An Improvement on Data-Driven Pole Placement for State Feedback Control and Model Identification

The recently proposed data-driven pole placement method is able to make use of measurement data to simultaneously identify a state space model and derive pole placement state feedback gain. It can achieve this precisely for systems that are linear time-invariant and for which noiseless measurement datasets are available. However, for nonlinear systems, and/or when the only noisy measurement datasets available contain noise, this approach is unable to yield satisfactory results. In this study, we investigated the effect on data-driven pole placement performance of introducing a prefilter to reduce the noise present in datasets. Using numerical simulations of a self-balancing robot, we demonstrated the important role that prefiltering can play in reducing the interference caused by noise.


Introduction
In state feedback pole placement, the state feedback gain must be determined for a given system such that the closed-loop poles coincide with the desired locations.This is a well-known problem, and the pole placement methods have been extensively discussed in the literature [1] [2] [3] [4].In standard pole placement methods, a state space model is assumed to be given by a system identification technique using data from past experiments.Whereas the traditional approach combines the identification of the state space model with the standard pole placement method; an alternative approach called "data-driven pole placement" has recently been proposed [5].In this approach, the state space model and pole placement feedback gain are identified simultaneously from the set of state measurements and control input sequences.The method proposed in [5] is based on the data-driven control framework ( [6] and references therein) such as unfalsified control [7], virtual reference feedback tuning (VRFT) [8] [9], or fictitious reference iterative tuning (FRIT) [10] [11] [12] [13].In the data-driven control framework, where no explicit mathematical plant model is used, a feedback controller must be derived that satisfies the prescribed closed-loop performance and fits to known experimental data.In contrast with traditional model-based controller designs, techniques such as controller identification [14] or a combination of plant model and controller identification must be applied [15] [16].
Many studies of data-driven control have focused on output feedback control and data-driven state feedback control [11] [12] [13], in which the prescribed closed-loop performance is achieved by applying a closed-loop reference transfer function.Such methods can be applied to the data-driven pole placement problem by choosing a reference transfer function with the desired poles.However, the zeros of the reference transfer function cannot normally be specified, because the zeros of the plant are unknown.In contrast, the data-driven pole placement method presented in [5] requires only a state space representation of the closedloop system to specify the prescribed closed-loop performance, as shown in Section 2. This avoids the zero assignment issue that arises in the transfer function approach used in [5].This data-driven pole placement method can, therefore, be applied to linear and time-invariant systems with measurable states.The method is briefly reviewed in Section 2. However, the capacity of the data-driven pole placement method to handle noise remains an open issue, though in [5], the total least square (TLS) method [17] was claimed to be effective.Measurement noise is one of the issues which may surely face in practical applications.Therefore, to resolve this, we introduced a prefiltering technique that reduces the effect of measurement noise in Section 3.More specifically, a finite impulse response (FIR) filter is used to prefilter the data, as this makes them easier to manipulate.In Section 4, by using the numerical example of a self-balancing robot, we discuss the effect of applying this prefiltering technique, together with the least square (LS) and TLS methods, to a self-balancing robot model.We investigate the ability of the data-driven pole placement method to produce a linearized model using numerical simulations as in [18].A nonlinear differential equation was used to represent the dynamics of a self-balancing robot there.Moreover, we evaluate the effects by two different exciting signals, the random and the chirp exciting signal, along with TLS and prefiltering.Finally, we compare all the results for the pole placement error and identification error when two exciting signals are applied.
Notation: Let A and B be m n × and p q × matrices, respectively.Then, the Kronecker product of A and B is a mp nq × matrix, defined as follow: where The vectorization of then stacks the columns into a vector: in which j a is the th j column of A .The Frobenius norm of matrix

