Penalized identification methods for self-tuning control

S. Bittanti Dipartimento di Elettronica e Informazione Politecnico di Milano Piazza Leonardo da Vinci 32 - 20133 Milano, Italy bittanti@elet.polimi.it

M.C. Campi Dipartimento di Elettronica per l’Automazione Universita’ degli Studi di Brescia Via Branze 38, 25123 Brescia, Italy campi@ing.unibs.it

Abstract

We consider the self-tuning control system based on the recursive least squares identification method and discuss various modifications of the identification algorithm all based on the introduction of penalization terms in the identification cost. It is shown that penalizing the identification cost may have beneficial effects in many situations, ranging from the exploitation of apriori knowledge on the system parameter to the possibility of forcing optimality in the resulting adaptive control scheme. A particular attention is devoted in this paper to the case in which the a-priori knowledge that the true parameter lies in a sphere is available. It is shown that a penalization term can be designed for this case so that the estimate tends in the long run to the sphere, while preserving the asymptotic properties of the standard least squares identification method. Moreover, the corresponding algorithm has a recursive, easy-to-implement, structure.

1 Penalized identification

Consider a discrete time stochastic SISO system described by the following ARX model

A(ϑ^{∘}; q^{−1}) y_{t} = B(ϑ^{∘}; q^{−1}) u_{t} + n_{t},  (1)

where A(ϑ^{∘}; q^{−1}) and B(ϑ^{∘}; q^{−1}) are polynomials in the unit-delay operator q^{−1} depending on the system parameter vector ϑ^{∘} = [ a_{1}^{∘} a_{2}^{∘} … a_{n}^{∘} b_{d}^{∘} b_{d+1}^{∘} … b_{d+m}^{∘}]^{T}. Precisely, they are given by

A(ϑ^{∘}; q^{−1}) = 1 − ∑_{i=1}^{n} a_{i}^{∘}q^{−i}

and

B(ϑ^{∘}; q^{−1}) = ∑_{i=d}^{d+m} b_{i}^{∘}q^{−i}.

As for the disturbance term {n_{t}}, we assume that it is described as a martingale difference sequence with respect to a filtration {F_{t}}, such that

sup_{t} E[|n_{t+1}|^{β}/F_{t}] < ∞, a.s., for some β > 2, (2)

and

lim inf_{t→∞} \frac{1}{t} ∑_{k=1}^{t} n_{k}^{2} > 0.  (3)

These are standard assumptions in adaptive control.

Letting

φ_{t} = [ y_{t} … y_{t−(n−1)} u_{t−(d−1)} … u_{t−(d+m−1)} ]^{T} (4)

be the observation vector, system (1) can be given the usual regression-like form

y_{t} = φ_{t−1}^{T}ϑ^{∘} + n_{t}.  (5)

The classical least squares identification criterion consists in minimizing the performance index:

V_{t}(ϑ) = ∑_{k=1}^{t}(y_{k} − φ_{k−1}^{T}ϑ)^{2}.  (6)

The properties of the minimizer of (6) have been deeply studied. One of the most general and powerful result, which does not assume any excitation assumption and thereby is often resorted to in an adaptive control context, is due to Lai and Wei [14]:

∥ϑ^{∘} − ϑ̂t^{LS}∥^{2} = O (\frac{log(∑_{k=1}^{t} ∥φ_{k−1}∥^{2})}{λ_{min}(∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T})}) a.s. (7)

Roughly, this formula relates the estimation error to the distribution of the information gleaned from data in the parameter space. This property has been used for obtaining stability and performance results for selftuning control systems based on the certainty equivalence principle, see e.g. [10, 9, 22].

On the other hand, there is a main drawback in the use of the raw least squares performance index: it does not incorporate any possible a-priori desire that the parameter estimate lies in a certain region in the parameter space. There are many possible motivations for desiring that the estimate lies in certain regions, as described below.

i) It is not rare the situation in which the system parameter has a physical interpretation and, consequently, must satisfy certain constraints. More generally, some a-priori knowledge on the system parameter can be available and one would like to take care of this knowledge;

ii) in order that the adaptive control system works correctly, it is well known that the estimated model has to satisfy certain restrictions; a typical one is that it has to be controllable;

iii) in adaptive control, the ideal goal one would like to achieve is that the adaptively tuned control system attains in the long run an optimal performance, namely the one that would be achieved if the system dynamics were known at the start. Also this objective can be reformulated in the requirement that the estimated system parameter lies in certain regions of the parameter space, as better explained in Section 2.

