Robust output tracking of constrained perturbed linear systems via model predictive sliding mode control

A robustifying strategy for constrained linear multivariable systems is proposed. A combination of tracking model predictive control with output integral sliding mode techniques is used to completely reject bounded matched perturbations. It can be guaranteed that all constraints on inputs, states, and outputs are satisfied although only output information is used. Finally, real‐world experiments with an unstable plant are presented in order to demonstrate the validity and the effectiveness of the proposed approach.


INTRODUCTION
Model predictive control (MPC) has become one of the most widely used control techniques due to its capability to explicitly handle constraints in the controller. [1][2][3][4] One of the important milestones is the paper of Mayne et al 5 that forms the basis for theoretical analysis in terms of stability and optimality. It is shown that terminal costs and terminal sets are the main ingredients to ensure stability for a huge number of different MPC formulations.
There are also extensions of MPC to track piecewise constant reference signals. 6 Generalized conditions for stability are given in the work of Mayne and Falugi 7 to increase the region of attraction for tracking equilibrium states. Further extensions deal with the robustness with respect to external disturbances and parameter variations. They come in very different flavors (see, eg, the work of Mayne 8 for an overview). The price to be paid for robust MPC formulations is an increasing complexity of the underlying optimization problems with the respective computational cost.
With regard to robustness, sliding mode control (SMC) is one of the most promising techniques that has been developed over many decades. [9][10][11] Especially, the ability to completely reject matched bounded perturbations is one of its main distinctive features. It comes together with the advantage of a low number of control parameters and simple implementation. The drawback of SMC is that constraint handling in general is nontrivial.
In the recent years, one line of research is to combine ideas from generalized predictive control 3 with SMC. In a paper by Garcia-Gabin et al, 12 a sliding variable is predicted for a SISO system modeled by its transfer function. This prediction is incorporated into the performance index of an optimization problem. The use of additional constraints guarantees that the sliding variable decreases asymptotically to zero. A first-order SISO system with constant delay is considered in the work of Perez et al. 13 In this work, additional conditions are imposed in the optimization problem to ensure discrete-time quasi-sliding with respect to a predicted sliding variable. Recently, first steps toward nonlinear generalized predictive control with SMC are published by Errouissi et al. 14 However, constraints are not taken into account.
A second line in literature uses the concept of integral SMC 9,15 (ISMC) to combine ideas from MPC with SMC. Rubagotti et al 16 propose a combination of discrete-time robust MPC with continuous time ISM to stabilize the origin under the presence of matched and unmatched perturbations. In the work of Raimondo et al, 17 a multirate sliding mode disturbance compensation for MPC is shown. It makes use of the properties of discrete-time SMC 18 to force the sliding variable to zero in one sampling step. Tightened MPC is used in the outer loop to handle unmatched perturbations. Both papers aim to stabilize the origin.
In contrast, the authors of this paper proposed first steps toward a scheme combining the strengths of SMC and MPC using output information only. 19 It exploits the properties of an output integral sliding mode observer (OISMO) and output integral sliding mode controller 20,21 (OISMC). Compared to this conference paper, 19 this article features the following main contributions: (a) A detailed scheme consisting of an inner and outer loop is proposed to ensure complete elimination of matched perturbations without violating the given constraints and using output information only. (b) The MPC formulation in the outer loop is extended and the stability of the overall feedback loops is proven. (c) An unstable mechanical system is used to show the properties of the proposed method in a real-world application.
Additionally, the effect of different sampling times that are used for discrete-time realizations of the inner loop are investigated with respect to the ideal sliding mode behavior.
The remainder of this paper is structured as follows: In Section 2, the problem to be tackled is formulated. After that, the proposed approach "model predictive sliding" (MPS) consisting of an OISMO, OISMC, and a tracking MPC is described in detail in Section 3. In Section 4, the properties of the method are exemplified using simulations as well as measurements on a lab experiment. Finally, conclusions are drawn and further extensions are discussed in Section 5.

