in the set {θ : y t − D(y t−1 − s t−1 , u t−1 )θ − d(y t−1 − s t−1 , u t−1 ) ∈ W ⊕ S}, and the smallest unfalsified parameter set based on this information is given by Thus Assumption 8 implies that the unfalsified set ∆ t is a convex polytope and the parameter set Θ t can be estimated using, for example, the update law (3.4), (3.5) if S is known.
Given Assumptions 8 and 9, the results of Sections 5.1 and 5.2 apply with minor modifications. Define ξ t = w t + s t , then Assumption 9 implies Pr ξ t − ξ 0 t < ǫ ≥ p w,s ǫ/ √ 2 for any given ξ 0 t = w 0 t + s 0 t with w 0 t ∈ ∂W and s 0 t ∈ ∂S. This implies the following straightforward extensions of Theorems 11 and 13 and Corollaries 12 and 14.

Numerical examples
This section presents simulations to illustrate the operation of the proposed adaptive robust MPC scheme. The section consists of two parts. The first part investigates the effect of additional weight γ in optimization problem P by using the example of a second-order system from [15]. The second part demonstrates the relationship between the speed of parameter convergence and minimal eigenvalue β from the PE condition.

Objective function with weighted PE condition
Consider the second-order discrete-time uncertain linear system from [15], with model parameters Simulations were performed in Matlab on a 3.4 GHz Intel Core i7 processor. Problem P was solved using Mosek [19]. For T with 9 rows, MPC prediction horizon N = 10, and PE window length N u = 2, the average solve time for each iteration was 0.0259 s. Figure 1 shows the effect of the weighting coefficient γ in the objective function (3.23) on the parameter set Θ t when the same initial conditions, x 0 , Θ 0 and disturbance sequences {w 0 , w 1 , . . .} are used. Larger values of γ place greater weighting on β in the MPC cost (3.23), and thus on satisfaction of the PE condition (3.21). Therefore increasing γ results in a faster convergence rate in the parameter set volume. The relationship between weighting coefficient γ and volume of parameter set Θ t is illustrated in Figure 2. For values of γ between 10 −3 and 10 3 , closed loop simulations were performed with the same initial conditions, disturbance sequences, and initial nominal model and parameter set. The parameter set volume after 20 timesteps is shown. Figure 2 also shows that increasing γ results in a faster parameter set convergence rate, in agreement with Figure 1. For the same set of simulations, Figure 3 plots the optimal value of β in (3.22) and (3.21) against γ. From (3.23), It is expected that a larger γ value will penalize the −γβ term more, which pushes β to be more positive. The left graph shows the value of β in the convexified constraint (3.22). As expected, the increase in γ leads to a smooth increase in β value initially, and after a certain point, any further increase in the weighing factor γ does not affect the calculated β value. The right graph shows the value of β 1 in the PE condition of (3.21). It is interesting to note that, although the optimal value of β in (3.22) levels off at γ = 1, the value of β 1 in the PE condition (3.21) increases monotonically between γ = 10 and γ = 10 3 . The smaller β values observed with (3.21) also explain the lower rates of parameter convergence for small values of γ in Figure 2. In practice, the bottom graph can be used as a guideline for the tuning of γ.

Relationship between PE coefficient and convergence rate
We next consider third-order discrete-time linear systems given by (2.1) with x ∈ R 3 , u ∈ R 2 , θ ∈ R 3 and W = {w : w ∞ ≤ 0.1}.
The system matrices A(θ), B(θ) satisfy (2.4) with randomly generated A i , B i , θ * parameters and initial parameter set Θ 0 = {θ : θ ∞ ≤ 0.25}. In each case the estimated parameter sets Θ t have fixed complexity, with face normals aligned with the coordinate axes in parameter space. A linear feedback law is applied, u t = Kx t , where K is a stabilizing gain. We use these systems to investigate the relationship between the coefficient β 1 in the PE condition (3.21) and rate of convergence of the estimated parameter set.
Taking the window length in (3.21) to be N u = 10, closed-loop trajectories were computed for 10 time steps and the parameter set Θ t was updated according to (3.5). Simulations were performed for 500 different initial conditions, and the average value of β 1 was computed for each initial condition using 100 random disturbance sequences {w 0 , w 1 , . . .}. Figure 4 illustrates the relationship between the average size of the identified parameter set Θ t and the average value of β 1 in the PE condition (3.21). Clearly, increasing β 1 results in a smaller parameter set on average, and hence a faster rate of convergence of Θ t , which is consistent with the analysis of Section 5.2. The inner and outer radii shown in the figure on the left are the radii of the smallest and largest spheres, respectively, that contain and are contained within the parameter set estimate after 10 time steps. A similar trend can also be seen between the average volume of the parameter set Θ and the ensemble average value of β 1 .

Conclusions
In this paper we propose an adaptive robust MPC algorithm that combines robust tube MPC and set membership identification. The MPC formulation employs a nominal performance index and guarantees robust constraint satisfaction, recursive feasibility and input-to-state stability. A convexified persistent excitation condition is included in the MPC objective via a weighting coefficient, and the relationship between this weight and the convergence rate of the estimated parameter set has been investigated. For computational tractability, a fixed complexity polytope is used to approximate the estimated parameter set. The paper proves that the parameter set will converge to the vector of system parameters with probability 1 despite this approximation.
Conditions for convergence of the estimated parameter set are derived for the case of inexact disturbance bounds and noisy measurements. Future work will consider systems with stochastic model parameters and probabilistic constraints. In addition, the quantitative relationship between convergence rates of the estimated parameter sets and conditions for persistency of excitation will be investigated further.