One way to take into account the desire that the estimate lies in a certain region is to add a suitable penalization term in the least squares performance index, so obtaining the penalized cost:

D_{t}(ϑ) = V_{t}(ϑ) + P_{t}(ϑ),  (8)

where P_{t}(ϑ) is the penalization term which may take a variety of forms depending on the problem at hand.

In the selection of P_{t}(ϑ) some general objectives should be taken into consideration:

1. the estimate must be driven into the desired region;

2. the nice identification properties of the least squares estimate should be preserved;

3. very importantly, the resulting algorithm should be easy-to-implement, and - possibly - recursive.

2 A glimpse through the penalization approach

The penalization approach can be used in different situations. Herein, we provide a quick glimpse through some significant examples.

The problem of designing the penalization term P_{t}(ϑ) such that the corresponding estimate is uniformly controllable (problem ii) above) has been addressed in [21]. The approach in [21] can be easily extended to other situations in which one desires to force the estimate to belong to a generic region in the parameter space. The minimization of the resulting cost is however not trivial since no recursive algorithm for the minimization exists. On the other hand, it is generally true that minimizing a penalized cost over the entire parameter space may be easier than minimizing a non-penalized cost over a given feasible region. Therefore, the penalized approach may gain an advantage over other methods based on a constrained minimization approach. We refer the reader to [21] for a more comprehensive discussion.

Problem iii) above (achieving optimality in adaptive control) has been addressed along the penalized approach in e.g. [6]. It is well known that the classical self-tuning control scheme can suffer from an identifiability problem and that this can result in a degradation of the control system performance; see [11], [2], [13]. The basic reason why this happens can be easily explained. Generically, it can be shown that the least squares parameter estimate converges to a limiting random variable. Such a limiting estimate results in a limiting controller and it is natural to expect that the least squares estimator will asymptotically identify, at a minimum, the closed-loop behavior of the system. Thus, one expects that the behavior of the true system with the loop closed by the adaptively tuned controller will be the same as the closed-loop estimated system. In turn, this implies that the achieved control performance will be the same as the performance of the closed-loop estimated system. The latter is, however, the optimal configuration for the estimated system (in fact, the controller is selected on the basis of the estimated system), while the former is not an optimal configuration for the true system. Thus, one has:

optimal control cost for the estimated system = achieved control cost ≥ optimal control cost for the true system.

The fact that the optimal control cost for the estimated system is in general greater than or equal to the optimal control cost for the true system means that the least squares estimator has a natural tendency to return estimates with larger optimal control cost than the optimal cost associated with the true system. This motivates the idea of introducing a penalization term in the least squares estimator so that it favors parameters with smaller values of the optimal control cost. In [6] it is shown that this penalization term can in fact be selected in such a way that this tendency is effectively counteracted, while preserving the least squares identification properties. The resulting adaptive control scheme exhibits optimality. However, again the identification cost has no recursive algorithm associated with it.

One peculiar aspect of the aforementioned optimality problem is that the ”desired region” in the parameter space is not known in advance in this particular case (it is the region where the optimal control cost is no larger than the optimal cost for the unknown true system). As a consequence, no identification method based on a feasible minimization region can be conceived in this case.

In other situations, one would simply like to incorporate in the identification algorithm the a-priori available information on the true parameter (problem i)). In this paper, we discuss in detail this problem in the case in which the a-priori information has a specific structure: the true parameter lies in a sphere. This is addressed in detail in the next section, whereas, due to space limitations, the reader is referred to the cited literature for the other aforementioned problems.

3 Penalized identification in the case when the true parameter lies in a sphere

Suppose that ϑ^{∘} belongs to a known open sphere with radius r. The center of this sphere is called the nominal parameterization and denoted by ϑ̄: ϑ^{∘} ∈ S(ϑ̄, r) = {ϑ : ∥ϑ − ϑ̄∥ < r}. This situation has been studied in [19].

A typical situation in which such a condition turns out to be natural is when some estimate ϑ̄ of ϑ^{∘} is a-priori obtained either by physical knowledge of the plant or by some coarse off-line identification procedure. The identification algorithm is then used to refine the parameter estimate during the normal operation of the control system so as to better tune the controller to the actual plant characteristics.