Data-Driven Pole Placement
In this section, we briefly review the data-driven pole placement method formulated in [5].
Consider a discrete-time linear time-invariant system and static state feedback where , , The data-driven pole placement problem was formulated in [5] as follows: Problem 1.We assume that the order of the plant n is known, state n is measurable, pair ( ) be a self-conjugate set of n complex numbers in the unit circle.Given the input and output measurement data sequence ( ) ( ) ( ) 0 0 , x k u k of (4), find a state feedback gain F from the observed data In a conventional approach, this problem is solved in two steps: A and B are identified from , x k u k , then F is derived using the standard pole placement algorithms.In contrast, the data-driven pole placement method solves the two steps simultaneously.To achieve this, the method uses the equivalency between the closed-loop system with the desired pole placement gain F and ( ) ( ) ( ) where ( ) is an appropriate controllable pair.This equivalency requires the nonsingular matrix T to exist.Then, we remove v from (7) by using (5), to obtain Then, using (8), we obtain where ( ) ( ) ( ) In [5], Equation ( 11) is cast into , .
Remark 1.The system in (7) can be interpreted as a reference model within VRFT (e.g., [8] [9]) and FRIT (e.g., [10] [11] [12] [13]).The idea of eliminating v in (9) is also based on FRIT.In [10] [11] [12], a similar state feedback control problem has been discussed within the FRIT framework.To apply these FRIT techniques to the data-driven pole placement problem, the desired transfer function must be specified from u to x , rather than d x .When precise values for ( ) , A B are not available, it becomes impossible to specify the zeros of the desired transfer function.
Remark 2. To obtain the datasets in (12) by applying state feedback in (5) to the system in (4), the initial feedback gain F should be based on ( ) Hence, in Problem 1, the exact value of ( ) A B is assumed to be unknown.
When applying the property of Kronecker product ( ) ( ) (see for example Th.2.13 in [19]) to the transpose of (15) to solve (15) for F and T , a further linear equation is derived, as follows: , η Χ = U (18) where , If T is nonsingular, the model coefficients can be obtained

Prefiltering Noisy Measurement
When the measurement of x is contaminated by noise ε , ( ) ( ) ( ) Then, (10) becomes Hence, if where Then, the resulting linear equation is given as where the effect of noise ∆Χ has the same structure as Χ in (20), then ( ) ( ) and ∆U is the equation error.Following [5], we can solve [17], by minimizing the Frobenius norm ∆Χ ∆U .It is known that the TLS solution is given as where these matrices are partitioned into blocks corresponding to Χ and U .
Here, we assume that there exists 0 M > such that ( ) for all i .This means that when , N M > for the matrix ( ) where each column has M elements of 1.Therefore, where This multiplication by Φ represents the prefiltering of signals via an th M order FIR filter.
When the systems ( 4) and ( 7) are driven by the exciting signal, we have ( ) By applying Φ to these systems, we obtain Here, if 0 V Φ ≈ , (34) cannot be satisfied.Hence, for all i, 0 V Φ ≠ , that is ( ) must be satisfied.

Numerical Example: Self-Balancing Robot
We next applied the data-driven pole placement method described above to the model of a self-balancing robot [21] [20] as shown in Figure 1.The robot is equipped with right and left wheels driven by direct current (DC) motors whose voltages r v and 1 v can be controlled.Because the motion dynamics can be decomposed by the input u , the control input to the robot was represented as We assume that the pitch angle b θ and the pitch angular velocity b θ  of the body could be measured, as well as the angles r θ and 1 θ of the right and left wheels, and their angular velocities r θ  and 1 θ  , respectively.We define the mean values of the right and left wheel angles r θ and 1 θ , and the yaw angle of the body as follows: ( ) where r is the radius of the wheel and is the distance between the two wheels.

Equation of Motion
The equation of motion for the self-balancing robot can be derived as ) The symbols are explained in Table 1.The parameters used in the simulations were taken from [20] [21].