PROBLEM STATEMENT
Consider a linear time-invariant multivariable system .
with state vector x(t) ∈ ℝ n , inputs u(t) ∈ ℝ m , and outputs (t) ∈ ℝ p that is subject to perturbations w(t) ∈ ℝ m . Matrices A, B, and C are constant, of proper dimensions and known. The following assumptions are imposed on system (1): Assumption 1 (Properties of plant, perturbation and constraints).
(a) Input and output matrices have full rank, ie, rank B = m and rank C = p. The pair (A, B) is controllable, the pair (A, C) is observable. (b) The initial states are unknown but bounded, ie, ||x(0)|| ≤ . (c) The perturbations are unknown but bounded in magnitude such that ||w(t)|| ≤ w max . (d) The dimensions of disturbances, outputs, and states fulfill m < p < n. (e) The system is of relative degree one, ie, rank(CB) = m. (f) The constraints for inputs u ∈ , states x ∈ , and outputs ∈ are compact sets.
The main goal of this work is to design an output tracking controller for system (1) capable of rejecting time-varying bounded perturbations w(t). It has to be ensured that all constraints on inputs, states, and outputs are satisfied and that stability of the closed loop is guaranteed. Figure 1 illustrates the structure of the proposed control architecture. 19 It consists of a continuous-time inner loop and a discrete-time outer loop coupled via sample (S) and hold (H) elements.

MODEL PREDICTIVE SMC
The plant states are estimated using output integral sliding mode (OISM) techniques in the inner loop. Additionally, a control signal u O is generated to counteract perturbations. In the outer loop, an output tracking MPC provides a nominal control signal u M for the system without perturbations. This control signal as well as u O are bounded in magnitude to make sure that neither constraints for states nor for outputs are violated. Their sum is applied at the plant. Detailed descriptions of the individual parts are given in the next sections.

Output ISMC and observation
The OISM block in the inner loop of Figure 1 consists of a Luenberger observer (LO), an OISMO, and an OISMC connected according to Figure 2.

Output integral sliding mode controller
In the first step, it is assumed that the estimatesx for the plant states x are known. 20,21 A sliding variable with G ∈ ℝ m×p is chosen such that det(GCB) ≠ 0. It represents the weighted deviation of the actual plant output from the output of the nominal system under nominal control u M . The so-called unit vector approach 10 is applied to determine the control signal of the inner loop in Figure 1, resulting in Sliding mode is enforced for sufficiently large (t) ∈ ℝ where all solutions of the closed-loop system are understood in the sense of Filippov. 22 In sliding mode, the resulting equivalent control 10,11 is As a result, system (1) in combination with (2) and (5) can be written as . whereÃ The estimated statesx as well as the nominal control signal u M generated by the outer loop are used in (6).

Proposition 1 (OISMC 20 ). Suppose that Assumptions 1(a, c-e) hold and controller (4) with (3) is used. Then, (a) sliding is enforced from the very beginning if the scalar gain fulfills
Remark 1. Note that (a) means that matched perturbations are exactly eliminated starting at t = 0. If the number of outputs is lower or equal to the number of inputs, the resulting system in sliding mode (Ã, C) is not observable.
In the work of Castanos and Fridman, 23 the goal is to stabilize the origin using measurements of the full state vector. Matrix G is chosen as the pseudoinverse B + of the input matrix to ensure that unmatched perturbations are not amplified by an integral sliding mode controller. In analogy, one can employ as a weighting matrix in (3) for the case when only output information is available. Hence, the system in sliding mode is given by In a next step, an observer is designed to provide estimatesx.

Output integral sliding mode observer
The observer structure proposed in the work of Bejarano et al 20 reconstructs in an hierarchical way the rows of the observability matrix (12) so that rank() = n. This matrix is used to determinex (see Figure 2). The basis for the hierarchical OISMO is formed by a Luenberger observer of the form for system (10). Matrix L is chosen such that the dynamics of the estimation error (t) = x(t) − x L (t) are exponentially stable. Hence, there exist constants and such that To enable reconstruction of CÃ x for = 1, … , l − 1, the following auxiliary dynamical systems 20 are introduced: MatrixL is constant an of proper dimension. In addition, sliding variables with initial values with sufficiently large M (t) are used as shown in the works of Bejarano et al. 20,21 The symbols v ( −1) eq represent the equivalent injection terms corresponding to v −1 . All rows of matrix  can be reconstructed using if sliding mode is attained. Then, (19) yields with v eq (t) = Proposition 2 (OISMO 20 ). Suppose that Assumptions 1(a-e) hold. Controller (3), (4) with (8) as well as observer (13), (15) with (16) and (17) are used. The gains in (18) are tuned such that eg, with > 0. Then, the plant states are exactly estimated from the very beginning, ie,

Realization of the inner loop
A realization of the hierarchical observer needs filtering to get v eq such that (25) turns intõ with using filter constant c 1 . In addition, v ( ) eq has to be replaced by v ( ) av in relations (16), (17), and (22). A choice of the filter constant may be c 1 = c with 0 < < 1, C > 0 that is proportional to the sampling time of the actual discrete-time realization. 11 One way to avoid filtering would be to make use of a technique presented in the work of Bejarano et al 24 with the drawback that the resulting control signal may be unbounded.