In order to incorporate the prior knowledge in the identification algorithm, a suitable penalization term can be introduced in the identification performance index. With the a-priori knowledge specified above, a natural way to pursue this objective is to select a penalization term of the form:

P_{t}(ϑ) = α_{t}∥ϑ − ϑ̄∥^{2}.  (9)

Here, α_{t} is a weighting factor. A suitable selection of this coefficient plays a crucial role in the determination of the algorithm properties; this issue is discussed in detail in Section 4.

The advantage of taking a penalization term which is quadratic in the unknown parameter is that it naturally leads to a recursive algorithm. Precisely, the recursive algorithm is:

1. Compute P_{t} according to the following steps:

set T_{0} = P_{t−1};

for i = 1 to n + m + 1, set ϕ_{i} = [0 … \begin{matrix}i \\ ↓ \end{matrix} ; 1 … 0]^{T} and compute

T_{i} = T_{i−1} − \frac{(α_{t} − α_{t−1})T_{i−1}ϕ_{i}ϕ_{i}^{T} T_{i−1}}{1 + (α_{t} − α_{t−1})ϕ_{i}^{T} T_{i−1}ϕ_{i}};

then,

P_{t} = T_{n+m+1} − \frac{T_{n+m+1}φ_{t−1}φ_{t−1}^{T}T_{n+m+1}}{1 + φ_{t−1}^{T}T_{n+m+1}φ_{t−1}} .

2. Compute the estimate ϑ̂_{t} according to the equation:

ϑ̂_{t} = ϑ̂_{t−1} + P_{t}φ_{t−1}(y_{t} − φ_{t−1}^{T}ϑ̂_{t−1}) + P_{t}(α_{t} − α_{t−1})(ϑ̄ − ϑ̂_{t−1}).

It is easy to see that the above recursive algorithm in fact minimizes the identification cost

D_{t}(ϑ) = ∑_{k=1}^{t}(y_{k}−φ_{k−1}^{T}ϑ)^{2}+(ϑ−ϑ̄)^{T} Q(ϑ−ϑ̄)+α_{t}∥ϑ−ϑ̄∥^{2},

provided that the algorithm is initialized as follows:

\{ \begin{matrix}ϑ̂_{0} = ϑ̄ \\ P_{0} = [Q + α_{0}I]^{−1}. \end{matrix}  (11)

The term (ϑ−ϑ̄)^{T} Q(ϑ−ϑ̄) comes from the initialization and can be interpreted as a regularization term. Coefficient α_{t} should be selected so as to achieve a fair compromise between two opposite goals. On the one hand, it should grow rapidly enough so as to ensure that the penalty for the estimates far away from the nominal parameter be effective indeed. On the other hand, one should be careful to avoid an over-penalization, since it would destroy the crucial property (7) of the least squares algorithm. The selection of the weighting factor α_{t} is the subject of the next section.

4 Choice of the weighting factor

Suppose that the weighting factor is selected as follows:

α_{t} = (log (∑_{k=1}^{t} ∥φ_{k−1}∥^{2} + tr(Q)))^{1+δ} ,  (12)

for some prespecified and arbitrary δ > 0. The weighting factor depends on the total amount of information conveyed by the data up to time t. From an algorithmic point of view, this is equivalent to complement point 2 of the algorithm above with: α_{t} = (log(r_{t}))^{1+δ}, where r_{t} = r_{t−1} + ∥φ_{t−1}∥^{2}, r_{0} = tr(Q).

The following result proves that the choice (12) provides a fair balancing between the two above mentioned contrasting objectives of asserting the penalization term and preserving the least squares identification properties.

Theorem Assume that u_{t} is F_{t}-measurable (this simply means that the control law is causal) and that n > 0 (i.e the autoregressive part in system (1) is nontrivial; this technical assumption can be removed). Then,

1. there exists a finite time instant t̄ such that ϑ̂_{t} ∈ S(ϑ̄, r), t ≥ t̄, a.s.; 2. the identification error satisfies the following

bound

∥ϑ^{∘}−ϑ̂_{t}∥^{2} = O (\frac{(log(∑_{k=1}^{t} ∥φ_{k−1}∥^{2}))^{1+δ}}{λ_{min}(∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T})} ) a.s.

(13)

Proof.

Part 1.)

