Data-Driven Inversion-Based Control: closed-loop stability analysis for MIMO systems
Carlo Novara Simone Formentin
cs.SY





Data-Driven Inversion-Based Control: 
closed-loop stability analysis
for MIMO systems
    Carlo Novara and Simone FormentinC. Novara is with Politecnico di Torino (Italy), S. Formentin is with
Politecnico di Milano (Italy). Email: ,
.
    December 30, 2023
====================================================================================================================================================


Data-Driven Inversion-Based Control (D^2-IBC) is a recently introduced
control design method for uncertain nonlinear systems, relying on
a two degree-of-freedom architecture, with a nonlinear controller
and a linear controller running in parallel. In this paper, extending
to the MIMO case a previous result holding for the SISO case, we derive
a finite-gain stability sufficient condition for a closed-loop system
formed by a nonlinear MIMO plant, connected in feedback with a D^2-IBC
controller.





§ INTRODUCTION


Data-Driven Inversion-Based Control (D^2-IBC) is a recently introduced
control design method for uncertain nonlinear systems, relying on
a two degree-of-freedom architecture, with a nonlinear controller
and a linear controller running in parallel, <cit.>.
Despite many different approaches for joint design of identification
and control have been already proposed (see <cit.>
and the reference therein for a comprehensive overview), as far as
we are aware, D^2-IBC is the first “identification for control
method for nonlinear dynamical systems, where also stability guarantees
are provided and enforced directly in the identification algorithm.
However, the underlying stability sufficient condition is valid for
Single Input Single Output (SISO) systems only. In this paper, we
extend this sufficient condition to Multiple Input Multiple Output
(MIMO) systems. 

Notation. A column vector x∈ℝ^n_x×1
is denoted as x=(x_1,…,x_n_x). A row vector
x∈ℝ^1× n_x is denoted as x=[x_1,…,x_n_x]=(x_1,…,x_n_x)^⊤,
where ⊤ indicates the transpose.

 A discrete-time signal (i.e. a sequence of vectors) is denoted with
the bold style: x=(x_1,x_2,…), where x_t∈ℝ^n_x×1
and t=1,2,… indicates the discrete time; x_i,t is the
ith component of the signal x at time t.

 A regressor, i.e. a vector that, at time t, contains n present
and past values of a variable, is indicated with the bold style and
the time index: x_t=(x_t,…,x_t-n+1).

 The ℓ_p norms of a vector x=(x_1,…,x_n_x)
are defined as 

    ‖ x‖ _p≐(∑_i=1^n_x|x_i|^p)^1/p,     p<∞,
    max_i|x_i|,     p=∞.

The ℓ_p norms of a signal x=(x_1,x_2,…)
are defined as 

    ‖x‖ _p≐(∑_t=1^∞∑_i=1^n_x|x_i,t|^p)^1/p,     p<∞,
    max_i,t|x_i,t|,     p=∞,

where x_i,t is the ith component of the signal x
at time t. 





§ THE D^2-IBC APPROACH




Consider a nonlinear discrete-time MIMO system in regression form:

    [ y_t+1=g(y_t,u_t,ξ_t);  y_t=(y_t,…,y_t-n+1);  u_t=(u_t,…,u_t-n+1);  ξ_t=(ξ_t,…,ξ_t-n+1) ]

where u_t∈ U=[-u̅,u̅]^n_u⊂ℝ^n_u
is the saturated input, y_t∈ℝ^n_y is the output,
ξ_t∈Ξ≐[-ξ̅,ξ̅]^n_ξ⊂ℝ^n_ξ
is a disturbance, and n is the system order. Both U and Ξ
are compact sets.

Suppose that the system (<ref>) is unknown, but a set of measurements
is available: 

    𝒟≐{ũ_t,ỹ_t} _t=1-L^0

where ũ_t∈ U, ỹ_t∈ Y, Y=[-y̅,y̅]^n_y,
y̅≐max_t,i|ỹ_t,i|<∞, and ỹ_t,i
is the ith component of ỹ_t. The tilde is used to
indicate the input and output samples of the data set (<ref>),
which is supposed to be available at time t=0 when the controller
needs to be designed. The input signals employed to generate (<ref>)
are also assumed to be such that the system output does not diverge.