Output tracking MPC
A tracking MPC formulation based on the nominal system (1) with w(t) = 0 is presented in this section. It will be used in the outer loop in Figure 1. A nonpathological choice of the sampling time T S in the sense of Kalman et al 25 yields a controllable and observable discrete-time linear time-invariant plant model with the same dimensions as (1). One can cast the considered tracking problem into a problem of stabilizing the origin by means of a transformation using a steady-state solution (x S , u S ) for system states and inputs. For that reason, new variables are introduced yielding a €-representation of system (28) To calculate steady-state solutions (x S , u S ) for given steady-state outputs S , so-called controlled variables with N Z m are used.
Assumption 2 (Rank of matrix EC). Matrix E ∈ ℝ N Z ×p is chosen such that rank(EC) = N Z , ie, EC, has full rank.

Proposition 3 (Existence of steady-state solutions). Let Assumption 2 hold for discrete-time system (28) with controlled variables (31). Let S be an arbitrary steady-state value for outputs k . Then, at least one steady-state solution
Proof. A proof is given in the Appendix. The following assumption is imposed for this paper.
Assumption 3 (Feasible steady-state). There exists at least one feasible steady-state solution (x S , u S ) such that x S ∈ and u S ∈ for desired steady-state controlled outputs z S and S = Ez S ∈ .
With the above assumption, an optimization problem is formulated aiming to force controlled outputs (31) in €-representation, ie, to zero. An iterative solution of (30) and (33) for a prediction horizon N and control horizon N C ≤ N yields where matrices F, F X , H, and H X only depend on A d , B d , C, and both horizons. 2,4 Predicted states and controlled outputs depend on the vector of optimization variables where Δx k+i = k+i (Δx k , Δū k ) is the solution of (30) at step k + i starting with initial condition €x k (29). The references r = z S ∈ ℝ N Z to be tracked are assumed to be constant over the prediction horizon. A quadratic cost function of form or alternatively can be used with terminal cost V (Δx k+N ) = Δx T k+N P Δx k+N . The matrices are positive semidefinite for Q ≥ 0, Q ≥ 0 and p < n. Therefore, cost function (38) with weights P is used instead of (37) with weights Q as will be shown later. Cost function (38) is minimized subject to hard constraintŝ u k+i ∈ ,x k+i ∈ ,̂k +i ∈ for all i ∈ ≥0 (40) according to Assumption 1(f) and a terminal constraint set to ensure stability as will be shown in the next section.

Analysis of the overall system
The stability of the closed loop consisting of the inner loop with an OISMC, an OISMO, and the outer loop with a tracking MPC as shown in Figure 1 is investigated in this section. • an OISM controller (3), (4) tuned such that relations (8) and (9) are fulfilled; • an OISM observer (13), (15)- (17) with L chosen such that matrix (Ã − LC) is Hurwitz and (24) is satisfied with > 0; (c) the outer loop with an output tracking MPC with prediction horizon N, control horizon N C , cost function (38) and • a nominal control signal u M,k constrained by • symmetric positive definite weighting matrices P > 0 and R > 0; • terminal constraint set for Δx k+N that is chosen as the maximal admissible set for the corresponding unconstrained infinite horizon problem • matrix P in the terminal cost is chosen as the solution of the discrete algebraic Riccati equation Then, independently of matched bounded perturbations w(t), exponential tracking of constant references is achieved.
Proof. A proof is given in the Appendix.

EXPERIMENTAL RESULTS
A second-order mechanical system, called Pendubot, 26 is used to show the properties of the proposed approach. Figure 3 shows a sketch as well as a photo of the considered lab experiment. It consists of an inner link that is actuated via a DC motor and an unactuated outer link. The desired motor current represents the input variable u(t) of the system. An additional loop, to control the motor voltage such that the desired current is achieved, is realized by means of an electronic circuit. This current control loop is neglected for the modeling given below. The lab experiment is connected via Quanser QuaRC real-time control software 27 to the computer, where MATLAB 28 is used to implement the presented observer and Using Lagrange equations, a nonlinear model can be derived as, eg, shown in the work of Altenbuchner. 26 Angles 1 and 2 shown in Figure 3 and the corresponding angular velocities are used as generalized coordinates, so that the state vector is given by

Mathematical model
with parameters listed in Table 1. The measurement signals are given by the position of the inner link and the velocity of the outer link such that Box constraints are used for all subsequent experiments.