Denote by ϑ̄t^{LS} the minimizer of the least squares performance index with regularization term V_{t}(ϑ) + (ϑ − ϑ̄)^{T} Q(ϑ − ϑ̄) and set

Q_{t} = ∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T} + Q.  (14)

It is then easy to show that ϑ̂_{t} = arg min_{ϑ∈R^{n+m+1}} D_{t}(ϑ)

can be expressed as a function of ϑ̄t^{LS} as follows

ϑ̂_{t} = (Q_{t} + α_{t}I)^{−1}Q_{t}ϑ̄t^{LS} + α_{t}(Q_{t} + α_{t}I)^{−1}ϑ̄.

By subtracting ϑ̄, we get

ϑ̂_{t} − ϑ̄ = (Q_{t} + α_{t}I)^{−1}Q_{t}(ϑ^{∘} − ϑ̄)

+(Q_{t} + α_{t}I)^{−1}Q_{t}(ϑ̄t^{LS} − ϑ^{∘}).

Thus, the norm of ϑ̂_{t} − ϑ̄ can be upper bounded as follows

∥ϑ̂_{t}−ϑ̄∥ ≤ ∥ϑ^{∘}−ϑ̄∥+∥(Q_{t}+α_{t}I)^{−1}Qt^{\frac{1}{2}} ∥∥Qt^{\frac{1}{2}} (ϑ̄t^{LS}−ϑ^{∘})∥.

(15)

We apply now Theorem 1 in reference [14] so as to upper bound the term ∥Qt^{\frac{1}{2}} (ϑ̄t^{LS} − ϑ^{∘})∥. Since u_{t} is assumed to be F_{t}-measurable, and also considering Assumption (2), by this theorem we obtain the following upper bound:

∥Qt^{\frac{1}{2}} (ϑ̄t^{LS} − ϑ^{∘})∥^{2} = O(log(tr(Q_{t}))), a.s.. (16)

The term ∥(Q_{t} + α_{t}I)^{−1}Qt^{\frac{1}{2}} ∥ can instead be handled as follows. Denote by {λ_{1,t}, …, λ_{n+m+1,t}} the eigenvalues of the positive definite matrix Q_{t}. Since Q_{t} is symmetric and positive definite, there exists an orthonormal matrix T_{t} such that Q_{t} = T_{t} diag(λ_{1,t}, …, λ_{n+m+1,t})T_{t}^{−1} and

Qt^{\frac{1}{2}} = T_{t} diag (λ1,t^{\frac{1}{2}}, …, λn+m+1,t^{\frac{1}{2}}) T_{t}^{−1}. Then,

(Q_{t} + α_{t}I)^{−1}Qt^{\frac{1}{2}}

= T_{t}(T_{t}^{−1}(Q_{t} + α_{t}I)T_{t})^{−1}T_{t}^{−1}Qt^{\frac{1}{2}} T_{t}T_{t}^{−1}

= T_{t} diag (\frac{λ1,t^{\frac{1}{2}}}{λ_{1,t} + α_{t}} , …, \frac{λn+m+1,t^{\frac{1}{2}}}{λ_{n+m+1,t} + α_{t}} ) T_{t}^{−1}.

This implies that

∥(Q_{t} + α_{t}I)^{−1}Qt^{\frac{1}{2}} ∥ = max_{i=1,...,n+m+1} (\frac{λi,t^{\frac{1}{2}}}{λ_{i,t} + α_{t}} ) . (17)

Consider now the function: f(x) = \frac{x^{\frac{1}{2}}}{x + α_{t}} , x ≥ 0. Such

a function has an absolute maximum value \frac{1}{2}α_{t}^{− \frac{1}{2}} in x = α_{t}. It then obviously follows from equation (17) that

∥(Q_{t} + α_{t}I)^{−1}Qt^{\frac{1}{2}} ∥ ≤ \frac{1}{2}α_{t}^{− \frac{1}{2}} .  (18)

Substituting the estimates (16) and (18) in equation (15), we obtain

∥ϑ̂_{t} − ϑ̄∥ ≤ ∥ϑ^{∘} − ϑ̄∥ + h_{1} (\frac{log(tr(Q_{t}))}{α_{t}})^{\frac{1}{2}} ,

h_{1} being a suitable constant. Observe now that from equation (5) it follows that n_{k}^{2} ≤ 2 max{∥ϑ^{∘}∥^{2}, 1}[ y_{k}^{2} + ∥φ_{k−1}∥^{2} ]. Taking into account that the autoregressive part of model (5) is not trivial (n > 0), this in turn implies that n_{k}^{2} ≤ 2 max{∥ϑ^{∘}∥^{2}, 1}[ ∥φ_{k}∥^{2} + ∥φ_{k−1}∥^{2} ], from which it is easily shown that

∑_{k=1}^{t} n_{k}^{2} ≤ h_{2} ∑_{k=1}^{t+1} ∥φ_{k−1}∥^{2},

where h_{2} is a suitable constant. From Assumption (3) and definition (14) of Q_{t}, we then get

lim_{t→∞} tr(Q_{t}) = ∞.