Let R≐[-r̅,r̅]^n_y, with 0≤r̅≤y̅,
be a domain of interest for the trajectories of the system (<ref>).
The aim is to control the system (<ref>) in such a way that,
starting from any initial condition y_0∈𝒴^0≐ R^n× n_y⊂ℝ^n× n_y,
the system output sequence y=(y_1,y_2,…)
tracks any reference sequence r=(r_1,r_2,…)∈ℛ⊆ R^∞⊂ℓ_∞.
The set of all possible disturbance sequences is defined as ≐{ξ=(ξ_1,ξ_2,…):ξ_t∈Ξ,∀ t}.

To accomplish this task, we consider a feedback control structure
with two controllers, K^nl and K^lin, working in parallel:K^nl
is a nonlinear controller used to guide the system (<ref>)
along the trajectories of interest, while K^lin is a linear controller
aimed to enhance the tracking precision.



 §.§ Nonlinear controller design


The first step needed to design the nonlinear controller is to identify
from the data (<ref>) a model for the system (<ref>)
of the form 

    [        ŷ_t+1=f(y_t,u_t)≡ f(q_t,u_t); q_t=(y_t,…,y_t-n+1,u_t-1,…,u_t-n+1) ]

where u_t and y_t are the system input and output, and ŷ_t
is the model output. A parametric structure is taken for the function
f: 

    f(q_t,u_t)=∑_i=1^Nα_iϕ_i(q_t,u_t)

where ϕ_i:ℝ^n(n_y+n_u)→ℝ
are polynomial basis functions and α_i∈ℝ^n_y×1
are parameter vectors that can be identified by means of convex optimization.

Once a model of the form (<ref>) has been identified, the
current command action u_t^nl of the nonlinear controller K^nl
is computed by the on-line inversion of the obtained models, given
the available regressor q_t. This inversion is performed
by solving the following optimization problem: 

    u_t^nl=min_u̅∈𝐔J_t(u̅)

where the current objective function J_t is 

    J_t(u̅)=∑_i=1^n_yζ_i/ρ_y_i(r_t+1,i-f_i(q_t,u̅))^2+μ_i/ρ_u_iu̅^2+λ_i/ρ_u_iδu̅^2

where r_t+1,i is the ith component of r_t+1 and f_i
is the ith component of f; ρ_y_i≐||ỹ_̃ĩ_1-L,…,ỹ_̃ĩ_0||_2^2/L
and ρ_u_i≐||ũ_̃ĩ_1-L,…,ũ_̃ĩ_0||_2^2/L
are normalization constants computed form the data set (<ref>),
δu̅=u̅-u_t-1 are the control action rate of change
with respect to the past input, μ_i≥0 and λ_i≥0
are design parameters which allow us to set the trade off between
tracking precision and control action aggressiveness for every input
(control action magnitude and speed, respectively), and 1≥ζ_i≥0
are other design parameters which are needed to set the priority of
tracking for all the different output. The controller K^nl is
fully defined by the law (<ref>).



 §.§ Linear controller design


The linear controller K^lin is defined by the centralized extended
PID (Proportional Integrative Derivative) control law 

    u_t^lin=u_t-1^lin+∑_i=0^n_θB_ie_t-i

where e_t=r_t-y_t is the tracking error, n_θ is
the controller order and B_i∈ℝ^n_u× n_y
are the controller parameter matrices. All the entries of these matrices
are contained in a vector θ=(θ_1,…,θ_n_y× n_u× n_θ).
Note that, for n_θ=1 and n_θ=2, the standard PI
and PID controller are selected, respectively. The goal of K^lin
in the proposed architecture is to compensate for model-inversion
errors and boost the control performance by assigning a desired dynamics
to the resulting nonlinearly-compensated system.