Parameters for observer and controllers
For the performed experiments, a disturbance consisting of two sinusoidal components and offset is used, where w max = 4. The OISM controller and OISM observer are tuned as = 5 and M = 50, c 1 = 50 4∕5 respectively. The parameters for the tracking MPC are N = 6, N C = 4, Q = diag ([100 1 100 1]), and R = 1. Matrix E = [1 0] is chosen such that (3) is fulfilled and steady states (x S , u S ) can be calculated. The resulting quadratic program is solved by means of qpOASES. 29 The maximal admissible set for the unconstrained problem for system , Δu k ∈ M , and Δ k ∈ is calculated using algorithm 3.2 from the work of Gilbert and Tan 30 for three different cases: system matrix A d + B d K and (i) output matrix K for constraints in €u M = u k,M − u S , (ii) output matrix I for constraints in €x k = x k − x S , and (iii) output matrix C for constraints in € k = k − S . Finally, the intersection of these sets from (i) to (iii) gives denoted by For the construction of the region  N where a solution of the MPC problem exists, constraints for predicted inputs, states, and outputs (34) as well as the terminal set are combined to where F and H are submatrices of F X and H X (last n rows). Symbols I r×c and 0 r×c stand for identity and zero matrices with r rows and c columns. Vectorsū S respectivelyx S correspond to the N C -times concatenation of u S respectively the N-times concatenation of x S . A projection onto x k gives  N . The multiparametric toolbox 3.0, 31 was used to remove redundant inequalities and perform set projections. Figure 4 shows examples for sets (red) and  N (green). Since the considered system is of order n = 4, always one component of the state vector is kept constant, while regions for the other three states are drawn in the subfigures for z S = 10 • . Please note that the volume of  N is not significantly larger than the volume of due to the properties of the Pendubot in the unstable upward position (up-up position). The volume of set  N is 41.7% larger compared to the volume of . In the next sections, the proposed approach is tested in simulation as well as on the lab experiment.

Simulation results
The effect of the chosen sampling time for the inner control loop (see Figure 1) and filter constant c 1 in (27)    filtration (27) for the calculation of the equivalent injection terms in the observer. Variables e 1,L = x 1 − x 1,L respectively e 2,L = x 2 − x 2,L represent the error using a Luenberger observer only (see Figure 2). Figure 7 shows the dependency of e 2 on different settings for sampling time with fixed filter constant c = 50, which is chosen as trade-off between convergence time and the amplitude of the oscillations. As expected, a smaller sampling time significantly improves the performance leading to the exact compensation for → 0. In the left plot of Figure 7, signal x 2,L depends on as well. This is due to (13) wherex is considered in the Luenberger observer as an external input to be able to observe the inner loop while sliding. This effect is not visible when no inner control loop is used (see left plot in Figure 7).

Experimental results
The combination of tracking MPC, OISM controller, and OISM observer is tested at the Pendubot lab experiment shown in Figure 3 (right). In all subsequent experiments, a set point changes between z S = −10 • and z S = −10 • starting from z S = −10 • is performed. Figure 8 (left) shows results using MPC only, where estimates of the states are provided by the presented OISM observer. The blue curve illustrates the behavior without additional perturbation, ie, only unmodeled parts and external vibrations act as disturbances. Spikes in the opposite directions during abrupt changes of the set point appear due to the nonminimum phase property of the considered system. When disturbance (49) is used in addition, one can see clear deviations ( Figure 8, red) from the desired values shown in black. The experiment is stopped at time t ≈ 17.5 seconds because x 1 leaves the admissible region.
In the right plot of Figure 8, these results are compared to experimental data achieved with the proposed MPS algorithm. The results are clearly better even in the case without additional perturbation, ie, w(t) = 0 (red). Note that the tracking performance is quite similar in the case of w(t) ≠ 0. The corresponding control signals u 0 for the OISM controller, u M for the MPC, and their sum u is shown in Figure 9 (left) for a smaller time interval. The right plot of Figure 9 depicts the  effects in the current control loop that is implemented in hardware at the lab experiment. The blue curve represents the input to the actuator, and the red curve represents the actual current at the DC motor. This difference due to actuator dynamics is neither modeled nor taken into account for controller design.
Finally, Figure 10 shows a comparison of the tracking performance between simulation and real-world measurements for the case with additional perturbation. This good matching is achieved even though multiple effects, such as the actuator dynamics of the DC motor and the friction in the joints, are neglected in the modeling.