Since by definition (12) α_{t} = (log(tr(Q_{t})))^{1+δ}, we then obtain that ∀ ϵ > 0 there exists a time instant τ such that ∥ϑ̂_{t} − ϑ̄∥ ≤ ∥ϑ^{∘} − ϑ̄∥ + ϵ, ∀ t ≥ τ. By the a-priori assumption that ϑ^{∘} belongs to the sphere S(ϑ̄, r), this implies that there exists a finite time instant t̄ such that ϑ̂_{t} ∈ S(ϑ̄, r), ∀t ≥ t̄. This proves point 1.

Part 2.) Let us rewrite the performance index D_{t}(ϑ) as a function of the least squares estimate ϑ̄t^{LS} as follows:

D_{t}(ϑ) = (ϑ − ϑ̂t^{LS})^{T} [∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T} + Q](ϑ − ϑ̄t^{LS})

+α_{t}∥ϑ − ϑ̄∥^{2} + V_{t}(ϑ̄t^{LS}).

From the definition of ϑ̂_{t}, it follows that

(ϑ̂_{t} − ϑ̄t^{LS})^{T} [∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T} + Q](ϑ̂_{t} − ϑ̄t^{LS})

+α_{t}∥ϑ̂_{t} − ϑ̄∥^{2}

≤ (ϑ^{∘} − ϑ̄t^{LS})^{T} [∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T} + Q](ϑ^{∘} − ϑ̄t^{LS})

+α_{t}∥ϑ^{∘} − ϑ̄∥^{2}

= O(α_{t}),  (19)

almost surely, where the last equality is a consequence of the already cited Theorem 1 in [14] and of the boundedness of ϑ^{∘}. Consider now the equation

(ϑ^{∘} − ϑ̂_{t})^{T} [∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T} + Q](ϑ^{∘} − ϑ̂_{t})

≤ 2 {(ϑ^{∘} − ϑ̄t^{LS})^{T} [∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T} + Q](ϑ^{∘} − ϑ̄t^{LS})

+ (ϑ̄t^{LS} − ϑ̂_{t})^{T} [∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T} + Q](ϑ̄t^{LS} − ϑ̂_{t})} .

Since in view of equation (19) both terms in the righthand-side are almost surely O(α_{t}), we get

(ϑ^{∘} − ϑ̂_{t})^{T} [∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T} + Q](ϑ^{∘} − ϑ̂_{t}) = O(α_{t})

almost surely. From this,

∥ϑ^{∘} − ϑ̂_{t}∥^{2} = O (\frac{α_{t}}{λ_{min}(∑_{k=1}^{t} φ_{k−1}φ_{k−1}^{T} + Q)}) a.s.,

which is the desired bound. □

Remark 1 Note that this result holds in open or closed loop. Indeed, the only assumption is that the input signal is measurable with respect to the past sigma algebra F_{t}, and this allow for any causal control law.

Remark 2 Condition (13) is slightly weaker than (7) due to the presence of exponent 1 + δ, with δ > 0.

Remark 3 In adaptive control, many well-established stability and performance results are stated under a uniform controllability assumption of the estimated model (see e.g. [8, 23, 7, 12, 18, 17, 15, 16, 20, 4, 1, 5, 3]). If one has the a-priori knowledge that the true system parameter belongs to a given sphere such that no pole-zero cancellation occurs for all parameters in the sphere, then the presented algorithm can be used and no pole-zero cancellation will occur in the estimated model. Stability and performance results can then be established thanks to the Property 2 proven in the Theorem.

5 Conclusion

We have provided an overview of the penalized approach in self-tuning adaptive control. This method turns out to be effective for many purposes as indicated in the paper and it is the personal belief of the present authors that it can be used in many more situations. Many open problems remain. The major one is certainly that in many cases no recursive algorithm is available for the on-line implemetation of the penalized identification procedure. Designing such recursive algorithms is a challenge for future research.

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