In the considered setting, where most of the information about the
system is that inferrable from data, finding a good control-oriented
model of the error system, i.e. the system describing the relationship
between u_t^lin and y_t, is not an easy task. Therefore,
in this paper, the Virtual Reference Feedback Tuning (VRFT) method,
originally developed in <cit.> and extended to the
MIMO case in <cit.>, is employed and adapted to
the present setting. Its rationale is briefly recalled next for self-consistency
of the paper.

First, define the desired behavior for the closed loop system by means
of a discrete transfer function M. In the present setting, this
function is used to assign a specific desired dynamic to the nonlinearly-compensated
resulting system. Typically, for n_y=n_u=m, M is a diagonal
m× m transfer function composed by m asymptotically stable
low-pass filters, synthesizing the desired closed loop behavior for
each output. The virtual reference rationale permits the design of
K^lin without identifying any model of the system, based on the
following observation: in a virtual operating condition, where the
closed-loop system behaves exactly as M, the virtual reference
signal r_t^v would be given by the filtering of the output
y_t by the model M^-1. Since the inverse model M^-1
is non-causal, the filtering task must be accomplished off-line using
a set of available data. The optimal controller is the one giving
the measured u_t^lin as output when fed by the virtual error
e_t^v=r_t^v-y_t, as to minimize the cost function

    J_VR=∑_t||u_t^lin-K^lin(θ)e_t^v||_2^2.

Two versions of the VRFT method in the D^2-IBC setting are available,
corresponding to two cases, related to bijectivity of the function
f in (<ref>). The details of these two versions are not
discussed here for the sake of brevity.



§ CLOSED-LOOP STABILITY ANALYSIS




The closed-loop system formed by the plant (<ref>), controlled
in feedback by the parallel connection of K^nl and K^lin,
is described by the following equations: 

    [             y_t+1=g(y_t,u_t,ξ_t);               u_t=u_t^nl+u_t^lin;  u_t^nl=K^nl(r_t+1,y_t,u_t-1^nl); u_t^lin=K^lin(r_t-y_t,u_t-1^lin) ]

where K^nl and K^lin are defined in (<ref>)
and (<ref>), respectively, and u_t∈ U, ∀ t. The
reference initial condition is chosen as r_0=y_0.

In the following, we study the stability properties of such a close-loop
system. The following stability notion is adopted. 
A nonlinear system (possibly time-varying), with
inputs r_t and ξ_t, and output y_t, is finite-gain
ℓ_∞ stable on (𝒴^0,ℛ,)
if finite and non-negative constants Γ_r, Γ_ξ
and Λ exist such that 

    ‖y‖ _∞≤Γ_r‖r‖ _∞+Γ_ξ‖ξ‖ _∞+Λ

for any (y_0,r,ξ)∈𝒴^0×ℛ×.□


Note that this finite-gain stability definition is more general than
the standard one, which is obtained for ℛ=ℓ_∞
and =ℓ_∞, see e.g. <cit.>.

In order to study how to guarantee finite-gain stability of the feedback
system (<ref>), some additional assumptions are introduced.

[Lipschitzianity]The function
g in (<ref>) and (<ref>) is Lipschitz continuous
on Y^n× U^n×Ξ^n. Without loss of generality,
it is also assumed that Y^n× U^n×Ξ^n contains
the origin.□  

This assumption is mild, since most real-world dynamic
systems are described by functions that are Lipschitz continuous on
a compact set. 

