دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي استاد درس دكتر فرزاد توحيدخواه بهمن 1389

دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي استاد درس دكتر فرزاد توحيدخواه بهمن 1389 MPC Stability-1 کنترل پيشبين-دکتر توحيدخواه

Discrete-time MPC with Prescribed Degree of Stability کنترل پيشبين-دکتر توحيدخواه

Finite Prediction Horizon: Re-visited Example 4.1. کنترل پيشبين-دکتر توحيدخواه

Condition number of the Hessian matrix increases as the prediction horizon Np increases. کنترل پيشبين-دکتر توحيدخواه

Origin of the Problem Using Laguerre functions (for real time): کنترل پيشبين-دکتر توحيدخواه

When there is an integrator in the system matrix A, the norms of the matrix power ||Am|| and the convolution sum ||φ(m)|| do not decay to zero, as m increases. Thus, the magnitudes of the elements in Ω increase as the prediction horizon Np increases. Hence, if the prediction horizon Np is large, a numerical conditioning problem occurs. This problem exists in the majority of the classical predictive controllers formulations, including GPC and DMC. کنترل پيشبين-دکتر توحيدخواه

Traditional solution (previous chapter): Use of an inner-loop state feedback stablization that may compromise the closed-loop performance when constraints become active, or the use of prediction horizon Np and control horizon Nc as the tuning parameters کنترل پيشبين-دکتر توحيدخواه

Idea basis: For a large Np, a large number is divided by another large number. This numerical problem becomes severe when the plant model itself is unstable, or when the dimension of the matrix A is large. کنترل پيشبين-دکتر توحيدخواه

Solution: 1- Improving the numerical condition of MPC algorithms without guaranteeing closed-loop stability. 2- Asymptotic stability 3- Create a prescribed degree of closed-loop stability for the predictive control algorithm. کنترل پيشبين-دکتر توحيدخواه

Use of Exponential Data Weighting کنترل پيشبين-دکتر توحيدخواه

Continuous-time (in the LQR design): eλt Discrete-time: {αj, j = 0, 1, 2 . . .}, α = eλt with t being the sampling interval. کنترل پيشبين-دکتر توحيدخواه

Cost Function: α = 1 the cost function becomes identical to the traditional cost function. کنترل پيشبين-دکتر توحيدخواه

Exponentially Increasing Weight (α < 1): Exponential weights α−2j , j = 1, 2, . . . ,Np, de-emphasizes the state x(ki + j | ki) at the current time and places emphasis on those at the future time. کنترل پيشبين-دکتر توحيدخواه

Exponentially Decreasing Weight (α >1): Exponential weights α−2j , j = 1, 2, . . . ,Np, more emphasizes the state x(ki+ j | ki) at the current time and less emphasis on those at the future time. کنترل پيشبين-دکتر توحيدخواه

Optimization of Exponentially Weighted Cost Function کنترل پيشبين-دکتر توحيدخواه

کنترل پيشبين-دکتر توحيدخواه

Weighted incremental control: Weighted state variable: کنترل پيشبين-دکتر توحيدخواه

Theorem 4.1. The minimum solution of the exponentially weighted cost function J can be found by minimizing: کنترل پيشبين-دکتر توحيدخواه

Example 4.2. Consider the same double-integrator system given in Example 4.1. Examine how the parameter α used in the weighting affects the numerical condition and closed-loop control performance with constraints on the amplitude of the control signal as (only impose constraints on the first sample of the control) α = 1/1.2 (exponentially increasing weight), α = 1 (no exponential weighting) and α = 1.2 (exponentially decreasing weighting) کنترل پيشبين-دکتر توحيدخواه

1- with exponentially increasing weighting, the Hessian matrix is poorly conditioned even for short prediction horizon; 2- without exponential weighting the condition number increases rapidly as the prediction horizon increases. 3- with exponentially decreasing data weighting, the condition number converges to a finite value and is much smaller than the one obtained without using exponential weighting. کنترل پيشبين-دکتر توحيدخواه