CONCLUSION AND OUTLOOK
This paper presents a method to robustly track the outputs of constrained linear systems exploiting the properties of OISMs as well as MPC. An excess of the number of outputs compared to the number of matched perturbations is necessary to be able to exactly reconstruct the system states independent of matched (time-varying) perturbation. Issues due to the actual implementation of the algorithm were shown in simulation and using an unstable mechanical lab experiment.
Future work will focus on the extension of the approach to unmatched perturbations and on methods to smooth the control signal provided by the inner loop using, eg, the saturated super-twisting algorithm 32,33 granted that the actuators are fast enough, as shown in the work of Pérez-Ventura and Fridman. 34 Also explicit MPC formulations 1 are of great interest for further improvement of the real-time capability.

PROOF OF PROPOSITION 3
A steady-state solution of (28) has to fulfill According to Rawlings and Mayne, 2 such a solution exists iff the rows of Z C are linearly independent which is possible only for p ≤ m. If p > m at least two rows are linearly dependent. As a consequence, only references for controlled variables (31) have to be chosen such that where z S symbolizes the steady-state solution of the controlled variables. This is true in the general case in which the vector on the right-hand side can be arbitrary. In the present formulation, the first n elements are zero. Therefore, the mentioned condition is sufficient but not necessary. In the first n rows, a rank deficiency n − rank( is admissible because of the zeros on the right-hand side. As a consequence, matrix Z must have rank n + N Z minus the rank deficiency in the upper part, ie, (A5) APPENDIX B

PROOF OF THEOREM 1
Stability of the inner loop follows immediately by Propositions 1 and 2 as outlined in the work of Bejarano et al. 20 Because of Proposition 2,x(t) ≡ x(t) and therefore (10) turns into .
which is the original system without perturbation. A pathological choice of the sampling time T S yields a controllable and observable discrete-time linear time-invariant plant model in the €-representation, ie, with €u k = u M,k − u S . In addition, the constraints are reformulated in terms of this €-representation leading to The set of states in for which a solution exists is given by This means that  N is a set of states €x k can be steered in N steps or less by an admissible control sequence to . All -sets are also compact due to Assumption 1(f). As a consequence, one can follow the line of the proof stated in the work of Mayne et al. 5 The continuity of system and cost together with the properties of constraint sets ensure the existence of solutions to the optimal control problem (see the work of Rawlings and Mayne 2 ). Value function V * N (Δx k ) = V N (Δx k , Δū * k ), ie, the cost function in the optimum, is used as a Lyapunov function candidate with V * N (0) = 0, V * N (Δx k ) ≥ Δx T k PΔx k > 0 for all Δx k ≠ 0 and V * N (Δx k ) → ∞ for ||Δx k ||∞. Vector represents the optimal control sequence for the current iteration. This yields V * N (Δx k ) = V N (Δx k , Δū * k ) = Δx T k PΔx k + Δû * T k RΔû * k (B9) Δû * T k+i RΔû * k+i + Δx * T k+N P Δx * k+N .
The value function for the next iteration can be upper bounded 2 by where and with R N as shown in (44). Figure B1 shows a graphical representation of that change in the cost. If there exists Δû = (Δx k ) such that is fulfilled, then the difference in the value functions for subsequent iterations can be written as ∀Δx k ∈ ℝ n and V = Δx T k P Δx k is a global CLF. Because of constraints, relations (B13) and (B14) are only requested locally, ie, ∀Δx k ∈ . With a linear ansatz Δû = (Δx k ) = KΔx k relation (B13) turns into (43), if and is chosen as the maximal admissible set 30 for the infinite horizon problem (see the work of Mayne et al 5 ). Assumption 1(a) together with Proposition 1 ensure that in the inner control loop is observable in sliding, which starts at the very beginning. As a result, Δu k → 0, Δ k → 0 and Δx k → 0. 2 This means that the closed-loop system consisting of an OISM controller, an OISM observer, and the tracking MPC (see Figure 1) is exponentially stable with a region of attraction N and x → x S respectively z → z S is achieved.