From Assumption <ref>, it follows that g can be written
as 

    g(y_t,u_t,ξ_t)=g^o(y_t,u_t)+g_t^ξξ_t

where g^o(y_t,u_t)≐ g(y_t,u_t,0)
and g_t^ξ∈ℝ^n_y× n is a time-varying
parameter (dependent on y_t, u_t
and ξ_t) bounded on Y^n× U^n×Ξ^n
as ‖ g_t^ξ‖ _∞≤γ_ξ,
for some γ_ξ<∞. Assumption <ref>, together
with (<ref>), implies that the residue function 

    (y_t,u_t)≐ g^o(y_t,u_t)-f(y_t,u_t)

is Lipschitz continuous on Y^n× U^n. Hence, a finite and
non-negative constant γ_y exists, such that 

    ‖(y,u)-(y',u)‖ _∞≤γ_y‖ y-y'‖ _∞

for all y,y'∈ Y^n and all u∈ U^n. 
[Model accuracy]The following
inequality holds: γ_y<1.□ 


The meaning of this assumption is clear: it requires f to accurately
describe the variability of g with respect to y_t. 

Now, consider that

    [                  ê_t+1≐ r_t+1-ŷ_t+1=r_t+1-f(y_t,u_t); ≡ r_t+1-f(y_t,u_t(r_t+1,r_t,y_t,u_t-1^nl,u_t-1^lin));                                    ≐ F(y_t,r_t,v_t), ]

where v_t≐(r_t+1,u_t-1^nl,u_t-1^lin)∈ V
and V is a compact set. Then, for any (y_t,r_t,v_t)∈ Y^n× R^n× V^n,

    |ê_t+1|≤Γ_y‖y_t‖ _∞+Γ_s‖r_t‖ _∞+Λ_e

where Γ_y,Γ_s,Λ_e<∞. This inequality
directly follows from the fact that the model function f is Lipschitz
continuous on Y^n× U^n. Note that (<ref>) does
not imply that y∈ Y^∞. 
[Effective model inversion]
The following inequality holds: Γ_y≤1-γ_y.□


This assumption is not restrictive: it is certainly satisfied if μ=0
and the reference r=(r_1,r_2,…) is a model
solution (i.e. r_t+1 is in the range of f(y_t,·)
for all t). Indeed, in this case, ŷ_t+1=r_t+1, ∀ t,
since K^nl performs an exact inversion of the model, see again
(<ref>) (K^lin gives a null input signal in this
case). This implies that Γ_y=0, Γ_s=0 and Λ_e=0.
Hence, if a sufficiently small μ is chosen and the reference
is sufficiently close to a system solution, supposing that inequality
(<ref>) holds with a sufficiently small Γ_y is reasonable. 

To formulate our last assumption, define 

    e̅≐1/1-λ_y(λ_rr̅+γ_ξξ̅+Λ_g)

where λ_y≐Γ_y+γ_y<1, λ_r≐λ_y+Γ_s
and Λ_g≐Λ_e+max_u∈ U^n‖(0,u)‖ _∞.
Note that e̅ is bounded, being the sum of bounded quantities.
In particular, for null disturbances (ξ̅=0), exact modeling
(f=g, Δ=0, γ_y=0) and reference signals properly
chosen (Γ_y=0, Γ_s=0, Λ_e=0), we have
e̅=0. In realistic situations, with reasonable disturbances,
sufficiently accurate modeling and reference signals properly chosen,
e̅ can be reasonably small (that is, e̅≪r̅).
[Output domain exploration]
The following inequality holds: y̅≥r̅+e̅.□


This assumption requires that the set Y explored by the output
data is somewhat larger than the set R where the trajectory of
interest are defined. Note that it can always be met just collecting
data that sufficiently enlarge the set Y.

Closed-loop stability of the system (<ref>) is stated by the
following result, which also provides a bound on the tracking error.

Consider the system (<ref>)
and let Assumptions <ref>-<ref> hold. Then:

(i) The feedback system (<ref>), having inputs r_t and
ξ_t and output y_t, is finite-gain ℓ_∞ stable
on (𝒴^0,ℛ,). 

(ii) The tracking error signal e≐r-y
is bounded as 

    ‖e‖ _∞≤e̅.



Proof. The proof of the theorem is structured as follows.
Firstly, the tracking error is proven to be upper bounded by a suitable
combination of the norms of the output and the reference. Secondly,
it is shown that, under the assumption of an effective model inversion,
such a bound is equivalent to a bound on the tracking error, whatever
the output is (claim (ii)). Claim (i) is derived as a straightforward
consequence of claim (ii).