Linear Model and Feedback Gain
We linearized the equations of motion ( 50 where By defining the state vector the linear state space model can be derived as Then, the feedback can be independently designed as Note that this can be more succinctly represented as When the parameters in Table 1 are used and the sampling period is 0.1 s h = , the discrete-time model after discretizing ( 55) is where Here, we assume that the exact values of (60) are not available, but that uncertain values are available: The coefficients can be derived from 1 2 , J J , with an assumed uncertainty of 10%.By applying linear quadratic optimal control theory to (61), the desired closed-loop pole locations can be chosen as 1 1 1 6.0355 10 , 0.5253, 0.5745, 0.7630 6.0426 10 , 0.7835 and the initial feedback gains needed to obtain datasets for the data-driven pole placement as

Comparison of Methods
Next, simulations were conducted and comparisons were made from the obtained results when using different methods and exciting signals.
Measurement noise was prepared with the Gaussian distribution    , , The eigenvalues j λ were sorted by magnitude using the MATLAB command "sort".This further sorts elements of equal magnitude by the phase angle on the , where x are the impulse responses of ( ) ( ) , respectively.
From the perspective of system control, smaller is better, particularly in the case of The following key results were contras- tively found in Table 2: 1) The initial model and feedback gain were affected by uncertainty: The model errors and pole placement errors are shown in Table 2 (initial).
2) The results when using the LS method to solve linear Equation ( 27) for noiseless data are shown in Table 2(a).All errors were reasonably small, confirming that the data-driven method performs well when the measurement data 3) The results when using the LS method to solve linear Equation ( 27) for noisy data are shown in Table 2(b).All errors became larger when noise was added, suggesting that LS analysis is inadequate when the measurement data are contaminated by noise.
4) The results when using the TLS method to solve linear Equation ( 27) are shown in Table 2(c).The errors were significantly smaller than those reported in [5], using the LS method.
5) The results when applying prefiltering (PF) and using the TLS method to   6) The results when applying PF and using the TLS method to solve the linear Equation ( 27), but with v as the chirp signal, are shown in Table 2(e).
No significant improvement in error rates was found with respect to 2 A when using the chirp exciting signal.However, the errors with respect to 1 A became significantly worse than when a random exciting signal was used.This was assumed to be because 1 A has an unstable eigenvalue of 1.7838.
conclude that a random exciting signal is more appropriate than a chirp exciting signal when using data-driven methods.
Finally, we compare the pole locations obtained as shown in Figure 5.As can be seen, a better performance was achieved when using the random exciting signal.

Conclusion
In this study, we evaluated the different approaches reducing the effect of measurement noise in data-driven pole placement methods for deriving a state space model and pole placement state feedback.Using numerical simulations of a self-balancing robot, which is a nonlinear system, we demonstrated the important role that prefiltering can play in reducing the interference caused by noise.Again using numerical simulation, we compared the use of two exciting signals: a random signal and a chirp signal.The use of a random exciting signal was found to be more effective with our proposed method.Further developments are needed in the methods used to cope with noise.A method such as that used in [9] may be appropriate for use in practical applications where noise is present, and adaptive control based on real-time updating [22] is a future promising approach.
is the external input to the closed loop system.
center to center of gravity of robot body [m] K = electromotive force constant of DC motor [V•s/rad] t e K K = torque constant of DC motor [N•m/A] d = coefficient of friction between wheel and DC motor w 0 d = coefficient of friction between wheel and floor

Figure 2 .
Figure 4(b), respectively.Of particular note is that the responses of b θ , w θ  , and b θ  in Figure 4(b) show the high-pass filter-like gain characteristics of the transfer function from v to x .For comparison, the dataset for the data-driven pole placement was chosen as ( ) ( ) ( ) { } 0 0 50, ,450 ,

Figure 4 .
Figure 4. Closed-loop response by an initial state feedback via (a) random exciting signal , v (b) chirp ex- citing signal v .To evaluate the obtained model ( ), A B  , the following identification errors were used:

:
to evaluate the model obtained, as follows:

Figure 5 .
Figure 5.Comparison of pole locations ("+" indicates the desired poles, "." those obtained by the random exciting signal and "o" those obtained by the chirp exciting signal.).

Table 2 .
Comparison of errors.