Model-error Control Synthesis Using Approximate Receding-horizon Control Laws

Model-Error Control Synthesis employs an optimal real-time nonlinear estimator to determine model error corrections to a nominal controller. Control compensation is achieved by using the estimated model error as a signal synthesis adaptive correction to the nominal control input so that maximum performance is achieved in the face of extreme model uncertainty and disturbance inputs. In this paper the capability of the model-error control synthesis is expanded by combining a nonlinear predictive filter using an approximate receding-horizon optimal solution with a Kalman filter in the overall control design. A robust stability analysis using the interlacing property from the HermiteBiehler theorem is presented for the new approach. Simulation results for linear systems are shown to verify theoretical predictions. INTRODUCTION Robust control of dynamic systems is usually achieved using one of two schemes. The first scheme involves the design of a controller that is insensitive as possible to model uncertainty and/or disturbance inputs, e.g., H∞ and μ-synthesis. The second scheme involves updating model parameters or control gains in real-time in order to achieve desired performance specifications. Adaptive control methods fall into this category. These control schemes can be used to provide robustness in a dynamic system with uncertainties, each with its own advantages and disadvantages. Model-Error Control Synthesis (MECS) is a signal synthesis adaptive control method. Robustness Graduate Student, Student Member AIAA. Assistant Professor, Senior Member AIAA. Copyright c © 2001 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. is achieved by applying a correction control to eliminate the effect of model error at the output. The error is estimated by a one step ahead predictive filter technique. The main advantage of this technique is that the model error is determined during the estimation process. More details on the predictive filter can be found in Refs. [2] and [3]. In Ref. [1] MECS was first applied to suppress the wing rock motion of a slender delta wing, which is described by a highly nonlinear differential equation. In Ref. [4] a simple approach to test the stability of the closed loop system was presented using a Padé approximation. As shown by the benchmark problem example in Ref. [4], the one-step ahead prediction technique inherent in MECS could not stabilize the system, which has one pole at the origin and two poles on the imaginary axis. However, when using a different approach to determine the model error, given by an Approximate RecedingHorizon Control (ARHC) law, the closed loop system can be stabilized. The main topic of this paper is to expand the previous results so that the MECS can stabilize non-minimum phase unstable systems with uncertainty. As shown in Ref. [4], since the future state is unknown in general, a trade-off between performance and stability exists. Hence, the weighting factor, W , for the model error has to be set to non-zero constant, i.e., perfect cancellation of the model error cannot be achieved. In addition, since the system measurement is given before calculating the model error, there is always the possibility that the predictive filter may have some bias error. To overcome this problem, we combine the predictive filter approach using an approximate receding-horizon optimal solution with a Kalman filter in the overall MECS approach. The closed form 1 American Institute of Aeronautics and Astronautics solution of the approximate receding-horizon control using Quadratic Programming (QP) was first presented by Lu. Also, in Ref. [5] the receding-horizon control concept was extended to the output tracking control problem. Although the problem is solved from a control standpoint, the algorithm can be reformulated as a filter and estimator problem. In this paper the model error is determined by the approximate receding-horizon optimal solution and the error is subtracted from the nominal control input. Finally, we obtain a new MECS scheme which has better characteristics than the one derived in Ref. [4]. To optimally design and analyze the new MECS scheme, we adopt the Hermite-Biehler theorem which gives the necessary and sufficient conditions for the closed loop system to be Hurwitz stable. By the theorem, for the system to be Hurwitz stable, the closed loop characteristic polynomial has to satisfy an interlacing property, which can be used to define some useful graphical relations as stability checks. Specifically, we concentrate on sixth-order polynomials, which can be easily applied for any polynomial less than sixth-order. The organization of this paper is as follows. First, we summarize the nonlinear predictive filter using an approximate receding-horizon optimal solution, and then combine the predictive filter with a Kalman filter. For robust design and analysis, we introduce the Hermite-Biehler theorem and derive some graphical relations for sixth-order characteristic equations to be Hurwitz stable. Using these relations we introduce a method to choose the optimal receding-horizon subinterval and weight matrices. Finally, we verify the results by some linear examples. CONTROL DESIGN In this section the predictive filter using an approximate receding-horizon scheme with a QP solution is summarized (see Ref. [4] for more details). In the nonlinear filter it is assumed that the state and output estimates are given by a preliminary model and a tobe-determined model error vector, given by ̇̂ x(t) = f̂ [x̂(t)] + Ĝ [x̂(t)]u(t) + Ĝ [x̂(t)] û(t) (1a) ŷ = ĉ [x̂(t)] (1b) where f̂ : R → R is the assumed model vector, Ĝ : R → R is assumed control input and model error distribution matrix, x̂ ∈ R is the state estimate vector, u ∈ R is the control input, û ∈ R is the model error vector, ĉ : R → R is the measurement vector, and ŷ ∈ R is the estimated output vector. Both f̂ and Ĝ are C, i.e., a function itself, where the first and the second derivatives are continuous and ĉ (x̂) is sufficiently differentiable. In addition, f̂ (0) = 0 (if not, we can transform the states x̂ to some new states so that this condition holds). Also, we assume that a unique solution for x̂ exists. The receding-horizon optimization problem is set up as follows: min û J [x̂(t), t, û] = ∫ t+T t [ e (ξ)R e(ξ) +û (ξ)W û(ξ) ] dξ (2) subject to the state equations (1a), (1b) and e(t+T ) = 0, where the residual error is defined by e(t) = ỹ(t)− ŷ(t) (3) where ỹ(t) is the measurement, and R and W are positive definite and positive semi-definite weighting matrices, respectively. Note that T is the recedinghorizon subinterval, which in general is not the sampling interval. At each time t the optimal model error solution, û, over a finite horizon [t, t + T ] is determined online. Then, the current model error û(t) is set equal to û and this process is repeated for every instant of time t, continuously. Define h ≡ T/N for some integer N ≥ n/m, where N is the number of subintervals on [t, t+T ]. Now, ŷ(t+kh) for each k = 1, 2, · · · , Nh = T is approximated by an iterative first-order Taylor series. For simplicity and avoiding the cross-product terms of û(t+ih) and û(t+jh), Ĝ(x̂(t+kh)) ≈ Ĝ(x̂(t)) and F̂ (x(t + kh)) ≈ F̂ (x̂(t)), where F̂ ≡ ∂ f̂/∂x̂. In addition, since future values of ỹ(t) and u(t) are not known, we assume that ỹ(t) and u(t) are constant over the finite horizon [t, t + T ]. Then, we obtain the following expression for 1 ≤ k ≤ N : ŷ(t+ kh) ≈ ŷ(t) + h Ĉ [{ k−1