To start with, consider that 

    e_t+1≐ r_t+1-y_t+1=ê_t+1-δ y_t

where 

    [ ê_t+1=r_t+1-ŷ_t+1=F(y_t,r_t,v_t);        δ y_t=(y_t,u_t)+g_t^ξξ_t. ]

The term ê_t+1 is bounded according to (<ref>).
Note that ‖y_t‖ _∞ could
be unbounded. In order to derive a bound on δ y_t, we can
use Assumption <ref> and observe that, for any y_t∈ Y^n,

    [          ‖(y_t,u_t)‖ _∞-‖(0,u_t)‖ _∞; ≤‖(y_t,u_t)-(0,u_t)‖ _∞≤γ_y‖y_t‖ _∞. ]

The following inequality thus holds for any y_t∈ Y^n:

    [ ‖δ y_t‖ _∞≤‖(y_t,u_t)‖ _∞+γ_ξ‖ξ_t‖ _∞;          ≤γ_y‖y_t‖ _∞+γ_ξ‖ξ_t‖ _∞+Δ̅, ]

where Δ̅≐max_u∈ U^n‖(0,u)‖ _∞<∞.
Hence, 

    [ ‖ e_t+1‖ _∞≤λ_y‖y_t‖ _∞+Γ_s‖r_t‖ _∞+Λ_e;                        +γ_ξ‖ξ_t‖ _∞+Δ̅, ]

which proves that the tracking error e is bounded by a suitable
combination of the norms of y_t and r_t,
for any y_t∈ Y^n. Note that (<ref>)
in this form is of no use, since ‖y_t‖ _∞
could be unbounded and thus the condition y_t∈ Y^n
may not hold. However, (<ref>) can be rewritten as 

    [ ‖ e_t+1‖ _∞≤λ_y‖e_t‖ _∞+λ_y‖r_t‖ _∞+Γ_s‖r_t‖ _∞;                               +γ_ξ‖ξ_t‖ _∞+Λ_g, ]

that means, 

    [ ‖ e_t+1‖ _∞≤λ_y‖e_t‖ _∞+w, ]

where w≐λ_rr̅+γ_ξξ̅+Λ_g.
Inequality (<ref>), again, holds only if y_t∈ Y^n.

 Consider now that, by assumption, y_0∈ R^n⊆ Y^n.
This implies that, at time t=0, inequality (<ref>) holds.
Being e_0=r_0-y_0=0
for the selected initialization of r_0, we have

    ‖ e_1‖ _∞≤λ_y‖e_0‖ _∞+w=w≤e̅.

Since ‖ e_1‖ _∞≤e̅ and |r_1|≤r̅,
it follows from Assumption <ref> that y_1∈ Y.
Consequently, the Lipschitzianity assumption holds and (<ref>)
can be used also for t=1, giving 

    [ ‖ e_2‖ _∞≤λ_y‖ e_1‖ _∞+w≤λ_yw+w;      ≤ w∑_k=0^∞λ_y^k≤w/1-λ_y=e̅ ]

where the geometric series sum has been obtained thanks to the fact
that, by Assumption <ref>, λ_y<1. It follows
that y_2∈ Y and, consequently, (<ref>) can be used
also for t=2. Iterating the above reasoning, 

    [ ‖ e_3‖ _∞≤λ_ymax{‖ e_2‖ _∞,‖ e_1‖ _∞} +w;         ≤λ_y‖ e_2‖ _∞+w≤λ_y^2w+λ_yw+w≤e̅;                                        ⋮;               ‖ e_t+1‖ _∞≤ w∑_k=0^tλ_y^k;              ≤ w∑_k=0^∞λ_y^k≤w/1-λ_y=e̅. ]

Then, y_t∈ Y, ∀ t≥0 and (<ref>) holds (claim
(ii)). Claim (i) is a direct consequence of claim (ii) and the relation
y=r-e.□ 

IEEEtran