Obviously, it is not feasible to use exponentially increasing weighting in this context, as the numerical condition rapidly deteriorates as prediction horizon increases, when α < 1. کنترل پيشبين-دکتر توحيدخواه

Interpretation of Results from Exponential Weighting کنترل پيشبين-دکتر توحيدخواه

The key point is that by transforming the exponentially weighted cost function to the traditional cost function, the augmented state-space model: maximum modulus of all eigenvalues < 1 If کنترل پيشبين-دکتر توحيدخواه

With this simple modification, intuitively we understand that there is no guarantee on the closed-loop stability with an arbitrary choice of α > 1. However, when α is chosen to be slightly larger than one for the class of stable plants with embedded integrator, the closed-loop predictive systems are often found to be stable with Q = CTC and a diagonal R matrix with small positive elements. کنترل پيشبين-دکتر توحيدخواه

For the first time, the prediction horizon Np can be selected to be sufficiently large to approximate the infinite prediction horizon case. Thus with Q ≥ 0 and R > 0, and sufficiently large (Np→∞), minimizing is equivalent to the discrete-time linear quadratic regulator (DLQR) problem. کنترل پيشبين-دکتر توحيدخواه

The traditional DLQR problem is solved using the algebraic Riccati equation controllable observable کنترل پيشبين-دکتر توحيدخواه

closed-loop system: Because کنترل پيشبين-دکتر توحيدخواه

if closed-loop system is stable. کنترل پيشبين-دکتر توحيدخواه

Second method: For stability کنترل پيشبين-دکتر توحيدخواه

By choosing α > 1, there is no guarantee that the closed-loop of the original system will be stable. But, if α is chosen to be slightly larger than unity, then the closed-loop system A−BK would often be stable. Indeed, a large number of simulation tests show that this simple modification usually produces a stable closed-loop system, if the unstable modes from the augmented model come from the embedded integrators. However, a proper choice of the weight matrices Q and R is important to create the degree of stability 1 − ε for the transformed system. کنترل پيشبين-دکتر توحيدخواه

Asymptotic Closed-loop Stability with Exponential Weighting کنترل پيشبين-دکتر توحيدخواه

Modification of Q and R Matrices Basic idea: The exponentially decreasing weight α > 1 increased the magnitudes of the actual closed-loop eigenvalues by the α factor. If the new Q and R matrices are selected to decrease the magnitudes of the eigen-values of the exponentially weighted system by a factor of α−1, then the magnitudes of the actual closed-loop eigenvalues become unchanged کنترل پيشبين-دکتر توحيدخواه

Theorem 4.2. کنترل پيشبين-دکتر توحيدخواه

Interpretation of the Results The essence of the results lies in the fact that the two cost functions lead to the same optimal control. However, the commonly used cost function is limited to a finite prediction horizon for the class of predictive control algorithms that have embedded integrators. In contrast, the exponentially weighted cost function removes the problem because the model used in the prediction is modified to be stable using the factor α. As a result, the prediction horizon Np can be selected to be sufficiently large without numerical problems. Hence, asymptotic closed-loop stability is guaranteed کنترل پيشبين-دکتر توحيدخواه

Example 4.3. Consider the simple double-integrator system described in 4.1 Design a MPC with an integrator for disturbance rejection, Calculate the closed-loop eigenvalues, gain matrix via the cost function using exponential data weighting with α = 1.6 and compare the results with the case without weighting (α = 1) کنترل پيشبين-دکتر توحيدخواه

With exponential data weighting کنترل پيشبين-دکتر توحيدخواه

Without exponential data weighting (α = 1) کنترل پيشبين-دکتر توحيدخواه

MIMO system کنترل پيشبين-دکتر توحيدخواه

Example 4.4. کنترل پيشبين-دکتر توحيدخواه

دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي استاد درس دكتر فرزاد توحيدخواه بهمن 1389